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Carlangelo Liverani, Maciej Wojtkowski

arXiv:math/9210229v1 [math.DS] 29 Oct 1992ERGODICITY INHAMILTONIAN SYSTEMS.Carlangelo Liverani, Maciej Wojtkowski9-20-92Abstract. We discuss the Sinai method of proving ergodicity of a discontinuousHamiltonian system with (non-uniform) hyperbolic behavior.CONTENT0.

Introduction . .

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p. 31. A Model Problem .

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p. 42. The Sinai Method .

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. p. 103.

Proof of Sinai Theorem (for piecewise linear maps of the two torus). .p.

144. Sectors in linear symplectic space .

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p. 185. The space of Lagrangian subspaces contained in a sector.

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236. Unbounded Sequences of Monotones Maps .

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. p. 287.

Properties of the system and the formulation of the results . .

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358. Construction of the neighborhood and coordinate system .

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p. 459. Unstable manifolds in the neighborhood U .

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Local ergodicity in the smooth case . .

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. p. 5411.

Local ergodicity in the discontinuous case . .

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. p. 5612.

Proof of Sinai Theorem . .

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. p. 6013.

‘Tail Bound’ . .

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p. 6514. Applications .

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. p. 69References .

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. p. 79We would like to thank N. Chernov, L. Chierchia, V. Donnay, A. Katok, N.Sim´anyi, D. Sz´asz and L.-S. Young for helpful and enlightening discussions.Thefirst author wishes to thank the the Mathematics Department of the University ofTucson and the Center for Applied Mathematics at Cornell University, in particularits director J. Guckenheimer, where he was visiting during part of this work, he alsoacknowledge the partial support received by CNR, grant n.203.01.52, and by theGNFM.

The second author gratefully acknowledges the hospitality of Forschungsin-stitut f¨ur Mathematik at ETH Z¨urich, where the first draft of this paper was written.He also acknowledges the partial support from NSF Grant DMS-9017993.Tt bAMS T X

2CARLANGELO LIVERANI, MACIEJ WOJTKOWSKISYMBOLS USED IN THE PAPER.αamount of long leaves in a connecting squareB(p; r)Ball of radius r and center pcamount of overlap in neighboring squaresCsectorsddistancek(c)maximal number of overlapping squaresLlinear mapMSymplectic manifoldM±Symplectic boxesµinvariant measureωsymplectic formQquadratic form defining a sectorRrectanglesGcollection of rectanglesS±singularity setsTmapUbig neighborhood in the smooth caseU(x)neighborhood of xVside of a sectorWlinear symplectic spaceWstable and unstable manifoldsIn the Figuresthe stable direction is verticalthe unstable direction is horizontal

ERGODICITY IN HAMILTONIAN SYSTEMS.3§0. INTRODUCTION.The notion of ergodicity was introduced by Boltzman as a property satisfied by aHamiltonian flow on its energy manifold.

The emergence of the KAM (Kolmogorov-Arnold-Moser) theory of quasiperiodic motions made it clear that very few Hamil-tonian systems are actually ergodic. Moreover, those systems which seem to beergodic do not lend themselves easily to rigorous methods.Ergodicity is a rather weak property in the hierarchy of stochastic behaviorof a dynamical system.

The study of strong properties (mixing, K-property andBernoulliness) in smooth dynamical systems began from the geodesic flows on sur-faces of negative curvature. In particular, Hopf [H] invented a method of provingergodicity, using horocycles, which turned out to be so versatile that it endured alot of generalizations.

It was developed by Anosov and Sinai [AS] and applied toAnosov systems with a smooth invariant measure. With the advances of the theoryof Kolmogorov - Sinai entropy the Hopf method turned out to be also a basis forproving the K-property of Anosov systems.The key role in this approach is played by the hyperbolic behavior in a dynamicalsystem.

By the hyperbolic behavior we mean the property of exponential divergenceof nearby orbits. In the strongest form it is present in Anosov systems and Smalesystems.

It leads there to a rigid topological behavior. In weaker forms it seems tobe a common phenomenon.In his pioneering work on billiard systems Sinai [S] showed that already weakhyperbolic properties are sufficient to establish the strong mixing properties.

Eventhe discontinuity of the system can be accommodated.The Multiplicative Ergodic Theorem of Oseledets [O] makes Lyapunov exponentsa natural tool to describe the hyperbolic behavior of a dynamical system with asmooth invariant measure.Pesin [P] made the nonvanishing of Lyapunov exponents the starting point forthe study of hyperbolic behavior. He showed that, if a diffeomorphism preservinga smooth measure has only nonvanishing Lyapunov exponents, then it has at mostcountably many ergodic components and (roughly speaking) on each component ithas the Bernoulli property.Pesin’s work raised the question of sufficient conditions for ergodicity or, moremodestly, for the openness (modulo sets of measure zero) of the ergodic components.In his work, spanning two decades, on the system of colliding balls (gas ofhard balls) Sinai developed a method of proving (local) ergodicity in discontinu-ous systems with nonuniform hyperbolic behavior.

We will refer to it as the Sinaimethod. It was improved by Sinai and Chernov [CS] and by A.Kr´amli, N.Sim´anyiand D.Sz´asz [KSS].

In both papers the discussion is confined to the realm of semidis-persing billiards.The purpose of the present paper is to recover the Sinai method as a part of thetheory of hyperbolic dynamical systems. In the process we have simplified some ofthe aspects of the method, and we have revealed its logical structure and limitations.We rely on two developments.

The first is the work of Katok and Strelcyn [KS]in which they generalized Pesin Theory to discontinuous systems. The other is thedevelopment of criteria for nonvanishing of Lyapunov exponents in Hamiltoniansystems in papers [W1], [W2] and [W3].

In the language of these criteria Burnsand Gerber [BG] found a sufficient condition for (local) ergodicity in the smoothcase of lowest dimension (3 for flows preserving a smooth measure)It was later

4CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIgeneralized by Katok [K1] to arbitrary dimension. As a byproduct of our generalapproach, which includes discontinuous systems, we obtain a similar theorem (MainTheorem in the smooth case) and a new proof.Let us give some advice to the reader on how to use our paper.

The first threeSections demonstrate what the Sinai method is and how it works. The discussionis conducted in the simplest possible environment of a linear discontinuous systemon the two dimensional torus.

It is reasonable to stop here, especially if the readeris only interested in two dimensional uniformly hyperbolic systems. But we do notrecommend trying to read the heart of the paper without going through the firstthree Sections.In Sections 4,5 and 6 we develop the linear symplectic language in which weformulate our results.

We suggest that the reader skips these sections and goesstraight to Section 7 where we formulate the multitude of hypotheses and the twoMain Theorems on local ergodicity, one for smooth systems and the other (muchharder) for discontinuous systems. The reading of Section 7, and the following Sec-tions, will require numerous trips back to Sections 4-6 for the necessary definitionsand theorems.If the reader does not care about the discontinuous case, she needs to read onlySections 8, 9 and 10 with significant leaps (since everything is simpler in the smoothcase).

Sections 11 and 12 contain almost the whole proof of the Main Theorem inthe discontinuous case (it also relies on the results of Sections 8-10). The remainingpart of the proof is contained in Section 13.

It stands out by the level of technicalcomplications.Section 14 contains some classes of examples where all the hard work can be put touse, and one class where it cannot. The interest in this last example comes from thefact that it is multidimensional and all the Lyapunov exponents are different fromzero.

Unfortunately, it does not satisfy an important property (proper alignmentof singularity sets). It points towards the need for a more flexible scheme.§1.

A MODEL PROBLEM.We will discuss here a very simple model problem in which the important featuresof the Sinai’s method are not obscured by technical details. Our discussion will bevery careful so that in the future when the technical details will cloud the horizonwe will be able to refer the reader to these basic clarifications.We consider a family of linear maps of the plane defined byx′1 = x1 + ax2x′2 = x2,where a is a real parameter.

We use these linear maps to define (discontinuous)maps of the torus by restricting the formulas to the strip {0 ≤x2 ≤1} and furthertaking them modulo 1. In this way we define a mapping T1 of the torus T2 = R2/Z2which is discontinuous on the circle {x2 ∈Z} (except when a is equal to an integer)and preserves the Lebesgue measure µ.Similarly we define another family of maps depending on the same parameter aby restricting the formulasx′1 = x1x′ax + x

ERGODICITY IN HAMILTONIAN SYSTEMS.5Figure 1 The map.to the strip {0 ≤x1 ≤1} and then taking them modulo 1. Thus for each a we geta mapping T2 of the torus which is discontinuous on the circle {x1 ∈Z} (exceptwhen a is equal to an integer) and preserves the Lebesgue measure µ.Finally we introduce the composition of these maps T = T2T1 which depends onone real parameter a.

An alternative way of describing the map T is by introducingtwo fundamental domains for the torus M+ = {0 ≤x1 + ax2 ≤1, 0 ≤x2 ≤1} andM−= {0 ≤x1 ≤1, 0 ≤−ax1 + x2 ≤1, } (see Fig.1).The linear map defined by the matrix1aa1 + a2=10a1 1a01takes M+ onto M−thus defining a map of the torus which is discontinuous at moston the boundary of M+ and preserves the Lebesgue measure. This is our map T.Let S± = ∂M± be the boundary of M±.

Except for integer values of a themapping T is discontinuous on S+ and its inverse T −1 is discontinuous on S−. Letus stress that the map T is well defined in the closed domain M+ but two differentpoints on the boundary S+ which correspond to the same point on the torus willbe mapped onto two different points on the boundary S−which correspond to twodifferent points on the torus (except for the corner).

We adopt the convention thatthe image under T of a point from S+ is the pair of image points in S−. With thisconvention we can apply T or any of its powers to any subset in the torus.For integer values of a ̸= 0 we have a hyperbolic algebraic automorphism of thetorus, a prime example of an Anosov system.

It is thus a Bernoulli system and hasa nice Markov partition [AW]. We restrict ourselves to the study of ergodicity andwe repeat the proof of ergodicity by the Hopf method, since the Sinai method isbuilt upon it.Let f : T2 →R be a continuous functionWe want to prove that for almost

6CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIevery x ∈T2 the time averagesf(x) + f(Tx) + · · · + f(T n−1x)nconverge as n →+∞to the average value of f, i.e.,Rfdµ. Once this is establishedone can obtain the same property for all integrable functions by an approximationargument.

From BirkhoffErgodic Theorem (BET) we know that the time averagesconverge almost everywhere to a function f + ∈L1(T2, µ) which is invariant onthe orbits of T, i.e., f + ◦T = f +, and has the same average value as f, i.e.,Rf +dµ =Rfdµ. Further applying BET to f and T −1 we obtain that the timeaverages in the pastf(x) + f(T −1x) + · · · + f(T −n+1x)nconverge almost everywhere as n →+∞to f −∈L1(T2, µ) for which f −◦T = f −andRf −dµ =Rfdµ.It is the usual magic of the ergodic theory which forces the functions f + and f −to coincide almost everywhere.

(Let us recall the argument: letA+ = {x ∈T2 | f+(x) > f−(x)};by definition A+ is an invariant set, henceZA+[f+(x) −f−(x)] dµ(x) =ZA+f(x)dµ(x) −ZA+f(x)dµ(x) = 0which implies µ(A+) = 0 and f+ ≤f−µ-almost everywhere. The same argument,this time applied to the set A−= {x ∈T2 | f−(x) > f+(x)}, implies the converseinequality.

)For a ̸= 0 the matrix1aa1 + a2is a hyperbolic matrix with eigenvalues λ = λ(a) > 1 and1λ < 1. For x ∈T2let us denote by W u(x) (W s(x)) the line in T2 passing through x and having thedirection of the unstable eigenvector (the stable eigenvector), i.e., the eigenvectorwith eigenvalue λ ( 1λ).

We call W u(x) (W s(x)) the unstable (stable) leaf of x. Theleaves of x have the following property. If y ∈W u(x) (y ∈W s(x)) then the distanced(T ny, T nx) = λ−|n|d(y, x) →0 as n →−∞(+∞).Hence for y, z ∈W u(s)(x)|f(T ny) −f(T nz)| →0 as n →−∞(+∞).It follows that for y, z ∈W u(s)(x) either f ±(y) and f ±(z) are both defined andequal or they are both undefined.

Lifting the functions f + and f −to R2 and usingthe directions of the eigenvalues as coordinate directions we can say that f + is a

ERGODICITY IN HAMILTONIAN SYSTEMS.7function of one coordinate alone and f −is a function of only the other coordinate.Since the two functions coincide almost everywhere they must be constant.Let us examine what can be saved of this argument when a is not an integer. Insuch a case, we still have the stable and unstable directions but a line parallel to,say, the unstable direction is cut by S−into pieces and if y and z belong to twodifferent pieces the distance d(T ny, T nz) does not decrease to zero as n →−∞.Since this last property is of crucial importance in the Hopf method, the unstable(and stable) leaves have to be much shorter than before.

Here is how we constructthem. For simplicity of notation we will formulate everything for the unstable leavesalone.We proceed inductively.

Thus, for x ∈intM−, we define W u1 (x) as the opensegment of the line through x with the direction of the unstable eigenvector whichcontains x and has both endpoints on S−. The preimage T −1W u1 (x) is by a factorof λ shorter than Wu1 (x) and, in general, is cut into two or three pieces by S−.

Wepick the piece which contains T −1x and take its image under T; this is our secondapproximate unstable leaf W u2 (x), i.e.,W u2 (x) = TT −1W u1 (x) ∩W u1 (T −1(x)).Unless T −1x ∈S−the second approximate unstable leaf W u2 (x) is again an opensegment containing x with endpoints on S−∪TS−and naturally W u2 (x) ⊂W u1 (x).Given W un (x), n = 1, 2, . .

., we define the n + 1 approximate unstable leaf of xW un+1(x) byW un+1(x) = T n T −nW un (x) ∩W u1 (T −n(x)).If x /∈S+∞i=0 T iS−then this inductive procedure will yield a nested sequence ofopen segments containing xW u1 (x) ⊃W u2 (x) ⊃. .

.with endpoints on+∞[i=0T iS−.We can also describe this construction in the following way. First we consider afairly long segment W u1 (x).

Then we look at TS−, if it does not intersect W u1 (x)then we do not change it, if it splits W u1 (x) into several segments, then we keepthe segment which contains x.We repeat it with T 2S−and further images ofS−, so that the segment may be cut shorter infinitely many times. The propertyx /∈S+∞i=0 T iS−ensures that x stays always strictly inside the segment.

It is quiteremarkable that, for almost every x, this inductive process shortens the segmentonly finitely many times. More precisely we haveProposition 1.1.

For almost all x ∈M−\S+∞i=0 T iS−the sequence of approximateunstable leaves of x stabilizes, i.e., there is a natural N = N(x) such that+∞\W ui (x) =N\W ui (x).

8CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIProof. For t > 0, letXt = {x ∈M−| d(x, S−) ≤t}where d(·, ·) is the distance of a point form a set.

Because S−is a finite union ofsegments we haveµ (Xt) ≤const t.Choosing tn =1n2 we get+∞Xn=1µ (Xtn) < +∞,hence also+∞Xn=1µ (T nXtn) < +∞.It follows by the Borel-Cantelli Lemma that almost every x belongs to only finitelymany of the setsTXt1, T 2Xt2, . .

.,which means that except for finitely many values of nd(T −nx, S−) > 1n2 .Choosing c(x) > 0 sufficiently small we can take care of the finite number of excep-tional values of n so thatd(T −nx, S−) > c(x)n2for each n = 1, 2, . .

. .

Each time W un+1(x) is shorter than W un (x) we must haved(T −nx, S−) < length (Wun(x))λn.But thenc(x)n2< length (Wun(x))λn≤length (W u1 (x))λn,which can hold for at most finitely many values of n.□We define the unstable leaf only for points x in the set of full measure describedin Proposition 1.1, by taking the intersectionW u(x) =+∞\i=1W ui (x).In view of Proposition 1.1, for each W u(x), there are natural numbers nl(x) andnr(x) such that T nl(x)W u(x) has the left endpoint on S−and T nr(x)W u(x) has theright endpoint on S−. Most importantly we have the exponential contraction ofW u(x), i.e., for y ∈W u(x) the distanced(T −ny, T −nx) = d(y, x) →0 as n →+∞.

ERGODICITY IN HAMILTONIAN SYSTEMS.9Everything that we have done to construct the unstable leaves can be repeatedfor the stable leaves and they have analogous properties. Once we have the stableand unstable leaves we are ready to do the Hopf argument.For any continuous function f : T2 →R the forward ergodic average f + isconstant on the stable leaves and the backward ergodic average f −is constant onthe unstable leaves.

Let us call a point x ∈T2 f-typical, if f +(x), f −(x), W u(x)and W s(x) are well defined and f +(x) = f −(x). The set of f-typical points has fullmeasure, so a stable (or an unstable) leaf contains a set of f-typical points of fullarc-length, except for a family of leaves of total measure zero.

If W s(x) is not oneof those exceptional leaves, then the setC1 =[y∈W s(x)y is f−typicalW u(y)has positive measure and f −= f + = const on C1. We can proceed by addingall the stable leaves through f-typical points in C1 to obtain C2, etc., but a priorithere is no reason to expect that we will be able to cover all of the torus in thisway.

(Indeed one can imagine that there is a dividing line between two ergodiccomponents of our system and that all the stable and unstable leaves stop short ofcrossing this line.) That is where the Hopf method breaks down.

It can only tell usthat the ergodic components have positive measure and, therefore, that there are atmost countably many of them. (To be more precise, we cannot really claim that C1belongs to one ergodic component.

To argue this we have to modify our argumentby taking a sequence of continuous functions dense in L1 and considering the setof points which are f-typical for all the functions f in the sequence. This set, asthe intersection of countably many sets of full measure, has full measure.

We canthen use it in the definition of C1 and claim that f −= f + = const on C1 for allthe functions in our dense sequence. This implies that such C1 does belong to oneergodic component.

It follows easily that every invariant subset of positive measurecontains an ergodic component of positive measure. Hence all ergodic componentshave positive measure.)§2.

THE SINAI METHOD.We have seen, in the previous section, that the Hopf method is not sufficient toprove the ergodicity of a discontinuous map because the stable and unstable leavesmay be short. The Sinai method amounts to establishing that most of the stable andunstable leaves are, in a certain sense, sufficiently long.

The first (highly nontrivial)step in this method is to formulate precisely what is meant by “sufficiently long”.As before, we do it only for the unstable leaves; the changes necessary in the caseof stable leaves are automatic.Let U ⊂T2 be a (small) square with the sides parallel to unstable and stabledirections respectively (to make the geometry simpler let us think that the unstabledirection is horizontal and the stable direction vertical). For any 0 < c < 1 weconstruct a sequence Gn(c), n = 1, 2, .

. ., of coverings of U in the following way.Without loss of generality we can letU{(u v) |b < u < bb < v < b}

10CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIFigure 2 The covering.We consider the net N (n, c) defined byN (n, c) = { cn(m, k) ∈U | m, k ∈Z}.Now the covering Gn(c) is the collection of squares having centers at points fromN (n, c) and sides, of length 1n, parallel to the sides of U. If c < 12 then Gn(c) is acovering of U (otherwise Gn(c) may cover only a smaller square).

The parameterc will be chosen later to be very small, so that many squares in Gn(c) overlap.However, once c is fixed, a point in U may belong, at most, to a fixed number,independent of n = 1, 2, . .

., of squares in Gn(c); we denote this number by k(c)(one can easily establish that k(c) ≤( 12c + 1)2, but we will not use any explicitestimate).We call two squares, in Gn(c), immediate neighbors if the distance between theircenters is cn.Two immediate neighbors overlap on 1 −c part of their areas.One can naturally define a column of squares and a row of squares as specialcollections of squares in Gn(c) (see Figure 2). For example, a sequence {Ri}li=1 ofsquares from Gn(c) is called a column of squares if, for every i = 1, .

. ., l −1, Ri andRi+1 are immediate neighbors, Ri+1 is above Ri, and there is no square in Gn(c)below R1 or above Rl.For each square R ∈Gn we introduce the stable, ∂sR, and unstable, ∂uR, bound-aries of R; ∂sR is the union of the two boundary segments of R which have thestable (vertical) direction and ∂uR is the union of the two boundary segments of Rwhich have the unstable (horizontal) direction.

Given a point x ∈R, the unstableleaf W u(x) may intersect both segments in ∂sR or it may be too short to reach oneof them (or both). In the first case we say that W u(x) is long in R, or that it isconnecting in R , in the second that it is short in R or that it is not connecting inR.Definition 2 1Given α 0 < α < 1 we call a square R ∈G (c) α connecting if

ERGODICITY IN HAMILTONIAN SYSTEMS.11the measure of the set of points x ∈R whose unstable leaf W u(x) is long in R is atleast α part of the total area of R.Sinai formulates the property that most of unstable leaves are sufficiently longin the following way.Sinai Theorem 2.2. There is α0 < 1 such that for any α, 0 < α ≤α0 and anyc, 0 < c < 1,limn→+∞n µ[{R ∈Gn(c) | R is not α-connecting }= 0.In other words, the theorem says that if α is sufficiently small, then the union ofthe squares in Gn(c) which are not α-connecting has measure o( 1n).Before proving the Sinai Theorem let us show how it can be used to get informa-tion about ergodic components.

Notice that Definition 2.1 and the Sinai Theoremcan be repeated for stable leaves.Proposition 2.3. The square U ⊂T2 (for which the Sinai Theorem holds for bothunstable leaves and stable leaves) belongs to one ergodic component of T.In view of the arbitrariness of the square U to which we can apply this Theoremwe obtain immediatelyCorollary 2.4.

The map T is ergodic.Proof of Proposition 2.3. Let us fix α sufficiently small so that the Sinai Theoremholds for α-connecting squares both in the unstable and stable versions.

Next wefix c smaller than α. As a consequence two α-connecting squares in Gn(c), whichare immediate neighbors, contain in their intersection a set of connecting leaves ofpositive measure.

The reason is that immediate neighbors intersect over 1 −c partof their areas and hence the guaranteed α part of the square covered by connectingleaves cannot fit into the remaining c part of the square. In the following we willnot change the values of α or c and, for simplicity, we will call an α-connectingsquare simply a connecting square.

Thus a connecting square is α-connecting bothwith respect to stable and unstable leaves.Consider any continuous function f on the torus. We call a point y ∈T2 f-typicalif the forward time average f + and the backward time average f −are well definedat y and f +(y) = f −(y).

The set of f-typical points has full measure. We call astable (unstable) leaf f-typical if its points, except for a subset of zero arc-length,are f-typical.

The union of leaves which are not f-typical is a set of measure zero.For any connecting square R let us defineW u(s)(R) = {x ∈R|W u(s)(x) is f-typical and long in R}.Although we cannot apply the Hopf argument to the whole torus we can use itin a connecting square R to claim that f + is constant on all of W s(R) and f −isconstant on all of W u(R) with the two constants coinciding. Note that we say here(and we mean it) “all of W s(u)” and not almost all.

Indeed, first of all f + is constanton each of the stable leaves in W s(R). Further let us fix an ustable leaf in W u(R).The stable leaves from W s(R) intersect this unstable leaf in f-typical points, exceptfor a set of stable leaves of total measure zero.

Hence excluding these exceptionalstable leaves the value of f + on the stable leaves has to coincide with the constant

12CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIvalue of f −on the distinguished unstable leaf. We conclude that f + is constantalmost everywhere on W s(R) and the constant is equal to the constant value of f −on the unstable leaf.

Since we could have used any other unstable leaf in W u(R)it follows that f −is constant on all of W u(R). By symmetry f + is constant on allof W s(R).

(The reader must have noticed the implicit use of the Fubini Theoremin the arguments above. It is only natural since the stable and unstable leaves areparallel segments.

In the nonlinear case one has to use the “absolute continuity” ofthe foliations into stable and unstable manifolds. This property is all that we need,to make the present argument work.

)Further for two connecting squares R1 and R2 which are immediate neighbors f +is constant on W s(R1)∪W s(R2) and f −is constant on W u(R1)∪W u(R2) with thetwo constants coinciding. Indeed at least one of the intersections W u(R1)∩W u(R2)(if one square is above the other) or W s(R1) ∩W s(R2) (if one square is next tothe other) must have positive measure and hence is nonempty, forcing the constantvalue of f + or f −to be the same for both squares.After this observation we proceed to prove that the time average of f is almosteverywhere constant in U.

To that end let y, z ∈U be two f-typical points withf-typical leaves, W u(y) and W s(z) respectively. Our goal is to prove that f −(y) =f +(z).We say that W u(y) (Ws(z)) intersects completely a column (row) of squaresin Gn(c) if it is connecting in one of the squares of the column (row).

The SinaiTheorem allows us to claim that, for sufficiently large n, W u(y) intersects completelyat least one column of connecting squares in Gn(c), i.e.a column in which allthe squares are connecting, and W s(z) intersects completely at least one row ofconnecting squares. Indeed, suppose to the contrary that every column of squares inGn(c) intersected completely by W u(y) contains at least one non-connecting square.Since the number of columns intersected completely by W u(y) grows linearly withn and the measure of one square in Gn(c) is1n2 , we obtain that the measure ofthe union of non-connecting squares would be O( 1n) which contradicts the SinaiTheorem.

(Here we have used the fact that the squares in Gn(c) cannot overlapmore than k(c) times. )Let us fix a column and a row of connecting squares which are intersected com-pletely by W u(y) and W s(z) respectively.Let R be the (unique) square whichbelongs both to the column and the row.

Let further R1 denote a square in whichW u(y) is connecting and R2 denote a square in which W s(z) is connecting. Bythe construction y ∈W u(R1) and f −is constant on the, possibly disjoint, setW u(R1) ∪W u(R).

Similarly z ∈W u(R2) and f + is constant on W s(R2) ∪W s(R).It follows that f −(y) = f +(z). In view of the arbitrariness in the choice of thef-typical leaves W u(y) and W s(z) we obtain that the time average of f must beconstant in U.To finish the proof let us consider a T-invariant measurable subset A.

Let g bethe indicator function of A andfn →g in L1(T2, µ)be a sequence of uniformly bounded continuous approximations to the indicatorfunction. We will use the fact that the time average is continuous with respect tothe L1 norm to establish that the time average of g must be constant on UIndeed

ERGODICITY IN HAMILTONIAN SYSTEMS.13if we denote by ∥· ∥1 the L1(T2, µ) norm, then∥f +n −g+1 = limN→∞1NNXi=1fn ◦T i −g ◦T i1= limN→∞1NNXi=1fn ◦T i −g ◦T i1by the Lebesgue Dominated Convergence Theorem.Using the invariance of the measure we get∥f +n −g+1 ≤limN→∞1NNXi=1fn ◦T i −g ◦T i1=fn −g1Since the time averages f +n of fn are all constant (almost everywhere) on U theabove inequality implies that the time average g+ is constant (almost everywhere)on U. But the invariance of A forces g+ = g so that either U \ A or U ∩A hasmeasure zero.

In view of the arbitrariness of the invariant set A it follows that Umust belong to one ergodic component.□§3. PROOF OF THE SINAI THEOREM.The proof of the Sinai Theorem does not require a rigid geometric structure ofthe coverings Gn(c); it holds for any sequence of coverings by squares with side 1nas long as there is a uniform bound on the number of squares covering one point.However, the lattice structure of the centers of the squares in Gn(c) allows to workwith columns and rows of squares, as we did in the above application of the SinaiTheorem.The first step in the proof is the choice of α0.

To that end we consider the smallestsector C in R2 symmetric about the horizontal (unstable) line which contains thelines with the two directions of the sides of M−, i.e., the directions of the segmentsin S−. LetC = {(ξ, η) | |η| ≤κ(a)|ξ|}.It can be checked that κ(a) < 1 for any a ̸= 0.

We put α0 = 12(1−κ(a)). The reasonfor this choice is that, for any square with vertical and horizontal sides crossed by aline with the direction contained in C, the shaded area in Figure 4 does not exceed1 −2α part of the area of the square.Let us observe that all of the segments in S+∞i=0 T iS−have directions containedin the sector C. Indeed a linear hyperbolic map pushes lines towards the unstabledirection except for the stable line, which stays put.It follows from the construction of the unstable leaves (Proposition 1.1) that anunstable leaf has endpoints on forward images of S−under T. Hence if an unstableleaf is short in a square then the square must be intersected by+∞[T iS−.

14CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIFigure 3 Leaves cut by a line with direction contained in the sector.Although this does not look like a severe restriction, since we can expect that thelast set is dense, it has far reaching consequences. The reason being, heuristically,that the singularity lines T iS−become more and more horizontal as i →+∞andthey cannot cut effectively unstable leaves which are themselves horizontal.We claim that, for any fixed M ≥1, the singularity linesS−M =M[i=0T iS−by themselves can produce only few squares which are not α-connecting so thattheir total measure is O( 1n2 ).

To make this precise (and clear) we introduce anauxiliary notion of an M-bad square in a covering Gn(c). We say that a squareR ∈Gn(c) is M-bad if the measure of the set of points y ∈R such that the unstableleaf W u(y) has an endpoint in R ∩S−M (so that it is short in R) is greater than1 −2α part of the measure of the square.

(Loosely speaking a square is M-bad if itis not connecting because of the singularity lines in S−M. )If a square R intersects only one segment in S−M then the measure of points inR whose unstable leaves have endpoints on the intersection of this segment with Rdoes not exceed 1−2α0 = κ(a) part of the measure of the square since the directionof the segment is in the sector C. Hence an M-bad square has to intersect at leasttwo segments in S−M.

But the singularity set S−M is a fixed finite collection of closedsegments with only fixed finite number of intersection points (i.e., belonging toseveral segments). Away from the intersection points the segments are fairly wideapart and a small square cannot extend from one to another, see Figure 5.

Hence,for sufficiently large n, an M-bad square in Gn(c) cannot be farther from one of theintersection points than constn. It follows that the total measure of M-bad squaresdoes not exceed constn2 , where the constant depends only on a, c, α and M.In this way we took care (in some sense) of the finite number of singularity linesin S−; we now face the problem of controlling the effects of the ‘tail’ S+∞T iS−

ERGODICITY IN HAMILTONIAN SYSTEMS.15Figure 4 Singularity lines.Let us suppose that a square R ∈Gn(c) is not α-connecting and it is not M-bad.Hence at least α part of its area is covered by short leaves with endpoints inR ∩+∞[i=M+1T iS−.Let Wu(y) be such a leaf short in R with an endpoint on T iS−. ThenT −i (W u(y) ∩R) ⊂Xtiwhere ti = n−1λ−i and, as before, Xt = {x ∈M−| d(x, S−) ≤t}.

Indeed, underthe action of T −1, an unstable leaf contracts by a factor of λ and the length of thepart of W u(y) in R does not exceed 1n.In view of this observation we can claim that each square which is not α-connecting and which is not M-bad has at least α part of its area covered by+∞[i=M+1T iXti.Since each point in U is covered by, at most, k(c) squares from Gn(c), then themeasure of the union of squares in Gn(c) which are not α-connecting and which arenot M-bad does not exceedk(c) × 1α+∞Xi=M+1constnλi= 1n k(c)α+∞Xi=M+1constλi!,(here the constant is equal to the total length of S−). We have thus estimated themeasure of the union of squares in G (c) which are not α connecting and which are

16CARLANGELO LIVERANI, MACIEJ WOJTKOWSKInot M-bad, by the size of an individual square times the M-tail of a fixed convergentseries. Some of the readers may have noticed that this completes the proof.

Forclarity, let us do it explicitly.Let us take an arbitrary ǫ > 0. We choose and fix M = M(ǫ) so large that thelast series does not exceedǫ2n, i.e.,k(c)α+∞Xi=M+1constλi< ǫ2.Given M we can still choose n0 = n0(ǫ, M) so large that, for any n ≥n0, themeasure of the union of M-bad squares in Gn(c) is less thanǫ2n.

To estimate themeasure of the union of squares in Gn(c), for n ≥n0, which are not α-connecting wesplit them into those which are M-bad and those which are not. For both familiesof squares the measure of their union is less thanǫ2n.

This proves our claim.□Remark 3.6.Let us point out that the property that the sector C, defined by the directions ofthe segments in S−, is sufficiently narrow (κ(a) < 1) can be relaxed. For a generalhyperbolic piecewise linear map it is sufficient that the segments in S−are notparallel to the stable direction.

In such a case we can find a natural N such that allthe segments in S+∞i=N+1 T iS−have directions contained in a chosen narrow sectorC ( N is the number of iterates of T which do not put the singularity lines S−intothe chosen sector C). Then the argument above applies to any square neighborhoodU which does not intersectS−N =N[i=0T iS−.Similarly in the version of the Sinai Theorem for the stable leaves we would havearrived at a natural N ′ such that the claim holds for any square U which does notintersectS+N′ =N′[i=0T −iS+ .Hence, it follows from Proposition 2.3 that any open square, with horizontal andvertical sides, which does not intersect S−N ∪S+N′ belongs to one ergodic component.This implies that the partition of T2 into ergodic components is coarser than thepartition into (open) connected components ofT2 \S−N ∪S+N′.Since S−N ∪S+N′ is a finite collection of segments we obtain that there are at mostfinitely many ergodic components.To argue that there is only one componentlet us note that S−N−1 ∪S+N′ and T NS−intersect in at most finitely many pointswhich split the segments in T NS−into finitely many segments {Ik}KNk=1 so that theinterior of every Ik lies in the boundary of at most two connected components ofT2 \S−N ∪S+N′, i.e., it has only one connected component on each side.

Supposethat for such a segment Ik is in the boundary of two different ergodic components.Then TI is also in the boundary of two different ergodic components But TI and

ERGODICITY IN HAMILTONIAN SYSTEMS.17S−N ∪S+N′ have only finitely many points of intersection, so that whole open sub-intervals of TIk must end up inside one connected component of T2\S−N ∪S+N′andthus it must have the same ergodic component on both sides. This contradictionimplies that Ik does not take part in the splitting of T2 into ergodic components sowe can drop it.

In this way we can drop all of T NS−and claim that the partitioninto ergodic components is coarser than the partition into connected components ofT2 \S−N−1 ∪S+N′.It is now clear that we can proceed by dropping T N−1S−and T −N′S+ as possibleboundaries for the ergodic components and arriving eventually at S+ ∪S−as theonly possible boundaries we see that even these can be dropped. Hence there isonly one ergodic component.Let us spell out the property of T which is basic in this argument:Although some points of S−return to S−under iterates of T, no interval in S−can do it.§4.

SECTORS IN A LINEAR SYMPLECTIC SPACE.For the convenience of the reader we will repeat here some of the material from[W3] and [LW].Let W be a linear symplectic space of dimension 2d with the symplectic form ω.For instance we call W = Rd × Rd the standard linear symplectic space ifω(w1, w2) = ⟨ξ1, η2⟩−⟨ξ2, η1⟩,where wi = (ξi, ηi), i = 1, 2,and ⟨ξ, η⟩= ξ1η1 + · · · + ξdηd.The symplectic group Sp (d, R) is the group of linear maps of W (2d×2d matricesif W = Rd × Rd) preserving the symplectic form i.e., L ∈Sp (d, R) ifω(Lw1, Lw2) = ω(w1, w2)for every w1, w2 ∈W.By definition a Lagrangian subspace of a linear symplectic space W is a d-dimensional subspace on which the restriction of ω is zero (equivalently it is amaximal subspace on which ω vanishes).Definition 4.1. Given two transversal Lagrangian subspaces V1 and V2 we definethe sector between V1 and V2 byC = C (V1, V2) = {w ∈W | ω(v1, v2) ≥0 for w = v1 + v2, vi ∈Vi, i = 1, 2}Equivalently, if we define the quadratic form associated with an ordered pair oftransversal Lagrangian subspaces,Q(w) = ω(v1, v2)where w = v1 + v2, is the unique decomposition of w with the property vi ∈Vi, i =1, 2, then we haveC{w ∈W | Q(w) ≥0}

18CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIIn the case of the standard symplectic space, V1 = Rd × {0} and V2 = {0} × Rdwe getQ ((ξ, η)) = ⟨ξ, η⟩andC = {(ξ, η) ∈Rd × Rd | ⟨ξ, η⟩≥0}.We will refer to this C as the standard sector. Since any two pairs of transver-sal Lagrangian subspaces are symplectically equivalent we may consider only thiscase without any loss of generality.

In the following we will alternate between thecoordinate free geometric formulations and this special case. On the one hand, co-ordinate free formulations are important because we need to apply these concepts tothe case of the derivative map which in general acts between two different tangentsubspaces, each one with its preferred sector.

On the other hand, it turns out thatmany arguments are greatly simplified by resorting to these special coordinates.It is natural to ask if a sector determines uniquely its sides. It is not a vacuousquestion since, for d > 1, there are many Lagrangian subspaces in the boundary ofa sector.

The answer is positive.Proposition 4.2. For two pairs of transversal Lagrangian subspaces V1, V2 andV ′1, V ′2 ifC (V1, V2) = C (V ′1, V ′2)thenV1 = V ′1and V2 = V ′2.Moreover V1 and V2 are the only isolated Lagrangian subspaces contained in theboundary of the sector C (V1, V2).The proof of this Proposition can be found in [W3].Based on the notion of the sector between two transversal Lagrangian subspaces(or the quadratic form Q) we define two monotonicity properties of a linear sym-plectic map.

By intC we denote the interior of the sector, i.e.,intC = {w ∈W|Q(w) > 0}.Definition 4.3. Given the sector C between two transversal Lagrangian subspaceswe call a linear symplectic map L monotone ifLC ⊂Cand strictly monotone ifLC ⊂intC ∪{0}.A very useful characterization of monotonicity is given in the followingTheorem 4.4.

L is (strictly) monotone if and only if Q (Lw) ≥Q (w) for everyw ∈W (Q (Lw) > Q (w) for every w ∈W, w ̸= 0).The fact that monotonicity implies the increase of the quadratic form definingthe cone is a manifestation of a very special geometric structure of a sector and doesnot hold for cones defined by general quadratic forms. The proof of the theoremrelies on the factorization (4.7), we postpone then the proof until such factorizationhas been established

ERGODICITY IN HAMILTONIAN SYSTEMS.19For a pair of transversal Lagrangian subspaces V1 and V2 and a linear mapL : W →W we can define the following ‘block’ operators:A : V1 →V1, B : V2 →V1C : V1 →V2, D : V2 →V2.They are uniquely defined by the requirement that for any v1 ∈V1, v2 ∈V2L (v1 + v2) = Av1 + Bv2 + Cv1 + Dv2.We will need the following Lemma.Lemma 4.5. If L is monotone with respect to the sector defined by V1 and V2 thenLV1 is transversal to V2 and LV2 is transversal to V1.Proof.

Suppose that, to the contrary, there exists 0 ̸= ¯v1 ∈V1 such that L¯v1 ∈V2.We choose ¯v2 ∈V2 so thatQ (¯v1 + ¯v2) = ω (¯v1, ¯v2) > 0.We have alsoω (¯v1, ¯v2) = ω (L¯v1, L¯v2) = ω (L¯v1, B¯v2 + D¯v2) = ω (L¯v1, B¯v2) .Let vǫ = ¯v1+ǫ¯v2. We have that for ǫ > 0 vǫbelongs to intC.

Hence also Q (Lvǫ) ≥0for ǫ > 0. On the other handQ (Lvǫ) = ǫ2ω (B¯v2, D¯v2) −ǫω (L¯v1, B¯v2)which is negative for sufficiently small positive ǫ.This contradiction proves the Lemma.□It follows, from Lemma 4.5, that the operators A : V1 →V1 and D : V2 →V2 areinvertible.We switch now to coordinate language.

LetL =ABCDbe a symplectic map of the standard symplectic space Rd × Rd monotone withrespect to the standard sector. A, B, C, D are now just d × d matrices.Let us describe those symplectic matrices which are monotone in the weakestsense, namely they preserve the quadratic form Q.

We will call such matrices Q-isometries. Obviously a Q-isometry maps the sector onto itself.

The converse isalso true.Proposition 4.6. If L is a linear symplectic map andLC = CthenL =A00A∗−1.

20CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIIn particular it preserves the quadratic form QQ ◦L = Q.Proof. If LC = C then L maps also the boundary of the sector C onto itself.

Itfollows from Proposition 4.2 that both sides of the sector stay put under L. HenceB = C = 0. By symplecticity D = A∗−1.□By Lemma 4.5 given a monotone L we can always factor out the following Q-isometries on the leftL =ABCD=A00A∗−1 IRP·(P and R are uniquely determined).

Symplecticity of L forces R, P symmetricand RP −A∗D = I, which allows the further unique factorization(4.7)L =A00A∗−1 I0PI IR0I.Moreover monotonicity forces P and R to be positive semidefinite (P ≥0, R ≥0).Strict monotonicity means that P and R are positive definite (P > 0, R > 0).These claims follow from the followingProof of Theorem 4.4. Using the above factorization we get for w = (ξ, η)Q(Lw) = ⟨ξ, η⟩+ ⟨Rη, η⟩+ ⟨P(ξ + Rη), ξ + Rη⟩.Putting η = 0 we obtain that P ≥0.

To show that also R ≥0 let us consider aneigenvector η0 of R with eigenvalue λ and let ξ = aη0. We get that if a ≥0 thenw = (ξ, η0) ∈C so that Q(Lw) ≥0.

It follows that(a + λ)⟨η, η⟩+ (a + λ)2⟨Pη, η⟩≥0.This implies immediately that λ ≥0. This proves the monotone version of theTheorem.

The strictly monotone version is obtained in a similar way.□As a byproduct of the proof we get the following useful observationProposition 4.8. A monotone map L is strictly monotone if and only ifLVi ⊂int C ∪{0}, i = 1, 2.□The following Proposition simplifies computations with monotone maps.Proposition 4.9.

IfL =ABCDis a strictly monotone map then by multiplying it by Q-isometries on the left andon the right we can bring it to the formIITI + T

ERGODICITY IN HAMILTONIAN SYSTEMS.21where T is diagonal and has the same eigenvalues as C∗B.Proof. The factorization of the monotone map L yieldsA00A∗−1L =IRPI + PRwhere P > 0, R > 0 and PR = C∗B.We have furtherR−1200R12 IRPI + PR R1200R−12=IIKI + Kwhere K = R12 PR12 has the same eigenvalues as C∗B = PR.Finally if F is the orthogonal matrix which diagonalizes K, i.e., F −1KF is diag-onal, thenF −100F −1 IIKI + K F00F=IITI + Thas the desired form with T = F −1KF having the same eigenvalues as C∗B.□Let us note that in the last Proposition we can ask for the diagonal entries of Tto be ordered because any permutation of the entries can be accomplished by anappropriate Q-isometry.§5.

THE SPACE OF LAGRANGIAN SUBSPACES CONTAINED INA SECTOR.Let us fix a sector C = C(V1, V2) between two transversal Lagrangian subspacesV1 and V2. We say that a Lagrangian subspace E is strictly contained in C ifE ⊂int C ∪{0}.We denote by Lag(C) the manifold of all such Lagrangian subspaces and by dLag(C)its closure in the Lagrangian Grassmanian, i.e., dLag(C) is the set of all Lagrangiansubspaces contained in C.We will introduce a metric and a partial order into Lag(C).

This will allow us toextend to the multidimensional case (d > 1) the most relevant features of the twodimensional case (d = 1). Letπi : W →Vi, i = 1, 2,be the natural projections, i.e.,w = π1w + π2wfor every w ∈W.If a Lagrangian subspace E is strictly contained in C then πiE = Vi, i = 1, 2, soπi|E (the restriction of πi to the subspace E) is a one to on map of E onto Vi.With every subspace E ∈Lag(C) we can associate a positive definite quadraticform on V1 obtained by the formulaQ ◦(π1|E)−1 .It will turn out that this is actually a one-to-one correspondence between positivedefinite quadratic forms on V and Lagrangian subspaces contained strictly in C

22CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIDefinition 5.1. For two Lagrangian subspaces E1, E2 ∈Lag(C) we define the re-lation E1 ≤E2 (E1 < E2) by the inequality of the corresponding quadratic formsQ ◦(π1|E1)−1 ≤(<)Q ◦(π1|E2)−1 .We define the distance of two Lagrangian subspaces E1, E2 ∈Lag(C) byd(E1, E2) = 12sup0̸=v∈V1| ln Q ◦(π1|E1)−1 (v) −ln Q ◦(π1|E2)−1 (v)|.It is easy to see that d(·, ·) is indeed a metric.There are other ways to introduce the partial order and the metric.

The coordi-nate free definitions simplify some of the arguments in the following. For equivalentdefinitions of the metric see [LW], [Ve].

Theses definitions are justified by the fol-lowing theorem.Theorem 5.2. For two transversal Lagrangian subspaces E1, E2 ∈Lag(C)E1 < E2if and only ifC(E1, E2) ⊂C(V1, V2).Further if E1 < E2 then for a Lagrangian subspace E ∈Lag(C)E ⊂C(E1, E2)if and only ifE1 ≤E ≤E2.Corollary 5.3.

If E1, E2 ∈Lag(C) and E1 < E2 then the diameter of the setdLag (C(E1, E2)) in Lag(C) is equal to the distance of E1 and E2.□We will prove Theorem 5.2 at the end of this Section.Let us introduce a convenient parametrization of Lag(C) by symmetric positivedefinite matrices. We consider the standard sector C in Rd ×Rd with V1 = Rd ×{0}and V2 = {0} × Rd.

Let U : Rd →Rd be a linear map andgU = {(ξ, η) ∈Rd × Rd | η = Uξ}be its graph. The linear subspace gU is a Lagrangian subspace if and only if U issymmetric and further for a symmetric U its graph gU ⊂C if and only if U ≥0.Every Lagrangian subspace in Lag(C) is transversal to V2 so that it is a graph of alinear map as above.

We will find the following Lemma useful.Lemma 5.4. If a Langrangian subspace E ⊂C(V1, V2) is transversal to both V1and V2 then it is strictly contained in the sector.Proof.

We use the coordinate description of the standard sector.Thus the La-grangian subspace E being transversal to V2 is the graph of a symmetric positivesemidefinite matrix. Since E is also transversal to V1 the matrix is nondegenerateand hence positive definite.

It follows immediately that E is strictly contained inthe sector.□We have obtained a one-to-one correspondence between Lagrangian subspaces inLag(C) and symmetric positive definite matrices. The quadratic form on V1 intro-duced in Definition 5.1 becomes the form defined by the positive definite matrix.The partial order becomes the familiar partial order between symmetric matrices

ERGODICITY IN HAMILTONIAN SYSTEMS.23The image of a Lagrangian subspace under a symplectic linear map is again aLagrangian subspace. Moreover monotone maps take Lagrangian subspaces strictlycontained in C into Lagrangian subspaces strictly contained in C. Hence a monotonemap L defines a map of Lag(C) into itself.

We will denote it again by L : Lag(C) →Lag(C). To simplify notation we will also write U instead of gU.

We have thatL =ABCDacts on Lagrangian subspaces by the following M¨obius transformationLU = (C + DU) (A + BU)−1 .In particular the action of a Q-isometryL =A00A∗−1is given byLU = A∗−1UA−1.By putting A = U12 we see that any U > 0 can be mapped onto identity matrix I.Thus Q-isometries act transitively on Lag(C). Moreover it is not hard to see thatProposition 5.5.

The action of a Q-isometry on Lag(C) preserves the partialorder and the metric.□Let E0 = {(ξ, η) | ξ = η}. By straightforward computations we find that(5.6)C(V1, E0) = {(ξ, η) | ⟨ξ, η⟩−⟨η, η⟩≥0},C(E0, V2) = {(ξ, η) | ⟨ξ, η⟩−⟨ξ, ξ⟩≥0}.We get that(5.7)C(V1, E0) ⊂C(V1, V2),C(E0, V2) ⊂C(V1, V2),C(V1, E0) ∩C(E0, V2) = E0.Because the group of Q-isometries acts transitively on Lag(C) (5.7) holds not justfor the special Lagrangian subspace E0 from (5.6) but for any Lagrangian subspacefrom Lag(C).

(It just happens that the easiest way to establish (5.7) is to do thecalculation in the standard sector. )Proposition 5.8.

For two Lagrangian subspaces E1, E2 ∈Lag(C) the following areequivalent(1) E1 ≤E2,(2) E2 ⊂C(E1, V2),(3) E1 ⊂C(V1, E2).Proof. We will be using the coordinate description of the standard sector.

Since thegroup of Q isometries acts transitively on Lag(C) we can assume that Eis equal

24CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIto E0 from (5.6). Let U2 be the positive definite matrix defining E2.

We get from(5.6) that E2 ⊂C(E0, V2) if and only if U2 ≥I. Hence (1) is equivalent to (2).Similarly let E2 be equal to E0 and U1 be the positive definite matrix defining E1.Using (5.6) again we get that E1 ⊂C(V1, E0) if and only if U1 −U 21 ≥0 which isequivalent to U1 ≤I.

This proves the equivalence of (1) and (3).□Proof of Theorem 5.2. If E1 < E2 then, by Proposition 5.8 and Lemma 5.4, E2 isstrictly contained in C(E1, V2).

Using (5.7) we getC(E1, E2) ⊂C(E1, V2) ⊂C(V1, V2).Suppose now that C(E1, E2) ⊂C(V1, V2). By Proposition 5.8 it suffices to showthat E2 ⊂C(E1, V2).

If it is not so then there is e2 ∈E2 which does not belong toC(E1, V2). Let us consider v1 = π1e2 where π1 : W →V1 is the projection onto V1 inthe direction of V2.

Let further e1 be the unique element in E1 such that π1e1 = v1(i.e., e1 = (π1|E1)−1 v1). Clearly the difference between the two vectors v2 = e2 −e1belongs to V2.

Because e2 = e1 + v2 and e2 /∈C(E1, V2) we have ω(e1, v2) < 0 sothat ω(−e1, e2) > 0. It follows that v2 = e2 −e1 ∈int C(E1, E2) ⊂int C(V1, V2).We have then reached a contradiction, since v2 cannot belong simultaneously to V2and to int C(V1, V2).

The above contradiction proves that indeed E2 ⊂C(E1, V2)which by Proposition 5.8 implies that E1 < E2 (remember that E1 and E2 areassumed to be transversal). The first part of the Theorem is proven.To prove the second part let E1 < E2 and E ⊂C(E1, E2).

By Proposition 5.8we get E2 ⊂C(E1, V2). It follows in view of (5.7) that C(E1, E2) ⊂C(E1, V2) andhence E ⊂C(E1, V2) which is equivalent (again by Proposition 5.8) to E1 ≤E.Similarly we get E ≤E2.In the opposite direction if E1 ≤E < E2 then by Proposition 5.8 E1 and E arestrictly contained in C(V1, E2) and E1 ⊂C(V1, E).

Applying now the equivalenceof (2) and (3) in Proposition 5.8 to the case of E1, E ∈Lag(C(V1, E2)) we getimmediately E ⊂C(E1, E2). The case of E1 ≤E ≤E2 can be now treated bycontinuity.□Let us consider a special family of Lagrangian subspaces in the standard sector:the graphs of multiples of the identity matrix, i.e., for a real number u letZu = {(ξ, η) | η = euξ}.We have thatd(Zu1, Zu2) = 12|u1 −u2|.In the next Lemma we have chosen two numbers u2 > u1.Lemma 5.9.

If for a Lagrangian subspace E ∈Lag(C)d(Zu1, E) ≤12(u2 −u1)thenE ≤Zu2.Proof. Let the Lagrangian subspace E be the graph of a positive definite matrix U.For every nonzero ξ ∈Rd, we haveln⟨ξ Uξ⟩ln⟨ξ eu1ξ⟩≤uu

ERGODICITY IN HAMILTONIAN SYSTEMS.25It follows that, for every nonzero ξ ∈Rd,ln ⟨ξ, Uξ⟩⟨ξ, ξ⟩≤u2.We conclude that U ≤eu2I.□We will use the following consequence of the last Lemma.Proposition 5.10. Let E1 < E2 be two Lagrangian subspaces contained strictlyin C(V1, V2).

There is a symplectic map which maps the sector C(V1, V2) onto thestandard sector C and the sector C(E1, E2) into the sector C(Z−u, Zu) if and onlyif d(E1, E2) ≤u.Proof. By a symplectic map we can map the subspace V1 onto Rd×{0}, the subspaceV2 onto {0}×Rd and E1 onto Z−u (because Q-isometries act transitively on Lag(C)).It follows from Lemma 5.9 that the sector C(E1, E2) will be then automaticallymapped into C(Z−u, Zu).The converse follows from the Corollary 5.3.□For aesthetical reasons we will be using Proposition 5.10 in a different coordinatesystem obtained by the following linear symplectic coordinate changeξ′ =1√2(ξ −η),η′ =1√2(ξ + η).Let us introduce the family of sectorsCρ = {(ξ, η) | ∥η∥≤ρ∥ξ∥}for any real ρ > 0.Proposition 5.11.

Let E1 < E2 be two Lagrangian subspaces contained strictlyin C(V1, V2). There is a symplectic map which maps the sector C(V1, V2) onto thesector Cρ−1 and the sector C(E1, E2) into the sector Cρ if and only ifd(E1, E2) ≤ln 1 + ρ21 −ρ2 ,with 0 < ρ < 1.Proof.

It is enough to define the coordinate change L, defined byξ′ =1√2(ρ−12 ξ −ρ12 η),η′ =1√2(ρ−12 ξ + ρ12 η).A direct computation shows that, if ρ < 1, LCρ−1 = C and LCρ = C(Z−u, Zu), withu = log 1+ρ21−ρ2 . The result follows then from Property 5.10.□

26CARLANGELO LIVERANI, MACIEJ WOJTKOWSKI§6. UNBOUNDED SEQUENCES OF LINEAR MONOTONE MAPS.In this section we fix a sector C = C(V1, V2) between two Lagrangian subspaces.One can think that C is the standard sector.

We start by computing the coefficientof expansion of Q under the action of a monotone symplectic map.For a linear symplectic map L monotone with respect to the sector C we definethe coefficient of expansion at w ∈intC byβ (w, L) =sQ (Lw)Q (w) .We define further the least coefficient of expansion byσC (L) =infw∈intC β (w, L) .Let us note that, for any two monotone maps L1 and L2,σC (L2L1) ≥σC (L2) σC (L1) ,i.e., the coefficient of expansion σC is supermultiplicative.We will omit the index C in σC(L) when it is clear what sector we have in mind.We want to find the value of the expansion coefficient in coordinates. We willuse the fact that this infimum does not change if L is multiplied on the left or onthe right by Q-isometries.

So letL =ABCDbe a monotone matrix.By the factorization (4.7) C∗B = PR is equal to theproduct of two positive semidefinite matrices and so it has only real non-negativeeigenvalues. Let us denote them by 0 ≤t1 ≤· · · ≤td.

The monotone map L isstrictly monotone if and only if t1 > 0.Proposition 6.1. For a monotone map Lσ (L) =√1 + t1 + √t1 = exp sinh−1 √t1,moreover, if L is strictly monotoneσ (L) = β (w, L)for some w ∈int C.Proof.

Let us putm (L) =√1 + t1 + √t1 = min1≤i≤d√1 + ti + √ti.First we prove the inequality β (w, L) ≥m (L) for w ∈intC. Since both β (w, L)and m (L) are continuous functions of L it is sufficient to prove the inequality for

ERGODICITY IN HAMILTONIAN SYSTEMS.27strictly monotone maps only. In view of Proposition 4.9 we can restrict ourselvesto maps L of the formL =IITI + Twith diagonal T and t1, .

. ., td on the diagonal.

We compute β(w, L) directly, forw = (ξ, η) such that Q (w) = 1(β (w, L))2 =dXi=1tiξ2i + (1 + 2ti) ξiηi + (1 + ti) η2i=Xi:ξiηi≥0√tiξi −√1 + tiηi2 +√1 + ti + √ti2 ξiηi+Xi:ξiηi<0√tiξi +√1 + tiηi2 +√1 + ti −√ti2 ξiηi≥≥Xi:ξiηi≥0√1 + ti + √ti2 ξiηi +Xi:ξiηi<0√1 + ti + √ti−2 ξiηi ≥≥(1 + δ) m (L)2 −δm (L)−2 ≥m (L)2whereδ =Xi:ξiηi≥0ξiηi−1 =Xi:ξiηi<0ξiηi ≥0and all the inequalities become equalities forξ1 =1 + t1t1 14, η1 =t11 + t1 14, ξi = 0, ηi = 0, i = 2, . .

., d.Thus the Proposition is proven for strictly monotone matrices and for all mono-tone matrices we get the inequality σ(L) ≥m(L). To get the equality σ(L) = m(L)for all monotone matrices we proceed as follows.

For any ǫ > 0 we choose a strictlymonotone matrix Lǫ so close to the identity that m (LǫL) < m (L) + ǫ. Since LǫLis strictly monotone and our Proposition has been proven for strictly monotonematrices there is wǫ ∈intC such thatβ (wǫ, LǫL) = m(LǫL) = σ (LǫL) .But β (w, LǫL) > β (w, L) for any w ∈intC.

Hencem(L) ≤σ (L) ≤β (wǫ, L) < β (wǫ, LǫL) = m (LǫL) < m (L) + ǫwhich ends the proof.□For a given sector C = C(V1, V2) let C′ = C(V2, V1) be the complementary sector.We have

28CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIProposition 6.2. If L is (strictly) monotone with respect to C then L−1 is (strictly)monotone with respect to C′ and σC(L) = σC′(L−1).Proof.

We have that the unionC (V1, V2) ∪intC (V2, V1)is equal to the whole linear symplectic space W. Hence ifLC (V1, V2) ⊂C (V1, V2)thenC (V1, V2) ⊂L−1C (V1, V2)and finallyL−1intC (V2, V1) ⊂intC (V2, V1) .The last property is easily seen to be equivalent to the monotonicity of L−1.To obtain the equality of the coefficient of least expansion we will use the standardsector and the block description of L. Let (see (4.7))L =A00A∗−1 I0PI IR0I.The linear symplectic map0I−I0takes the standard sector C onto C′ andfurtherL1 =0−II0L−10I−I0has the same least coefficient of expansion with respect to C as L−1 with respect toC′. SinceL−1 =I−R0I I0−PI A−100A∗we getL1 =IPRI + RP A∗00A−1.Our claim follows now from the formula in Proposition 6.1 and the fact that PRhas the same eigenvalues as RP.□The next Proposition is a useful addition to the Corollary 5.3.Proposition 6.3.

For a strictly monotone map Ld(LV1, LV2) = ln σ(L)2 + 1σ(L)2 −1.Proof. Since Q −isometries preserve the distance between Lagrangian subspaces itfollows from Proposition 4.9 that we can restrict our calculations toL =IITI + T

ERGODICITY IN HAMILTONIAN SYSTEMS.29with diagonal T. By the Definition 5.1 we haved(LV1, LV2) = 12sup0̸=ξ∈Rd | ln⟨ξ, Tξ⟩−ln⟨ξ, (T + I)ξ⟩|= 12sup0̸=ξ∈Rd ln ⟨ξ, (I + T −1)ξ⟩⟨ξ, ξ⟩= maxiln1 + t−1i2= ln1 + t−112where t1 ≤t2 ≤· · · ≤td are the eigenvalues of T. The desired formula is nowobtained by a straightforward calculation.□We introduce now an important property of a sequence of monotone maps. Letus consider a sequence of linear symplectic monotone maps {Li}+∞i=1 .

To simplifynotation let us put Ln = Ln . .

. L1.Definition 6.4.

A sequence {L1, L2, . .

.} of monotone maps is called unbounded iffor all w ∈intCQ(Lnw) →+∞as n →+∞.It is called strictly unbounded if for all w ∈C, w ̸= 0,Q(Lnw) →+∞as n →+∞.Theorem 6.5.

A sequence {L1, L2, . .

.} of maps monotone with respect to C isunbounded if and only if+∞\n=1L−11 L−12.

. .

L−1n C′ = one Lagrangian subspacewhere C′ is the complementary sector.Corollary 6.6. If a sequence of monotone maps {L1, L2, .

. .} is unbounded thenthe sequence {L2, L3, .

. .} is also unbounded.□We were not able to find a proof of Corollary 6.6 independent of Theorem 6.5.Proof of Theorem 6.5.

We note that {L1, L2, . .

.} is unbounded if and only if forany strictly monotone L the sequence {L, L1, L2, .

. . } is unbounded.The next step is to prove that {L1, L2, .

. . } is unbounded if and only if for everystrictly monotone L(6.7)σC (LnL) →+∞as n →+∞.Indeed the last property implies immediately that {L, L1, L2, .

. .} is unboundedand so, if it holds for all strictly monotone L, then also {L1, L2, .

. .} is unbounded.To prove the converse we will need the following well known fact from point settopology:

30CARLANGELO LIVERANI, MACIEJ WOJTKOWSKILemma. Let f1 ≤f2 ≤.

. .

, be a nondecreasing sequence of real-valued continuousfunctions defined on a compact Hausdorffspace X. If for every x ∈Xlimn→+∞fn(x) = +∞thenlimn→+∞infx∈X fn(x) = +∞.If {L1, L2, .

. .} is unbounded and L is strictly monotone then we haveσC (LnL) =infw∈intCpQ(LnLw)pQ(w)≥inf0̸=w∈CpQ(LnLw)pQ(Lw)σC (L) .Applying the Lemma tofn(w) =pQ(LnLw)pQ(Lw), n = 1, 2, .

. .,which can be considered as a sequence of functions on the compact space of rays inC we obtain (6.7).Now we will be proving that (6.7) is equivalent to+∞\n=1L−1L−11 L−12.

. .

L−1n C′ = one Lagrangian subspacewhere C′ = C(V2, V1) is the complementary sector. The sectorsC′n = L−1L−11 L−12.

. .L−1n C′ = L−1 (Ln)−1 C′ = C(L−1 (Ln)−1 V2, L−1 (Ln)−1 V1)n = 1, 2, .

. ., form a nested sequence.

We consider the space Lag(C′) of all La-grangian subspaces contained strictly in C′ with the metric defined in Section 5.The sequence of subsets dLag(C′n) ⊂Lag(C′), n = 1, 2, . .

., . .

., is a nested sequenceof compact subsets. Hence its intersection contains one point (= Lagrangian sub-space) if and only if their diameters converge to zero.

By Corollary 5.3 the diameterof dLag(C′n) is equal to the distance of the Lagrangian subspaces L−1 (Ln)−1 V2 andL−1 (Ln)−1 V1. By Proposition 6.3 this distance is equal toln s2n + 1s2n −1where sn = σC′ L−1(Ln)−1.

But by Proposition 6.2σC′ L−1(Ln)−1= σC(LnL).This shows that indeed the set+∞\n=1dLag(C′n)contains exactly one point if and only if (6.7) holds.□We will use the following characterization of strict unboundedness

ERGODICITY IN HAMILTONIAN SYSTEMS.31Theorem 6.8. Let {Li}+∞i=1 be a sequence of linear symplectic monotone maps.The following are equivalent.

(1) The sequence {Li}+∞i=1is strictly unbounded,(2)inf0̸=w∈CpQ(Lnw)∥w∥→+∞as n →+∞,(3) σ(Ln) →+∞as n →+∞,(4) the sequence {Li}+∞i=1is unbounded and Ln0 is strictly monotone for somen0 ≥1.Proof. The Lemma from set topology used in the Theorem 7.5 can also be appliedto the sequence of functionsfn(w) =pQ(Lnw)∥w∥, n = 1, 2, .

. .,to shows that (1) ⇒(2).

Further (2) ⇒(3) becauseσ(Ln) =infw∈intCpQ(Lnw)pQ(w)≥inf0̸=w∈CpQ(Lnw)∥w∥infw∈intC∥w∥pQ(w).The implication (3) ⇒(4) is obvious (σ(Ln) > 1 if and only if Ln is strictlymonotone, cf. Proposition 6.1).

Finally let the sequence {Li}+∞i=1 be unbounded andLn0 be strictly monotone. By Corollary 6.6 also the sequence {Ln0+1, Ln0+2, .

. .

}is unbounded. It follows that {Li}+∞i=1 is strictly unbounded.□The following example plays a role in the study of special Hamiltonian systems.Example.LetLn =An00A∗−1n I0PnI IRn0I,n = 1, 2, .

. ., be a sequence of monotone symplectic matrices with nonexpandingAn, i.e., ∥Anξ∥≤∥ξ∥for all ξ.

We assume further that the symmetric matrices Rnsatisfyτ ′nI ≥Rn ≥τnI and τ ′nτn≤Cfor some positive constants C and τn, τ ′n, n = 1, 2, . .

.. We do not make any assump-tions about Pn (beyond Pn ≥0 which is forced by the monotonicity of Ln). Notethat if a symmetric matrix R satisfies τI ≤R ≤τ ′I then τ∥η∥≤∥Rη∥≤τ ′∥η∥.Indeed⟨Rη, Rη⟩=DRR12 η, R12 ηEDR12 η, R12 ηE⟨Rη, η⟩which yields the estimate

32CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIProposition 6.9. If P+∞n=1 τn = +∞then the sequence {L1, L2, .

. . } is unbounded.Proof.

Let w1 = (ξ1, η1) ∈intC and wn+1 = (ξn+1, ηn+1) = Lnwn, n = 1, 2, . .

..Our goal is to show thatqn = Q(wn) →+∞as n →+∞.We have ξn+1 = An (ξn + Rnηn) so that(6.10)∥ξn+1∥≤∥ξn∥+ ∥Rnηn∥≤∥ξn∥+ τ ′n∥ηn∥≤∥ξ1∥+nXi=1τ ′i∥ηi∥.At the same time qn = ⟨ξn, ηn⟩≤∥ξn∥∥ηn∥so that(6.11)∥ηn∥≥qn∥ξn∥and hence (see also the proof of Theorem 4.4)qn+1 ≥qn + ⟨Rnηn, ηn⟩≥qn + τn∥ηn∥2 ≥qn + τn∥ηn∥qn∥ξn∥.Using (6.10) we obtain from the last inequality(6.12)qn+1qn≥1 +τn∥ηn∥∥ξ1∥+ Pn−1i=1 τ ′i∥ηi∥≥1 + 1Cτ ′n∥ηn∥∥ξ1∥+ Pn−1i=1 τ ′i∥ηi∥.If P+∞i=1 τ ′i∥ηi∥< +∞then by (6.10) the sequence ∥ξn∥is bounded from above andhence by (6.11) the sequence ∥ηn∥is bounded away from zero which is a contradic-tion (in view of P+∞i=1 τ ′i = +∞).Hence+∞Xi=1τ ′i∥ηi∥= +∞.Now the claim follows from (6.12) and the followingLemma 6.13. For a sequence of positive numbers a0, a1, .

. ., if+∞Xn=1an = +∞then+∞Xn=1anPn−1i=0 ai= +∞.Proof of the Lemma.

We have for 1 ≤k ≤llXn=kanPn−1i=0 ai≥Pln=k anPln=0 an→1 as l →+∞.□□

ERGODICITY IN HAMILTONIAN SYSTEMS.33§7. PROPERTIES OF THE SYSTEM AND THE FORMULATION OFTHE RESULTS.In this section we define rigorously the class of systems to which the presentpaper applies.

We divide the conditions that the systems must satisfy into severalgroups. The multitude of conditions is justified by the fact that we want to includediscontinuous systems (there is only one way to be continuous but many ways tobe discontinuous !

). In the case of a symplectomorphism of a compact symplecticmanifold most of these conditions are vacuous.

Because of that we will single outthis case and we will refer to it as the smooth case. The bulk of our effort is devotedto the discontinuous case.A.

The phase space.In the smooth case the phase space M is a smooth compact symplectic manifold.In the discontinuous case it is a disjoint union of nice subsets of the linear sym-plectic space. More precisely, let us consider the standard linear symplectic spaceW = Rd × Rd equipped with a Riemannian metric uniformly equivalent to thestandard Euclidean scalar product and which defines the same volume element(measure) µ.

The measure µ is also equal to the symplectic volume element.By a submanifold of W we mean an embedded submanifold of W. Further wedefine a piece of a submanifold S to be a compact subset of S which is the closureof its interior (in the relative topology of the submanifold S).A piece X of asubmanifold has a well defined boundary which we will denote by ∂X (it is theset of boundary points with respect to the relative topology of the submanifold).Notice that at every point of a piece of a submanifold, including a boundary point,we have a well defined tangent subspace.A submanifold carries the measure defined by the Riemannian volume element,for this measure the boundary of a piece of a submanifold is not necessarily of zeromeasure.The phase space is made up of pieces of W which have regular boundaries in thesense of the following definition.Definition 7.1. A compact subset X ⊂W is called regular if it is a finite unionof pieces Xi, i = 1, .

. ., k, of 2d −1-dimensional submanifoldsX = X1 ∪· · · ∪Xk.The pieces overlap at most on their boundaries, i.e.,Xi ∩Xj ⊂∂Xi ∪∂Xj, i, j = 1, .

. .k;and the boundary ∂Xi of each piece Xi, i = 1, .

. .k, is a finite union of compactsubsets of 2d −2-dimensional submanifolds.To picture such sets one can think of the boundary of a 2d-dimensional cube.The faces are pieces of 2d −1-dimensional submanifolds and they clearly overlaponly at their boundaries.

The boundary of each face is a union of pieces of 2d −2dimensional submanifolds (actually it is a union of 2d −2 dimensional cubes). Letus stress that in the definition of a regular set we do not impose any requirementson the 2d −2 dimensional subsets in the boundary.

Due to the generality of thedefinition one cannot even claim that the union of two regular sets is regular

34CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIAs a consequence of Definition 7.1 the natural measures on the pieces Xi, i =1, . .

., k, of any regular subset X can be concocted to give a well defined measure µXon X (the 2d −1 dimensional Riemannian volume). It is so because the boundariesof the pieces being themselves finite unions of subsets of submanifolds of lowerdimension have zero measure.

Hence if we put∂X =k[i=1∂Xi,then(7.2)µX (∂X) = 0.Moreover, by the regularity of the measure µX, it follows from (7.2) that, if wedenote by (∂X)δ the δ-neighborhood of ∂X in X, then(7.3)limδ→0 µX(∂X)δ= 0.Further we have the following Proposition.Proposition 7.4. For a subset Y of X ⊂W let the δ-neighborhood of Y in W bedenoted by Y δ, i.e.,Y δ = {x ∈W | d(x, Y ) ≤δ}.If X is a regular (2d −1-dimensional) subset of W and Y ⊂X is closed thenlimδ→0µ(Y δ)2δ= µX(Y ).Although Proposition 7.4 holds as we formulated it, we will use only the weakerproperty(7.5)lim supδ→0µ(Y δ)δ≤constµX(Y ).We leave the proof of the Proposition or of the easier property (7.5) to the reader.Definition 7.6.

A compact subset M ⊂W is called a symplectic box if the bound-ary ∂M of M is a regular subset of W and the interior intM of M is connectedand dense in M.We can now formulate the requirements on the phase space of a discontinuoussystem.The phase space of our system is a finite disjoint union of symplectic boxes.To simplify notation we assume that the phase space consists of just one symplec-tic box M. It will be quite obvious how to generalize the subsequent formulationsto the case of several symplectic boxes.B. The map T (the dynamical system).In the smooth case the map T is a symplectomorphism T : M →M

ERGODICITY IN HAMILTONIAN SYSTEMS.35In the discontinuous case we assume that the symplectic box M is partitionedin two ways into unions of equal number of symplectic boxesM = M+1 ∪· · · ∪M+m = M−1 ∪· · · ∪M−m.Two boxes of one partition can overlap at most on their boundaries, i.e.,M±i ∩M±j ⊂∂M±i ∩∂M±j ,i, j = 1, . .

., m.The map T is defined separately on each of the symplectic boxes M+i , i =1, . .

., m. It is a symplectomorphism of the interior of each M+i onto the interiorM−i , i = 1, . .

., m and a homomorphism of M+i onto M−i , i = 1, . .

., m. We assumethat the derivative DT is well behaved near the boundaries of the symplectic boxes.Namely, we assume that it satisfies the Katok-Strelcyn conditions so that we canapply their results [K-S] on the existence of the foliation in (un)stable manifoldsand its absolute continuity.We will say that T is a (discontinuous) symplectic map of M. Formally T is notwell defined on the set of points which belong to the boundaries of several plus-boxes: it has several values. We adopt the convention that the image of a subset ofM under T contains all such values.Let us introduce the singularity sets S+ and S−.S± = {p ∈M | p belongs to at least two of the boxes M±i , i = 1, .

. ., m}.The plus-singularity set S+ is a closed subset and T is continuous on its com-plement.

Similarly T −1 is continuous on the complement of S−. Note that most ofthe points in the boundary ∂M of M do not belong to S−or S+.We have that S+ ∪∂M is the union of all the boundaries of the plus-boxes andS−∪∂M is the union of all the boundaries of the minus-boxes, i.e.,S± ∪∂M =m[i=1∂M±i .Note that most of the points in the boundary ∂M of M do not belong to S−orS+.

We assume that the singularity sets S± and the union of boundaries Smi=1 ∂M±iare regular sets.An important role in our discussion will be played by the singularity sets of thehigher iterates of T. We define for n ≥1S+n = S+ ∪T −1S+ ∪· · · ∪T −n+1S+.andS−n = S−∪TS−∪· · · ∪T n−1S−.We have that T n is continuous on the complement of S+n and T −n is continuous onthe complement of S−

36CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIRegularity of singularity sets. We assume that for every n ≥1 both S+n andS−n are regular.We will formulate, in Lemma 7.7, an abstract condition on the first power of Talone that guarantees the regularity of the singularity sets but it requires that themap is a diffeomorphism on every symplectic box up to and including its boundaryi.e., it can be extended to a diffeomorphism of an open neighborhood of M+i ontoan open neighborhood of M−i , i = 1, .

. ., m.Hence it is very appealing to restrict the discussion to such maps.

Unfortunately,such a restriction would leave out important examples: billiard systems where thederivative may blow up at the boundary. The conditions in the work of Katok andStrelcyn [K-S] were tailored for such systems.Nevertheless the reader is invited to be generous with the restrictions on theregularity of T, this will make it easier to follow the main line of the argument.C.

Monotonicity of T.In the smooth case we assume that two continuous bundles of transversal La-grangian subspaces are chosen in an open subset U ⊂M (U is not necessarilydense). We denote them by {V1(p)}p∈U and {V2(p)}p∈U respectively.In the discontinuous case we assume that two continuous bundles of transversalLagrangian subspaces are chosen in the interior of the symplectic box M. Theirlimits (if they exist at all) at the boundary ∂M are allowed to have nonzero inter-section.We consider the bundle of sectors (see Definition 4.1) defined by these LagrangiansubspacesC(p) = C(V1(p), V2(p)).LetC′(p) = C(V2(p), V1(p))be the complementary sector.We require that the derivative of the map and its iterates, where defined, ismonotone, if only monotonicity is well defined (cf.

Definition 4.3).More precisely, in the smooth case we require that, if p ∈U and T kp ∈U fork ≥1, thenDpT kC(p) ⊂C(T kp).In the discontinuous case we assume thatDpTC(p) ⊂C(Tp)for points p in the interior of every symplectic boxes M+i , i = 1, . .

., m .We call a point p ∈intM (p ∈U in the smooth case) strictly monotone in thefuture if there is n ≥1 such that DpT n is defined and it is strictly monotone ( inthe smooth case we require naturally that T np ∈U), i.e.,DpT nC(p) ⊂intC(T np) ∪{0}.Similarly a point p is called strictly monotone in the past if there is n ≥1 suchthat DpT −n is strictly monotone with respect to the complementary sectors, i.e.,DpT −nC′(p) ⊂intC′(T −np) ∪{0}.It is clear that if p is strictly monotone in the future then its preimages are alsostrictly monotone in the future. By Proposition 6.2 we also have that if p is strictlymonotone in the future then there is n ≥1 such that T np is strictly monotone inthe past

ERGODICITY IN HAMILTONIAN SYSTEMS.37Strict monotonicity almost everywhere. We assume that almost all points inM (U in the smooth case) are strictly monotone.This property implies that all Lyapunov exponents are non-zero almost every-where in M (in U in the smooth case).

The proof of this fact is quite simple andcan be found in [W1]. It will also follow easily from our Proposition 8.4.

Thusby the work of Pesin [P] in the smooth case and of Katok and Strelcyn [K-S] inthe discontinuous case through almost every point there are local stable and unsta-ble manifolds of dimension d and the foliations into these manifolds are absolutelycontinuous.The sectors C(p) contain the unstable Lagrangian subspaces (tangent to theunstable manifolds) and the complementary sectors C′(p) contain the stable La-grangian subspaces (tangent to the stable manifolds). The sectors can be viewed asa priori approximations to the unstable and stable subspaces.

We will refer to thesectors as unstable sector and stable sector respectively.This ends the list of required properties for the smooth case.The last threeproperties of our system are introduced only for the discontinuous case.D. Alignment of Singularity setsFor a codimension one subspace in a linear symplectic space its characteristicline is, by definition, the skeworthogonal complement (which is a one dimensionalsubspace).Proper alignment of S−and S+.

We assume that the tangent subspace of S−at any p ∈S−has the characteristic line contained strictly in the sector C(p) andthat the tangent subspace of S+ at any p ∈S+ has the characteristic line containedstrictly in the complementary sector C′(p). We say that the singularity sets S−andS+ are properly aligned.Let us note that if a point in S± belongs to several pieces of submanifolds thenwe require that the tangent subspaces to all of these pieces have characteristic linesin the interior of the sector.It will be clear from the way in which the proper alignment of singularity sets isused in Section 12 that it is sufficient to assume that there is N such that T NS−and T −NS+ are properly aligned.

We will show, in section 14, that for the systemof falling balls even this weaker property fails. Hence the study of ergodicity of thissystem would require some further relaxation of this property.Let us note that it is helpful in establishing the regularity of singularity sets S±nif the boundaries of M have tangent subspaces characteristic lines contained in theboundary of the sectors C(p).

It is so in some examples. More precisely we havethe following lemma.Lemma 7.7.

If the map T is a diffeomorphism up to and including the boundariesof the symplectic boxes M+1 , . .

. , M+m, satisfies properties C, D and the boundary∂M of M has all the tangent subspaces with characteristic lines contained in theboundary of the sectors then the sets S±n , n ≥1, are regular (i.e.

the property B isautomatically verified).Proof. Let us recall that, by assumption, S−and Smi=1 ∂M+i are properly alignedregular subsets.

Further the intersection of any properly aligned regular subset X(the characteristic lines of its tangent subspaces are contained strictly in the un-stable sector C) with any of the symplectic boxes M+M+ is a regular subset

38CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIIndeed let X1, . .

., Xp be the pieces of 2d −1 dimensional manifolds which makeup X (X = Spi=1 Xi) and Y1, . .

., Yq be the pieces of 2d −1 dimensional manifoldswhich make up the boundary of say M+1 (∂M+1 = Sqj=1 Yj). By the proper align-ment of the pieces we can assume that any Xi and any Yj are pieces of transversalsubmanifold.

Hence the intersection of the submanifolds is a submanifold of dimen-sion 2d−2, and therefore Xi ∩Yj are disjoint pieces of 2d−2-dimensional manifolds(allowed to intersect only at the boundary). It follows that the intersection of Xiwith M+1 is a piece of the 2d −1 dimensional manifold and also a regular subset.The same can be repeated for the other symplectic boxes M+2 , .

. ., M+m.Moreover we have that any (Xi∩M+1 )∪∂M+1 , i = 1, .

. ., p, is a regular subset andfurther (X ∩M+1 )∪∂M+1 is a regular subset.

It follows that T(X ∩M+1 ) ∪∂M+1= (TX ∩M−1 ) ∪∂M−1 is a regular subset and after repeating the argument for theother symplectic boxes we get that for any regular and properly aligned subset XTX ∪Smi=1 ∂M−i and therefore TX ∪S−are regular properly aligned subsets.Now the proof can clearly be completed by induction sinceS−n+1 = TS−n ∪S−.The argument for S+ is completely analogous.□The last two properties are rather technical. They are used only in Section 14in the proof of the ‘tail bound’.

It remains an open question if one can do withoutthem.E. Noncontraction property.There is a constant a, 0 < a ≤1, such that for every n ≥1 and for everyp ∈M \ S+n∥DpT nv∥≥a∥v∥for every vector v in the sector C(p).Notably the above condition holds in all the examples to which the other con-ditions apply (see §14), apart from the case of semi-dispersing billiards in morethen two dimensions (the case from which this type of strategy originated).

In fact,through a tangent collision a vector in the unstable direction can shrink by an arbi-trary amount. Instead of the present condition the original article of Chernov-Sinai[CS] was taking advantage of a special property of semi-dispersing billiard.

Namelythe existence of a semi-norm (the configuration norm) that is increased by the dy-namics for vectors in the unstable direction. Moreover, such norm is well alignedwith respect to the singularity manifolds and with respect to the cone bundle: onthe one hand a δ neighborhood of the singularity in this semi-norm is of measureO(δ), on the other hand the hyperplane of vectors on which the seminorm has valuezero is not contained in the interior of the cone (note that this two requirement,together with the requirement of the proper alignment of the singularities, implythat the singularity manifold is aligned with the boundary of the cone).

It wouldbe possible to generalize such setting and use the generalization of these propertiesinstead of the non-contraction property.The bold reader can see how it wouldbe possible to adapt §13 to this setting. We choose not to do this explicitly forreasons of clarity and also because we do not know of any example (apart fromsemi-dispersing billiards) to which such alternative condition could apply.F SinaiChernov Ansatz

ERGODICITY IN HAMILTONIAN SYSTEMS.39This is a property pertaining the derivatives of the iterates of T on the singularityset itself, of T −1 on S+ and of T on R−. Namely, we require that, for almost everypoint in R−with respect to the measure µS (µS is the 2d−1 dimensional Riemann-ian volume on R−∪R+), all iterates of T are differentiable and for almost everypoint in S+ all iterates of T −1 are differentiable.

Note that the last requirementholds automatically under the assumptions of Lemma 7.7. Moreover,we assume that for almost every point p ∈S−with respect to the measure µS,the sequence of derivatives {DT npT}n≥0 is strictly unbounded (cf.

Definition 6.4).Analogous property must hold for S+ and T −1.By Theorem 6.8 the forward part of Sinai - Chernov Ansatz is equivalent to thefollowing property. For almost every point p ∈S−with respect to the measure µSlimn→+∞σ(DpT n) = +∞,where the coefficient σ is defined at the beginning of Section 6.In several examples unboundedness holds for all orbits by virtue of Proposition6.9 but strict monotonicity is hard to establish.We have completed the formulation of the conditions.

Under these conditions wewill prove the following two theorems.Main Theorem (Smooth case). For any n ≥1 and any p ∈U such that T np ∈U and σ(DpT n) > 1 (i.e., p is strictly monotone) there is a neighborhood of p whichis contained in one ergodic component of T.It follows from this theorem that if U is connected and every point in it isstrictly monotone then S+∞i=−∞T iU belongs to one ergodic component.Such atheorem was first proven by Burns and Gerber [BG] for flows in dimension 3.

Itwas later generalized by Katok [K] to arbitrary dimension and recently also to anon-symplectic framework [K1]. Our proof is a byproduct of the preparatory stepsin the proof of the followingMain Theorem (Discontinuous case).

For any n ≥1 and for any p ∈M \ S+nsuch that σ(DpT n) > 3 there is a neighborhood of p which is contained in oneergodic component of T.Let us note that the conditions of the last theorem are satisfied for almost allpoints p ∈M. Indeed letMn,ǫ = {p ∈M | σ(DpT n) > ǫ}.Since almost all points are strictly monotone, then+∞[n=1[ǫ>0Mn,ǫhas full measure.

By the Poincare Recurrence Theorem and the supermultiplicativ-ity of the coefficient σ we conclude that+∞[Mn,3

40CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIFigure 5 The Baker Map and the Modified Baker Map.has also full measure.Hence the theorem implies in particular that all ergodic components are essen-tially open. The theorem allows also to go further since we assume that only finitelymany iterates of T are differentiable at p so that we can apply it to orbits that endup on the singularity sets both in the future and in the past (e.g.

p ∈S−andT np ∈S+). We need though a specific amount of hyperbolicity on this finite or-bit (σ(DpT n) > 3); note that in the smooth case any amount of hyperbolicity(σ(DpT n) > 1) is sufficient.This theorem gives a fairly explicit description of points which can lie in theboundary of an ergodic component.By checking that there are only few suchpoints (e.g.

that they form a set of codimension 2) one may be able to concludethat a given system is ergodic.Although the techniques used in the proof make it unavoidable to require morehyperbolicity in the non-smooth case, we do not know of any examples of non-ergodic systems satisfying all the conditions above where some points on the bound-aries of two ergodic components are strictly monotone, i.e., σ(DpT n) > 1 for somen ≥1.In all the examples that we know, any point with an infinite orbit (in the futureor in the past) has the unbounded sequence of derivatives (in the sense of Definition6.4).In such case, it follows from Theorem 6.8 that for any strictly monotonepoint with the infinite orbit in the future the condition σ(DpT n) > 3 is satisfiedautomatically, if only n is sufficiently large.There is no need to formulate the Main Theorem separately for a point p whichhas only the backward orbit (p ∈S+). We can simply apply the theorem to T −np(one can appreciate now the convenience of Proposition 6.2).Let us finish this Section with an example where the role of the proper alignmentof singularities is exposed.

The well known Baker’s Transformation maps the unitsquare as shown in Fig.5a and it is ergodic. Let us consider a variation of thisconstruction where the square is stretched and squeezed as before but now the

ERGODICITY IN HAMILTONIAN SYSTEMS.41Figure 6 The discontinuity lines of the Modified Baker Map.middle one half is left at the bottom and the quarters on the left and right aretranslated to the top as shown in Fig.5b. This time the map T is not ergodic.

Theergodic components are separated by the dotted line although for any point p onthe dotted line we have thatσ(DpT 2) = 4.Of all the conditions formulated in this Section only the proper alignment of singu-larity sets is violated; namely part of S−has stable (vertical) direction (all of S+has stable direction which is fine), see Fig.6 where S± are indicated by bold lines.For the standard Baker’s transformation the condition of the proper alignment isclearly satisfied.§8. CONSTRUCTION OF THE NEIGHBORHOOD AND THE COOR-DINATE SYSTEM.We will construct a convenient coordinate system in a neighborhood of a strictlymonotone point p ∈M.

There are two cases: strict monotonicity in the past andstrict monotonicity in the future but they are completely symmetric. Therefore, wewill discuss only one of them.

Namely we assume that there is N ≥1 such that(8.1)i)T −N is differentiable at p : p ̸∈S−N ∪∂M, (discontinuous case)T −Np ∈U, (smooth case)ii)DpT −N is strictly monotone.We will find a neighborhood U(p) in which there is an abundance of “long” stableand unstable manifolds. Let us emphasize that we have assumed only that p (andits N preimages) does not belong to S−but it may very well belong to S+.

Sucha level of generality is crucial in obtaining local ergodicity also for points in thesingularity sets S±

42CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIOur first requirement on the neighborhood is that T −N is a diffeomorphism ofU(p) onto a neighborhood of ¯p = T −Np (and in the smooth case both neighborhoodsare contained in U).By the Darboux theorem a symplectic manifold looks locally like a piece of thestandard linear symplectic space. Hence reducing U(p) further, if necessary, we canidentify it with a neighborhood U of the standard linear symplectic space Rd × RdU = Ua = Va × Va,whereVa = {x = (x1, .

. ., xd) ∈Rd | |xi| < a, i = 1, .

. ., d}.

(In the discontinuous case we have assumed from the very beginning that a sym-plectic box is a subset in Rd × Rd). We assume that the point p becomes the zeropoint and the symplectic structure is the standard one.

In particular all the tan-gent spaces in U(p) can be identified with Rd × Rd. The choice of a cube for theshape of the neighborhood is important only for some of the arguments in Section11 otherwise we want to stress that our neighborhood U is the cartesian product ofneighborhoods Va in the d-dimensional linear space and we will not use any specialdirections there.Let us further introduce for any positive ρ the following sectors in the tangentspace of U.Cρ = {(ξ, η) ∈Rd × Rd | ∥η∥≤ρ∥ξ∥}and the complementary sectorC′ρ = {(ξ, η) ∈Rd × Rd | ∥ξ∥≤ρ−1∥η∥}.By the assumption (8.1) the sector D¯pT NC(¯p) is strictly inside the sector C(p).We change coordinates in U in such a way that for some ˜ρ < 1C′(p) = C′˜ρ−1andD¯pT NC(¯p) ⊂C˜ρ.By Propositions 5.11, 6.2 and 6.3 this can be done with ˜ρ = (σ(D¯pT N))−1 .We pick ρ, ˜ρ < ρ < 1.

By the continuity of the sector bundle C(z), z ∈U, and ofthe derivative DyT N, y ∈T −NU, if we reduce the size of U appropriately, we canachieve that for any z ∈U (see Figure 7)(8.2)C′(z) ⊂C′ρ−1and for any y ∈T −NU(8 3)D T NC(y) ⊂C

ERGODICITY IN HAMILTONIAN SYSTEMS.43Figure 7 The cones at TzM.The properties (8.2) and (8.3) seem to be asymmetric in time, i.e., T plays inthem a different role than T −1. Nevertheless we can obtain from them the followingfundamental Proposition which is perfectly symmetric in time.We will say that a point z ∈U has k spaced returns in a given time interval ifthere are k moments of time in this intervali1 < i2 < · · · < ikat which z visits U, i.e.,T ijz ∈U for j = 1, .

. ., k,and the visits are spaced by at least time N, i.e.,ij+1 −ij ≥N for j = 1, .

. ., k −1.Proposition 8.4.

If T n is differentiable at z ∈U for n ≥N and z′ = T nz ∈Uthen(8.5u)DzT nCρ−1 ⊂Cρand(8.5s)Dz′T −nC′ρ ⊂C′ρ−1.Moreover for (ξ′, η′) = DzT n(ξ, η) if (ξ, η) ∈Cρ then(8.6u)∥ξ′∥≥bρ−k∥ξ∥and if (ξ′, η′) ∈C′ρ−1 then(8 6s)∥η∥≥bρ−k∥η′∥

44CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIwhere k is the maximal number of spaced returns of z in the time interval from Nto n andb =p1 −ρ4.Proof. It follows from (8.2) that for any x ∈UCρ ⊂Cρ−1 ⊂C(x).HenceDzT n−NCρ−1 ⊂C(T n−Nz).Now (8.5u) follows from (8.3).Let us further note that (8.3) implies that for any x ∈UDxT −NC′ρ ⊂C′(T −Nx).We obtain (8.5s) by applying first Dz′T −N, then DT −Nz′T −n+N and using (8.2)again.The properties (8.6u) and (8.6s) follow from (8.5u) and (8.5s) respectively inexactly the same way.

We will prove only the unstable version. To measure vectorsin Cρ we use the form Q associated with the sector Cρ−1.

It is equal toρ−1∥ξ∥2 −ρ∥η∥2and on every spaced return to U the value of this form on vectors from Cρ−1 getsincreased by at least the factor ρ−2, cf. Propositions 5.11 and 6.3 .

It remains tocompare the value of this form at (ξ, η) ∈Cρ with ∥ξ∥2. We haveρ−1∥ξ∥2 ≥ρ−1∥ξ∥2 −ρ∥η∥2 ≥(ρ−1 −ρ3)∥ξ∥2which immediately yields (8.6u).□Having achieved the symmetry with respect to the direction of time we willrestrict the discussion in the next section to the case of unstable manifolds usingthe unstable version of Proposition 8.4.It can be then repeated for the stablemanifolds with the use of the stable version.Remark 8.7.

If p is not a periodic point then by reducing the neighborhood U wecan guarantee that any successive visits to U are spaced by, at least, a time N. Insuch a case the number of spaced returns becomes simply the number of returns toU. It is so also if N = 1.§9.

UNSTABLE MANIFOLDS IN THE NEIGHBORHOOD U.Let us repeat the properties of T and U established in the previous section whichwe will rely upon. Note that the original point p does not appear explicitly.There is a positive number ρ < 1 such that for any z ∈U(9 1)C ⊂C⊂C(z)

ERGODICITY IN HAMILTONIAN SYSTEMS.45and for any y ∈T −NU(9.2)DyT NC(y) ⊂Cρ.It follows that if z ∈U and T nz ∈U for n ≥N then(9.3)DzT nCρ−1 ⊂Cρ.Moreover if(ξ, η) ∈Cρ and (ξ′, η′) = DzT n(ξ, η)then(9.4)∥ξ′∥≥bρ−k∥ξ∥where k is the maximal number of spaced returns to U between the times N and nand b =p1 −ρ4.By the Pesin theory [P] in the smooth case and the Katok-Strelcyn theory [K-S]in the general case for almost all z ∈U we have a local unstable manifold W uloc(z)through z. Further the tangent spaces of W uloc(z) ∩U are Lagrangian subspacescontained in Cρ.

Unfortunately the general theory does not give us a good hold ontheir size.Let πi : V ×V →V, i = 1, 2, be the projection on the first and second componentrespectively. We denote by B(c; r) the open ball with the center at c and the radiusr.Definition 9.5.

We say that an unstable manifold in U of a point z = (z1, z2) ∈Uhas size ε if it contains the graph of a smooth mapping from B(z1; ε) to V. Wedenote such a graph by W uε (z) and we will call it the unstable manifold of size ε.By the definition of an unstable manifold W uε (z) of size ε its projection onto thefirst component is the open ball with the center at π1z and radius ε.Lemma 9.6. The projection onto the second component of an unstable manifoldthrough z = (z1, z2) ∈U of size ε lies in the open ball with the center at z2 and theradius ρε, i.e.,π2 (W uε (z)) ⊂B(z2; ρε).Proof.

Let W uε (z) be the graph ofψ : B(z1; ε) →V.The subspace {(ξ, Dψξ)|ξ ∈Rd} is tangent to W uε (z) and hence is contained in Cρ.It follows that∥Dψ∥≤ρ.By the mean value theorem if z′ = (z′1, z′2) ∈W uε (z) then∥z′2 −z2∥= ∥ψ(z′1) −ψ(z1)∥≤sup ∥Dψ∥∥z′1 −z1∥< ρε.□In contrast to the model problem at the beginning where we had fairly long initialunstable leaves and then we cut them because of the discontinuity of our systemwe start here with small unstable manifolds and “grow” them until they are largeor until they hit the singularity whichever comes first. This is done in the proof ofthe following Theorem

46CARLANGELO LIVERANI, MACIEJ WOJTKOWSKITheorem 9.7. For any δ > 0 almost every point z in U1δ ,U1δ = Ua1(δ)where a1(δ) = a−b−1δ (Ua is defined in §8 and b =p1 −ρ4), either has an unstablemanifold of size δ or it has an unstable manifold of size δ′ < δ such that the closureof W uδ′(z) intersects Sj>N T jS−.Proof.

Let A(ε) ⊂U1δ be the set of points which have unstable manifolds of size ε.By the Katok-Strelcyn theory almost all points in U1δ belong to Sε>0 A(ε). Let usfix A(ε) of positive measure and let k be the smallest natural number such thatbρ−kε ≥δ.Almost all points in A(ε) have k spaced returns to A(ε) in the past.

Let z be sucha point and let−N ≥−i1 > · · · > −ik = −nbe the k times of spaced returns of this point, i.e.,T −ijz ∈A(ε), j = 1, . .

., k.The geometric idea for growing unstable manifolds is to take the unstable mani-fold of size ε through the point T −nz and map it forward under T n. The expansionproperty (9.4) guarantees then that the image contains the unstable manifold ofsize δ. There are two complications in this argument.

First it may happen that T nis not continuous on the unstable manifold W uε (T −nz), that isW uε (T −nz) ∩S+n ̸= ∅.The other problem occurs when parts of the images of the unstable manifold areoutside of U where the expansion property (9.4) may fail.To present clearly the core of the argument we ignore for the time being thesetwo difficulties and assume that T n is differentiable on W uε (T −nz) and thatT n−ijW uε (T −nz) ⊂U, j = 0, . .

., k,here we set i0 = 0. We can prove then that z has an unstable manifold of size δ.Indeed let W uε (T −nz) be the graph ofψ : B(π1(T −nz); ε) →Vand let us consider the mapϕ : B(π1(T −nz); ε) →Vdefined by ϕ(x) = π1 (T n(x, ψx)).

By (9.4) this map is an expanding map with thecoefficient of expansion not less than bρ−k, i.e.,∥Dϕξ∥≥bρ−k∥ξ∥

ERGODICITY IN HAMILTONIAN SYSTEMS.47Hence the image of B(π1(T −nz); ε) by ϕ contains the ball B(π1z; δ). Additionalcomplication is caused by the fact that ϕ is not necessarily one-to-one.

But sinceϕ is a local diffeomorphism we can define ϕ−1 on B(π1z; δ) as the branch of theinverse for which ϕ−1π1z = π1(T −nz). Therefore, T nW uε (T −nz) contains the graphof the mapπ2 ◦T n ◦(id × ψ) ◦ϕ−1which defines W uδ (z).Let us now address the general case.

We will construct the maximal subset ofW uε (T −nz) on which T n is differentiable and its images at the return times to Uare contained in U. Our first step is to consider the connected component ofW uε (T −nz) \ S+nwhich contains T −nz and denote it by ggW uε (T −nz).

Further the connected compo-nent ofk\j=0T ij−nT n−ij ggW uε (T −nz) ∩Uwhich contains T −nz will be denoted it by gW uε (T −nz). It is the part of the unstablemanifold which has the desired properties.Now we consider the imageT ngW uε (T −nz)and we let δ′ be the largest positive number such that W uδ′(z) is well defined andcontained in T ngW uε (T −nz).If δ′ ≥δ then we are done.

Let us hence assume that δ′ < δ.It follows from the maximality of δ′ that the boundary of W uδ′(z) contains, atleast, a point from the boundary of T ngW uε (T −nz). Let z′ be such a point.

If z′belongs to Sn−1i≥N T iS−then we are again done. If not then T −n is differentiable atz′ and hence T −nz′ belongs to the boundary of gW uε (T −nz) and it does not belongto S+n .

It follows now from the construction of gW uε (T −nz) that T −nz′ must belongto the boundary of W uε (T −nz) or for some j, 0 ≤j ≤k, T −ijz′ belongs to theboundary of U.We will obtain now a contradiction by using the expansion property (9.4) . LetW uδ′(z) be the graph ofχ : B(π1z; δ′) →Vand letγ0 : [0, 1) →B(π1z; δ′)be the segment connecting π1z and π1(z′).

We consider the preimages of the curve{(γ0(t), χγ0(t)) | 0 ≤t < 1} and obtain γj : [0, 1) →V, j = 0, . .

., k by the formulaγj(t) = π1T −ij(γ0(t), χγ0(t)).It follows from (9.4) that the length of γ0 is not smaller than the length of γj timesbρ−j. If T −nz′ belongs to the boundary of W uε (T −nz) then the length of γk is atleast ε and we get the contradictionδ′ ≥bρ−kε ≥δ

48CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIFinally if T −ijz′ belongs to the boundary of U for some j, 0 ≤j ≤k, then γj whichconnects π1(T −ijz) ∈U1δ and π1(T −ijz′) must have the length at least b−1δ. Weget again the contradictionδ′ ≥bρ−jb−1δ ≥δ.□Definition 9.8.

We say that the unstable manifold of size δ Wuδ (z) is cut byT iS−, i ≥0, if its boundary contains a point from T iS−.By Theorem 9.7 to guarantee that at least some points (and in the case of asmooth map almost all points) have unstable manifolds of size δ we need to stepaway from the boundary of U by at least b−1δ. In the following we fix a sufficientlysmall δ0 and restrict our discussions to U1 = U1δ0.

We can then claim that in U1almost every point has a uniformly large unstable manifold (of size δ0) or a smallerunstable manifold cut by some image of the singularity set S−.By ¯B(c; r) we denote the closed ball with the center at c and the radius r. Wedefine a rectangle R(z; δ) with the center at z = (z1, z2) and the size δ as theCartesian product of closed ballsR(z; δ) = ¯B(z1; δ2) × ¯B(z2; δ2).Definition 9.10. We say that the unstable manifold W uδ′(z′) of z′ = (z′1, z′2) ofsize δ′ is connecting in the rectangle R(z; δ) with the center at z = (z1, z2) and sizeδ if¯B(z1; δ2) ⊂B(z′1; δ′)andπ2 (W uδ′(z′) ∩R(z; δ)) ⊂B(z2; δ2).We can say equivalently that an unstable manifold W uδ′(z′) is connecting in therectangle R(z; δ) if the intersection of W uδ′(z′) with the rectangle is the graph of asmooth mapping from the closed ball ¯B(π1z; δ2) to the open ball B(π2z; δ2).

Clearlyit is necessary that δ′ > δ/2.Definition 9.11. For a given rectangle R(z; δ) with the center at z = (z1, z2) andsize δ we define its unstable core as the subset of those points z′ = (z′1, z′2) ∈R(z; δ)for whichρ∥z′1 −z1∥+ ∥z′2 −z2∥< (1 −ρ)δ2.The role of an unstable core is revealed in the following Lemma.Lemma 9.12.

If an unstable manifold W uδ′(z′) of size δ′ > ∥π1z′ −π2z∥+ δ2 inter-sects the unstable core of a rectangle R(z; δ) then it is connecting in the rectangle.Proof.Let z = (z1, z2) and z′ = (z′1, z′2), let W uδ′(z′) be the graph of ψ : B(z′1; δ′) →Vand let (x1, ψx1) be a point in the unstable core of the rectangle. By the conditionon δ′¯B(z1; δ ) ⊂B(z1; δ′).

ERGODICITY IN HAMILTONIAN SYSTEMS.49Figure 8. The core of a rectangle.We have to check only that if x ∈¯B(z1; δ2) then∥ψx −z2∥< δ2.We have∥ψx −z2∥≤∥ψx −ψx1∥+ ∥ψx1 −z2∥≤sup ∥Dψ∥∥x −x1∥+ ∥ψx1 −z2∥≤ρ∥x −z1∥+ ρ∥x1 −z1∥+ ∥ψx1 −z2∥< ρδ2 + (1 −ρ)δ2 = δ2.□The point of the above lemma is that a large unstable manifold may fail to beconnecting in a rectangle if it intersects the rectangle too close to the boundary.§10.

LOCAL ERGODICITY IN THE SMOOTH CASE.Contrary to the title of this section we will consider here several propositionsvalid in the general case. Incidentally they will suffice to obtain local ergodicity inthe smooth case.It is important to remember that all of Section 9 can be repeated for stablemanifolds.

In this section we will be using both stable and unstable manifolds.Lemma 10.1. If an unstable manifold and a stable manifold are connecting ina rectangle then there is a unique point of intersection of these manifolds in therectangle and it belongs to the interior of the rectangle.Proof.

Let the rectangle have the center at z = (z1, z2) and size δ. The intersectionsof the unstable and stable manifolds with the rectangle R(z; δ) are the graphs of

50CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIthe smooth mappingsψu : ¯B(z1; δ2) →B(z2; δ2)andψs : ¯B(z2; δ2) →B(z1; δ2)respectively.Since both ψu and ψs are contractions so is their compositionψsψu : ¯B(z1; δ2) →B(z1; δ2).Hence it has a unique fixed point x ∈B(z1; δ2). The point(x, ψux) = (ψsψux, ψux)is the desired intersection point.□For a rectangle R we denote by W (u)s(R) the union of the intersections with Rof all (un)stable manifolds connecting in R, i.e.,W (u)s(R) =[{R ∩W (u)sδ′(z′) | W (u)sδ′(z′) is connecting in R}.The union of the unstable core and the stable core of a rectangle will be in thefollowing called simply the core of the rectangle.Proposition 10.2.

For any rectangle R ⊂U1 if the sets W s(R) and W u(R) havepositive measure then W s(R) ∪W u(R) belongs to one ergodic component of T.Proof. The proof is done by the Hopf method as described in Sections 1 and 2.Let us fix a continuous function defined on our phase space.For all pointsin one (un)stable manifold the (backward) forward time averages are the same.As shown in Section 1 the forward and backward time averages have to coincidealmost everywhere.

Our goal is to show that they are constant almost everywherein W s(R) ∪W u(R).There is a technical difficulty stemming from the fact that the foliations intostable and unstable manifolds are not smooth in general. One has to use the absolutecontinuity of the foliations which was proven in [KS] under the conditions which fitour scheme.

(It is by far the hardest fact to prove in their theory. )It follows from absolute continuity of the foliation into unstable manifolds thatexcept for the union of unstable manifolds from W u(R) of total measure zero almostevery point (with respect to the Remannian volume in the manifold) in an unstablemanifold from W u(R) has equal forward and backward time averages.

Let us takesuch a typical unstable manifold.Again by the property of absolute continuitythe union of stable manifolds in W s(R) which intersect the distinguished unstablemanifold at points where the forward and backward time averages exist and areequal differs from W s(R) by a set of zero measure. Hence the time average of ourfunction is constant almost everwhere in W s(R).

Similarly the time average of ourfunction is constant almost everywhere in W u(R).Finally using the property of absolute continuity for the third time we can claimthat W u(R) and W s(R) intersect on a subset of positive measure. Hence the timeaverage of our function is constant almost everywhere in W s(R) ∪W u(R)

ERGODICITY IN HAMILTONIAN SYSTEMS.51To prove that W s(R) ∪W u(R) belongs to one ergodic component we proceed inthe same way as at the end of Section 2.□We are ready to prove the local ergodicity in the smooth caseProof of Main Theorem (smooth case).All the constructions started in Section 9 apply to our point p. We will provethat a neighborhood U2 only slightly smaller than U1 belongs to one ergodic com-ponent. Indeed according to Lemma 9.12 all the points in the (un)stable core of arectangle R ⊂U1 which have an (un)stable manifold of sufficiently large size belongto W (u)s(R).

By Theorem 9.7 in the smooth case almost every point in U1 has boththe unstable manifold and the stable manifold of size δ0. Hence by Lemma 9.12 forany rectangle R ⊂U1 of size δ < δ0 the set W s(R) contains at least the stable coreof R and W u(R) contains at least the unstable core of R. Clearly then the setsW s(R) and W u(R) have positive measure and we can apply Proposition 10.2.To end the proof we consider a family of rectangles of size δ ≤δ0 contained in U1whose cores cover a slightly shrunk neighborhood U2 ⊂U1.

By Proposition 10.2 wecan claim that each core belongs to one ergodic component. Since the cores form anopen cover of the connected set U2 we can conclude that U2 belongs to one ergodiccomponent.□Actually we can claim that under the assumptions of the Main Theorem thewhole neighborhood U constructed in Section 8 belongs to one ergodic component.Indeed by taking δ →0 the above argument applies to U2 →U1 so that actuallyU1 belongs to one ergodic component.

Again the δ0 in the definition of U1 canbe chosen arbitrarily small so that also the whole neighborhood U belongs to oneergodic component.This does not strengthen the theorem but it demonstratesthe usefulness of coverings with rectangles of size δ →0. It will be crucial in thetreatment of the discontinuous case.Let us outline the plan for proving local ergodicity in the general case.

We coverthe neighborhood U2 with rectangles of size δ. At least for some rectangles R thesets W s(R) and W u(R) will have positive measure.

We will be actually interestedin the property that these sets cover certain fixed (but otherwise arbitrarily small)percentage of the core of the rectangle and we will call such rectangles connecting.One may then expect to have more connecting rectangles as δ →0. The preciseformulation of such a property is the subject of Sinai Theorem.

The method of theproof requires that the size of the sector satisfies ρ < 13. In applying Sinai Theoremit is convenient to work with more structured coverings, namely the centers of therectangles will belong to a lattice with vertices so close that the cores of nearestneighbors rectangles will overlap almost completely.

Consequently, if both nearestneighbors R1 and R2 are connecting then the union of W s(R1) ∪W u(R1) andW s(R2) ∪W u(R2) belongs to one ergodic component (see Preposition 2.3).Itwill follows from Sinai Theorem that the network of connecting rectangles becomesmore and more dense as δ →0 so that we will be able to claim that one ergodiccomponent reaches from any place in the neighborhood U1 to any other place. Wewill conclude by using the Lebesgue Density Theorem to show that U2 belongs toone ergodic component.§11LOCAL ERGODICITY IN THE DISCONTINUOUS CASE

52CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIGiven δ > 0 we consider a shrunk neighborhood U2δ defined by the requirementthat a rectangle with the center in U2δ and size δ lies completely in U1. (One caneasily see that U2δ = Ua2(δ) where a2(δ) = a1(δ0) −δ2).

Let us note that U2δ →U1as δ →0.Let N (δ, c) be the net defined byN (δ, c) = {cδ(m, k) ∈U2δ | m, k ∈Zd}.We consider the family Gδ of all rectangles with the centers in N (δ, c) and size δGδ = {R(z; δ) | z ∈N (δ, c)}.If c is sufficiently small the family Gδ is a covering of U2δ . The parameter c will bechosen later to be very small so that many rectangles in Gδ overlap.

But once c isfixed a point may belong to at most a fixed number of rectangles, which we denoteby k(c) (it does not depend on δ).Definition 11.1. Given α, 0 < α < 1, we call a rectangle R ∈Gδ α-connecting inthe (un)stable direction (or simply connecting) if at least the α part of the measureof the (un)stable core of R is covered by W (u)s(R).Sinai Theorem 11.2.

If ρ < 13 then there is α, 0 < α < 1, such that for any climδ→0 δ−1µ[{R ∈Gδ | R is not α-connecting }= 0,i.e., the union of rectangles which are not α-connecting in either the stable or theunstable direction has measure o(δ)It is very important for the application of this theorem that given ρ < 13 we geta certain α (which may be very small if ρ is close to 13) and we are free to choose c(which determines the overlap of the rectangles in Gδ) as small as we may need.We will prove Sinai Theorem in Sections 12 and 13. In the remainder of thisSection we will show how to obtain the Main Theorem in the discontinuous casefrom Sinai Theorem.We start with some auxiliary abstract facts.The first one concerns MeasureTheory.

For any finite subset S we will denote by |S| the number of elements in S.Lemma 11.3. Let {As | s ∈S} be a finite family of measurable subsets of equalmeasure a in the measure space (X, ν) such that no point in X belongs to morethan k elements of the family.

For any subfamily {As | s ∈S1}, S1 ⊂S, we haveak |S1| ≤ν [s∈S1As!≤a|S1|.Further if for a measurable subset Y ⊂X and some α, 0 < α < 1,ν(As ∩Y ) ≥αν(As) for s ∈S1thenν(Y ) ≥ν [As ∩Y!≥αk ν [As!.

ERGODICITY IN HAMILTONIAN SYSTEMS.53□The second fact concerns Combinatorics. Let us consider the lattice Zd and itsfinite piecesLn = Ln(d) = {0, 1, .

. ., n −1}d ⊂Zd.Let K ⊂Ln be an arbitrary subset which we call a configuration.

We think ofelements of K as occupied sites and elements of Ln \ K as empty sites.For a given configuration K ⊂Ln we consider the graph obtained by connectingby straight segments all pairs of occupied sites which are nearest neighbors. LetgK ⊂K be the family of sites in the largest connected component of the graph.Proposition 11.4.

Let Kn ⊂Ln(d), n = 1, 2, . .

., be a sequence of configurations.Ifn|Ln \ Kn||Ln|→0 as n →+∞then|gKn||Ln| →1 as n →+∞.Proof. This proposition will follow immediately from the following combinatorialLemma.Lemma 11.5.

Let K ⊂Ln(d) be an arbitrary configuration. If|Ln \ K|nd−1< a < 1then|gK|nd≥1 −(d −1)a.Proof.

The proof is by induction on d. For d = 1 the statement is obvious. Supposeit is true for some d. We will establish it for d + 1.We partition Ln(d+1) into subsets Ln(d)×{i}, i = 0, .

. ., n−1 and we call themfloors.

We pick the floor with the fewest number of empty sites. Clearly the numberof empty sites there does not exceed and−1 so that we can apply to it the inductiveassumption.

We obtain in this floor a connected graph with at least (1−(d−1)a)ndelements.Now we partition Ln(d + 1) into subsets {z} × {0, . .

., n −1}, z ∈Ln(d) and wecall them columns. A column is called an elevator if all of its elements are occupied.The number of elevators is at least (1 −a)nd.

Hence the number of elevators whichintersect the connected graph in the floor considered above is at least (1 −da)nd.Adding these elevators to the graph we obtain a connected graph with at least(1 −da)nd+1 elements which ends the proof of the inductive step.□□Proof of Main Theorem (Discontinuous case). All the constructions of Sections 8through 10 apply with some ρ < 13.

We will be proving that the neighborhood U1belongs to one ergodic component.The Sinai Theorem gives us α < 1 which depends only on ρ and may have tobe very small if ρ is very close to 1Let us consider the lattice N (δ c) and the

54CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIcovering Gδ. We choose c so small that if the centers of two rectangles in Gδ arenearest neighbors in N (δ, c) then their unstable cores (and then automatically alsostable cores) overlap on more than 1 −α part of their measure.

Note that such aproperty depends on c but is independent of the value of δ. This choice of c hasthe following consequence.

If two rectangles R1 and R2 with centers at nearestneighbors in N (δ, c) are α-connecting in the unstable direction then W u(R1) andW u(R2) intersect on a subset of positive measure. If in addition we also know thatW s(R1) and W s(R2) have positive measure then using Proposition 10.2 we obtainthatW u(R1) ∪W u(R2) ∪W s(R1) ∪W s(R2)belongs to one ergodic component.We consider the configuration K(δ) in the lattice N (δ, c) which consists of thecenters of all rectangles in Gδ which are α-connecting both in the stable and unstabledirections.

As in the discussion proceeding Proposition 11.4 we consider the graphobtained by connecting with straight segments all pairs of nearest neighbors in K(δ).Let as before gK(δ) be the collection of vertices in the largest connected componentof this graph. By our construction the setY (δ) =[{W u(R(z; δ)) ∪W s(R(z; δ)) | z ∈gK(δ)}belongs to one ergodic component.

This set is crucial in our proof that U1 belongsto one ergodic component. It may be very small in measure (if α is small) but itcovers at least certain fixed α′ portion of the measure of each of the rectangles withcenters in gK(δ), i.e.,(11.6)µ (R(z; δ) ∩Y (δ)) ≥α′µ (R(z; δ))for any z ∈gK(δ) (α′ is smaller than α since α is only the part of the measure ofthe (un)stable core covered by the connecting (un)stable manifolds).

It remains toshow that the points in gK(δ) reach into all parts of U1. It will follow from SinaiTheorem.By Sinai Theorem the total measure covered by rectangles which are not α-connecting is o(δ).

Using Lemma 11.3 we can translate this estimate ask(c)−1|N (δ, c) \ K(δ)|δ2d = o(δ).Since in addition|N (δ, c)|(cδ)2d= O(1)we see that the assumptions of Proposition 11.4 are satisfied and we can claim that(11.7)|gK(δ)||N (δ, c)| →1 as δ →0.We are ready to finish the proof by a contradiction. Suppose there are two Tinvariant disjoint subsets E1 and E2 which have intersections with U1 of positivemeasure.

Let us pick two Lebesgue density points p1 and p2 for E1∩U1 and E2∩U1respectively. Next we fix cubes C1 and C2 with centers at p1 and p2 so small thatµ(Ci ∩Ei) ≥1 −α′2k( )µ(Ci), i = 1, 2.

ERGODICITY IN HAMILTONIAN SYSTEMS.55It follows from (11.7) that| (N (δ, c) \ gK(δ)) ∩Ci||N (δ, c)|→0 as δ →0, i = 1, 2.Since|N (δ, c)||N (δ, c) ∩Ci| = O(1), i = 1, 2,we conclude that| (N (δ, c) ∩Ci) \ gK(δ)||N (δ, c) ∩Ci|→0 as δ →0, i = 1, 2.Now we get immediately that(11.8)µ[{R(z; δ)|z ∈gK(δ) ∩Ci}△Ci→0 as δ →0, i = 1, 2,where △denotes the symmetric difference, i.e., for any two sets A and BA△B = (A \ B) ∪(B \ A).By (11.6) and Lemma 11.3µ[{R(z; δ)|z ∈gK(δ) ∩Ci} ∩Y (δ)≥α′k(c)µ[{R(z; δ)|z ∈gK(δ) ∩Ci},i = 1, 2.Comparing this with (11.8) and remembering how dense Ei is in Ci, i = 1, 2, weconclude that for sufficiently small δ the set Y (δ) must intersect both E1 and E2over subsets of positive measure which contradicts the fact that it belongs to oneergodic component.□§12. PROOF OF SINAI THEOREM.We will be proving only the unstable version of the theorem, i.e., we will estimatethe measure of the union of rectangles which are not α-connecting in the unstabledirection.

Everything can be then repeated for the stable manifolds.For a point y = (y1, y2) in the core of a rectangle R(z; δ) there are two possibil-ities:(1) the point y has an unstable manifold of size δ′ > ∥y1 −π1z∥+ δ2 (which isconnecting in R(z; δ) by Lemma 9.12),(2) the point y has an unstable manifold of size δ′ ≤∥y1 −π1z∥+ δ2 cut bySi≥0 T iS−.If a rectangle R(z; δ) is not connecting then the second possibility must occur forat least 1 −α part of its core.The neighborhood U was chosen so small that S−N = SN−1i=0 T iS−is disjoint fromU. It follows that, for points in U1, the unstable manifolds of size δ′ < δ0 cannotbe cut by these singularities.

For any M ≥N let us introduce the following specialcase of the second property:(2M) the point y has an unstable manifold of size δ′ ≤∥y1 −π1z∥+ δ2 cut bySMi=N T iS−.Further, we introduce the auxiliary notion of a M-nonconnecting rectangle.Roughly speaking, it is a rectangle which is not connecting because of the sin-gularity set SMT iS−

56CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIDefinition 12.1. Given α < 12 we say that a rectangle R of size δ is M-nonconnec-ting, if at least 1−2α part of the measure of the unstable core of R consists of pointswhich satisfy the property (2M).The plan of the proof is the following.

We fix an arbitrary positive ε > 0 and wedivide the argument in two parts. In one part we will prove that there is M = M(ε)and δε such that, for all δ < δε, the total measure of all rectangles in Gδ which arenot α-connecting and are not M-nonconnecting is less than δ ε2.

This is the subjectof the ‘tail bound’ (section 13) and it is by far the hardest part of the proof. Itwill require global considerations (i.e., outside of U).

The particular value of α isimmaterial there.We will start with the easier part proving that, for a given ρ < 13 and any M,there are α and δε such that, for all δ < δε, the total measure of all M-nonconnectingrectangles of size δ is less than δ ε2. Let us formulate it in a separate Proposition.Its proof will be completely confined to the neighborhood U.Proposition 12.2.

For any ρ <13, there is α, 0 < α < 1, such that, for anyM ≥N,limδ→0 δ−1µ[{R ∈Gδ | R is M-nonconnecting }= 0.Proof. We rely on our assumption that S−and its images are sufficiently ‘nice’.More precisely we have required that the singularity set S−M+1 = SMi=0 T iS−isregular.

The definition of regularity was tailored to the needs of this proof. Inparticular the singularity set S−M+1 is a finite union of pieces of submanifolds Ikof codimension one, with boundaries ∂Ik, k = 1, .

. ., p. The boundaries ∂Ik, k =1, .

. ., p are themselves also finite unions of compact subsets of submanifolds ofcodimension 2 .

What is moreIk ∩Il ⊂∂Ik ∪∂Il for any k, l.In each of the closed manifolds Ik, k = 1, . .

., p, we consider the open neigh-borhood of the boundary of radius r, and we denote by Jr the union of theseneighborhoods, i.e.,Jr =p[k=1{p ∈Ik | d(p, ∂Ik) < r}.For each δ let r(δ) be the smallest r such that, for any k ̸= l, the distance ofIk \ Jr and Il \ Jr is not less than 2δ. (In other words, for any k ̸= l, the sets Ik \ Jrand Il \ Jr are disjoint compact subsets, and their distance is at least 2δ.) Clearlylimδ→0 r(δ) = 0.Hence, by the property (7.3)(12.3)limδ→0 µS(Jr(δ)) = 0where µS is the natural volume element on S−M+1.Let us note that, if a rectangle R = R(z; δ) contains a point with the unstablemanifold of size δ′ < δ cut by S−then it intersects the 2δ neighborhood of S−

ERGODICITY IN HAMILTONIAN SYSTEMS.57but it does not necessarily intersect the singularity set itself. For technical reasons,we prefer to blow up every rectangle, so that the blown up rectangle must intersectS−M+1 itself, and not only its neighborhood.

For a fixed b0 < 13, to be chosen later,and for any rectangle R = R(z; δ), we introduce the blown up rectangleeR = B(π1z, (1 + 2b0)δ2) × B(π2z, δ2).The diameter of eR is less than 2δ, since we assume that b0 < 13.Let y belong to the core of R, satisfy the property (2M), and∥π1y −π1z∥≤b0δ2.This implies that the unstable manifold W uδ′(y) is contained in eR, so that eR in-tersects SMi=N T iS−. We conclude that, for α sufficiently small, if a rectangle Rof size δ is M-nonconnecting, then eR intersects at least one of the submanifoldsIk, k = 1, .

. ., p. If for a rectangle R of size δ the blown up rectangle eR intersectstwo submanifolds Ik and Il, k ̸= l then, by definition of r(δ) it must intersect Jr(δ),and so it must be contained in the neighborhood of Jr(δ) of radius 2δ.

By (12.3) andProposition 7.4 the measure of the neighborhood of Jr(δ) of radius 2δ is o(δ) (i.e.,when divided by δ, it tends to zero as δ tends to zero). It remains to consider thoseblown up rectangles which intersect only one of the submanifolds Ik, k = 1, .

. ., p.The proof will be finished when we prove that, for all sufficiently small δ, if ablown up rectangle eR intersects only one of the submanifolds Ik, k = 1, .

. ., p, (anddoes not intersect ∂Ik), then the rectangle R is not M-nonconnecting.Our first observation is that there is a constant K depending only on the mani-folds Ik, k = 1, .

. ., p, such that for any x, x′ ∈Ik there is v in the tangent space toIk at x (v ∈TxIk) for which(12.4)∥x′ −x −v∥≤K∥x′ −x∥2Here we consider the tangent space TxIk of Ik at x as a subspace in Rd × Rd.

Thisproperty is a formulation of the fact that smooth submanifolds are locally close totheir tangent subspaces and follows easily from the Taylor expansion.Further, in view of the proper alignment of the singularity manifolds, the tangentsubspaces TxIk, x ∈Ik ∩U1 must have their characteristic lines in Cρ.Let us now take a rectangle R = R(z; δ) such that the blown up rectangle eRintersects Ik. We will show that π2(Ik ∩eR) is contained in a fairly narrow layer.To show this, let x = (x1, x2), x′ = (x′1, x′2) ∈Ik ∩eR and let v = (ξ, η) ∈TxIk bethe vector for which (12.4) holds.

We pick a nonzero vector v0 = (ξ0, η0) ∈TxIkwith the direction of the characteristic line. For convenience, we scale it so that∥ξ0∥= 1.

We have, by the definition of a characteristic line,ω(v, v0) = ⟨ξ, η0⟩−⟨η, ξ0⟩= 0.It follows that|⟨η ξ ⟩||⟨ξ η ⟩| ≤ρ∥ξ ∥∥ξ∥ρ∥ξ∥

58CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIReplacing v by x −x′ in the last inequality and using (12.4), we get|⟨ξ0, x′2 −x2⟩| ≤ρ(∥x′1 −x1∥+ K∥x′ −x∥2) + K∥x′ −x∥2.Since both x and x′ are in eR, we have that∥x′1 −x1∥< (1 + 2b0)δand∥x′ −x∥< 2δ.Therefore, for any x, x′ ∈Ik ∩eR, we obtain the inequality(12.5)|⟨ξ0, x′2 −x2⟩| ≤ρ(1 + 2b0)δ + const δ2where the constant depends only on ρ and K. The inequality (12.5) shows thatπ2(Ik ∩eR) is contained in a layer perpendicular to ξ0 of width ρ(1+2b0)δ+const δ2.Hence, there is ¯x2 (in the ‘center’ of the layer) such that every x = (x1, x2) ∈Ik ∩eRmust belong to the layer defined by the inequality(12.6)|⟨ξ0, x2 −¯x2⟩| ≤ρ(1 + 2b0)δ2 + const δ2We want to estimate the width of the layer where all the points from the coreof the rectangle with ‘short’ unstable manifolds, cut by Ik, must lie. To that endlet us take a point y = (y1, y2) in the core of the rectangle R(z; δ) and such that∥y1−π1z∥≤b0 δ2.

If y satisfies the property (2M) then by Lemma 9.6 the projectionπ2W uδ′(y) of the unstable manifold lies in the ballB(y2; ρδ′) ⊂B(y2; ρ(1 + b0)δ2).Assuming that W uδ′(y) is cut by Ik, there is x = (x1, x2) ∈Ik ∩eR for which|⟨ξ0, y2 −x2⟩| ≤ρ(1 + b0)δ2Hence, by (12.6), the point y must belong to the layer defined by the inequality(12.7)|⟨ξ0, y2 −¯x2⟩| ≤ρ(1 + b0)δ2 + ρ(1 + 2b0)δ2 + const δ2The last step is to choose b0 so small that this layer cannot cover all of the core.We prefer, for convenience, to fit a Cartesian product into the unstable core, andto prove that a fixed part of this set is cover by connecting manifolds. We choosesuch set to beX(b0) = B(π1z; b0δ2) × B(π2z; s(b0)δ2)where s(b0) = 1 −ρ −ρb0.

By the definition of a core the set X(b0) is contained inthe core of R(z; δ), and its measure is not less than certain fixed part of the measureof the core depending on b (and the dimension d) but independent of δ

ERGODICITY IN HAMILTONIAN SYSTEMS.59If the layer (12.7) is sufficiently narrow, it cannot cover all of X(b0). The preciseinequality, which guarantees that, is easily transformed into(12.8)3ρ + const δ < 1 −4ρb0.After a moment of reflection the reader will realize that only if ρ < 13 we can chooseb0 so small that not only (12.8) is satisfied, but also certain fixed part of X(b0)(depending on b0 but independent of δ) is not covered by the layer (12.7).

Thus,there is α sufficiently small, depending on ρ and b0, such that more than 2α partof the measure of the core is free of points satisfying the property (2M). Hence therectangle R is not M-nonconnecting.□If the reader finds it hard to follow the above argument, it is because we strivedto use as little hyperbolicity as possible on our finite orbit.

The amount of hyper-bolicity is measured by the size ρ of the sector . We have managed to relax thecondition on ρ up to ρ < 13.

It is not hard to see that if the last condition is relaxedfurther Proposition 12.2 will not hold in general.§13. ‘TAIL BOUND’.We will be proving that for every ε > 0 there is M such that the measure ofpoints z ∈U1 with the unstable manifold of size δ′ < δ cut by Si≥M+1 T iS−doesnot exceed εδ.

Comparing this set with the union of rectangles in Gδ which are notα-connecting and not M-nonconnecting, we establish immediately that the measureof the union can be bigger by at most an absolute (=independent of δ) factor, madeup of ρ, α and the overlap coefficient k(c) (introduced prior to Definition 11.1). Toarrive at this conclusion it is important that we consider only the rectangles fromthe covering Gδ (and not all possible rectangles of size δ).We start by exploring some of the consequences of the Sinai - Chernov Ansatz.No reference to the neighborhood U will be made at this stage.

So we have assumedthat almost all points in S−(with respect to the measure µS) are strictly unboundedin the future. It follows from Theorem 6.8 that, for almost every point p ∈S−,limn→+∞inf0̸=v∈C(p)pQ(DpT nv)∥v∥= +∞.For a linear monotone map, let us putσ∗(L) =inf0̸=v∈C(p)pQ(Lv)∥v∥.Consequently, for any (arbitrarily small) h > 0 and any (arbitrarily large) t > 0,there is M = M(h, t) so large that the subseteEt = {p ∈S−| σ∗(DpT M) ≤t + 1}has measureµ ( eE ) ≤h

60CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIThe map T M is, in general, not even continuous in all of S−.The coefficientσ∗(DpT M) is defined only for almost every point p ∈S−. Hence, so far, the subseteEt is defined modulo subsets of measure zero.

We need a closed subset, since weplan to use Proposition 7.4.The map T M is discontinuous on S+M, which was assumed to be a regular set.Using the proper alignment of singularity sets and monotonicity of the system, weconclude that S+M is transversal to S−(in the natural sense). It follows that theset BM =S+M ∪∂M∩S−is a finite union of compact subsets of submanifoldsof dimension 2d −2.

Further, S−is decomposed into (possibly very large) finitenumber of pieces of submanifolds of dimension 2d−1 such that T M is differentiablein the interior of every piece, and their boundaries are subsets of BM. It followsthat the coefficient σ∗(DpT M) is continuous in the interior of every piece.Let us choose ζ so small that the closure of the ζ-neighborhood of BM in S−BζM = {p ∈S−| d(p, BM) < ζ}has small measureµSBζM≤h.Now the set Et defined byEt = eEt \ BζM = {p ∈S−\ BζM | σ∗(DpT M) ≤t + 1}is closed, and we haveµSEt ∪BζM≤2h.LetSt = {p ∈S−\ BζM | σ∗(DpT M) ≥t + 1}.St is a compact set and the coefficient σ∗(DpT M) is continuous in a neighborhoodof St in M. Hence, there is r > 0 such thatσ∗(DpT M) > t,for every point p in the r-neighborhood of St in M, letSrt = {p ∈M | d(p, St) < r}.Now we look at our neighborhood U.

Our goal is to estimate, for given δ, themeasure of the set Y (δ, M) of points in U1 which have the unstable manifold ofsize δ′ < δ cut by Si≥M+1 T iS−. We will achieve this by splitting Y (δ, M) intoconvenient pieces and showing that their preimages must end up in extremely smallneighborhoods of S−.For z ∈Y (δ, M) the unstable manifold Wuδ′(z) may be cut by several (possiblyinfinitely many) of the singularity sets T iS−, i = M + 1, .

. .. Let m(z) be thesmallest i ≥M + 1 such that Wuδ′(z) is cut by T iS−.

Let furtherk(z) = #{i | 1 ≤i ≤m(z) −M, T −iz ∈U1}.Roughly speaking k(z) is the number of times the point z visits in U1 in the pastin the time frame bounded by m(z). We put for k = 0, 1, .

. ., m = M + 1, .

. .,Y km = {z ∈Y (δ, M) | m(z) = m, k(z) = k}.We will now fix k and estimate the measure of[m≥M+1Y km.

ERGODICITY IN HAMILTONIAN SYSTEMS.61Lemma 13.1. For m ̸= m′T −mY km ∩T −m′Y km′ = ∅.Proof.

Let m < m′. If y ∈T −mY km ∩T −m′Y km′ then for z = T my and z′ = T m′y wehavek(z′) ≥k(z) + 1.It contradicts the fact that z ∈Y km and z′ ∈Y km′.□By Lemma 13.1 we haveµ([m≥M+1Y km) ≤Xm≥M+1µ(Y km) =Xm≥M+1µ(T −mY km) = µ([m≥M+1T −mY km).Let z ∈Y km and z′ ∈T mS−be a point in the boundary of Wuδ′(z).

We connectz and z′ by the curve γ in Wuδ′(z) which projects under π1 onto the linear segmentfrom π1z to π1z′. In the neighborhood U we have three ways of measuring thelength of γ.

We can use the quadratic form Q, or the length of the projection ontothe first component, or finally, we can use the Riemannian metric. All these metricsare equivalent in U and we will use the following coefficients defined by their ratiossup∥v∥∥ξ∥| 0 ̸= v = (ξ, η) ∈Cρ=p1 + ρ2,q = sup(pQ(v)∥ξ∥| 0 ̸= v = (ξ, η) ∈Cρ)where the last supremum is taken also over all of U.Our goal is to estimate the distance of T −mz and T −mz′ in the Riemannianmetric, such a distance clearly does not exceed the length of the curve T −mγ.

Tothat end, let n ≤m −M, be the time of the k-th visit in the past by z to U1, i.e.,T −nz ∈U1. By Proposition 8.4 on every spaced return to U the projection of thepreimage of γ is contracted by at least the coefficient ρ.

In the k visits there mustbe at leastkN −1 spaced returns. Hence, the projection of T −nγ has the lengthwhich, by (8.6u), does not exceedc1λkδ,whereλ = ρ1Nand c1 = 1ρb =1ρp1 −ρ4 .It follows that the Riemannian length of T −nγ does not exceedc2λkδ,wherec2 =1ρp1 −ρ2 ,and its length in the metric Q does not exceedc λkδ

62CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIwherec3 =qρp1 −ρ2 ,Now we apply T −(m−n) to T −nγ and we use the fact that m −n ≥M. Thereare two different cases.Case 1.T −mz′ ∈Et ∪BζMWe use the noncontraction property.Under the action of T −(m−n) the Rie-mannian length of γ can expand at most by the factor 1a.

We conclude that theRiemannian length of T −mγ does not exceedc2a λkδ.Thus T −mz belongs to the neighborhood of Et ∪BζM in M of this radius.ByProposition 7.4 its measure does not exceed(13.2)3h2c2a λkδ,if only δ is small enough (δ ≤δ0 and δ0 does not depend on k or m).Case 2.T −mz′ ∈StWe claim that, for sufficiently small δ the length of T −mγ does not exceed1t c3λkδ.Indeed, it is so if T −mγ is contained in Srt (the r-neighborhood of St in M). Sincem −n ≥M, we haveσ∗(DpT m−n) > t,for every point p ∈Srt .

Hence, the length in the metric Q of T −nγ is longer thanthe Riemannian length of T −mγ by at least the factor t. If T −mγ is not containedin Srt , then there must be a segment of this curve in Srt which has at least lengthr. It follows that the image of this segment under T m−n has the length in themetric Q not less than tr, which is more than the total length in the metric Q ofT −nγ for sufficiently small δ.

This contradiction shows that, for sufficiently smallδ, T −mγ ⊂Srt . We have proven our claim.

It follows that T −mz belongs to theneighborhood of S−of radius 1t c3λkδ. Using again Proposition 7.4, we can estimatethe measure of this neighborhood by(13.3)2µS(S−)2c3t λkδ,if only δ is sufficiently small (δ ≤δ0 and δ0 does not depend on k or m).Combining the estimates (13.2) and (13.3) we obtain that for any k = 0, 1, .

. .,µ([T −mY km) ≤h6c2a + 1t 2c3µS(S−)λk δ.

ERGODICITY IN HAMILTONIAN SYSTEMS.63It follows thatµ(Y (δ, M)) ≤h6c2a + 1t 4c3µS(S−)11 −λ δ.The last inequality tells us how we should choose a small h and a large t at thebeginning of our argument to guarantee thatµ(Y (δ, M)) ≤εδ.The ‘tail bound’ is proven.§14. APPLICATIONS.A.

Billiard systems in convex scattering domains.We assume that the reader is familiar with billiard systems. If it is not the case,we recommend [W4] for a quick introduction into the subject.

We will rely on theresults of that paper.Let us consider a domain in the plane bounded by a locally convex closed curvegiven by the natural equation r = r(s), 0 ≤s ≤l describing the radius of curvaturer as a function of the arc length s. We assume that the radius of curvature satisfiesthe condition(14.1)d2rds2 < 0,for all s, 0 ≤s ≤l.Curves satisfying this condition were called in [W4] strictly convex scattering.Examples.1. Perturbation of a circle.2.

Cardioid.Such a domain cannot be convex, and there is a singular point in the boundarywhere the curve intersects itself. (If you do not like playing billiards on a tablewhich is not convex, you may take the convex hull of our domain and everythingbelow still applies.

)The following theorem is a fairly easy consequence of the Main Theorem.Theorem 14.2. The billiard system in a domain bounded by a strictly convex scat-tering curve (i.e., satisfying (14.1)) is ergodic.Let us consider the map T describing the first return map to the boundary.

T isdefined on the set M of unit tangent vectors pointing inwards. We parametrize Mby the arc length parameter of the foot point s, 0 ≤s ≤l, and the angle ϕ, 0 ≤ϕ ≤π, which the unit vector makes with the boundary (oriented counterclockwise).

Inthese coordinates M becomes the rectangle [0, l] × [0, π]. The symplectic form (theinvariant area element) is given by sin ϕ ds ∧dϕ.

After we derive the formula forthe derivative of T, we will be able to check immediately that T preserves this areaelement

64CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIThe map T is discontinuous at those billiard orbits which hit the singular point ofthe boundary. They form a curve S+ in M which is a graph of a strictly decreasingfunction, decreasing curve for short.

This curve divides the rectangle M into twocurvilinear triangles, M+b with a side at the bottom and M+t with a side at thetop.To find the images of M+b and M+t we use the reversibility of our system. Namely,let S : M →M be defined by S(s, ϕ) = (s, π −ϕ).

We haveT ◦S = S ◦T −1.We can now claim that T −1 is continuous except on S−= SS+ which is an increas-ing curve (the graph of a strictly increasing function). S+ divides the rectangle Minto two curvilinear triangles M−b = SM+t and M−t = SM+b .

We have constructedour symplectic boxes. T is a diffeomorphism on their interiors and a homeomor-phism on the closure.

The derivative of T does blow up at least at one point ofthe boundary S+ (different for M+b and for M+t ) corresponding to the two billiardorbits tangent to one of the branches of the boundary at the singular point. Inthe case of the cardioid the derivative blows up at any point of S+ and also at thevertical boundaries because the curvature at the cusp is infinite (see the formulafor the derivative of T below).

It is very handy that we did not have to require inSection 7 that our map is a diffeomorphism on the closed symplectic boxes.The derivative of DT at (s0, ϕ0) has the form(14.2)τ−d0r0 sin ϕ1τsin ϕ1τ−d0−d1r0r1 sin ϕ1τ−d1r1 sin ϕ1where T(s0, ϕ0) = (s1, ϕ1), τ is the time between consecutive hits (i.e., the lengthof the billiard orbit segment) and di = ri sin ϕi, i = 1, 2. This derivative can beobtained by straightforward implicit differentiation but we do not recommend it.There is a more geometric (and safer) way to obtain the derivative by resorting tothe description of billiard orbit variations by Jacobi fields.

In our two dimensionalsituation it amounts to introducing coordinates (J, J′) in the tangent planes of M(14.3)J = sin ϕds,J′ = −1r ds −dϕ.The evolution of (J, J′) between collisions is given by the matrix(14.4)1τ01.At the collision (J, J′) is changed by(14.5) −102d1−1.Now the derivative (14.2) is obtained by multiplying the matrices (14.4) and (14.5)and taking into account (14 3)

ERGODICITY IN HAMILTONIAN SYSTEMS.65The geometric meaning of d0, d1, and the inequality(14.6)τ > d0 + d1is explained at length in [W4]. It was proven there that (14.6) holds for any billiardorbit segment, if the boundary curve is strictly convex scattering (actually thesetwo properties are essentially equivalent).

It follows from (14.6) that for a strictlyconvex scattering curve all elements in (14.2) are positive.We choose as our family of sectors the constant sector between the horizontal line{dϕ = 0} and the vertical line {ds = 0}. We see immediately that the derivativeDT is strictly monotone.We are now ready to argue that the singularity sets S−n = Sni=0 T iS−are regular.We claim that S−n is a finite union of increasing curves which intersect each otheronly at the endpoints.

It can be proven by induction. Indeed S−is an increasingcurve and so it is also properly aligned.

The singularity set S+ is a decreasingcurve, and as such it may intersect each of the increasing curves of S−n in at mostone point. Hence both M+b ∩S−n and M+t ∩S−n are finite unions of increasing curveswith intersections only at the endpoints.

Hence in view of the monotonicity of oursystem the images under T are also finite unions of increasing curves in M−b andM−t respectively. It is clear that we can safely add S−to these images.

We havethus checked that S−n+1 = S−∪TS−n is also a finite union of increasing curves whichintersect only at the endpoints. Note that the assumptions of Lemma 7.6 are toorestrictive to allow its application in this case.One can easily compute (and it was done explicitly in [W4]) that(14.7)σ(DT) =√1 + ω + √ω,where ω = (τ −d0 −d1)τd0d1.It follows from (14.7) and from the supermultiplicativity of the coefficient of expan-sion σ that the only way in which an orbit can fail to be strictly unbounded is whenthe lengths of the segments of the orbit go to zero.

It was shown by Halpern [Ha]that there are no such billiard orbits, if r(s) is a C1 function bounded away fromzero. Hence, under such an assumption, which excludes the cardioid, all orbits forwhich arbitrary power of T is differentiable are strictly unbounded.

To include thecardioid, or more generally the curves with the radius of curvature r(s) decreasingmonotonously to zero at the endpoints of the interval, 0 ≤s ≤l, (at the singularpoint), we shall argue that also for this class there is no accumulation of collisions atthe singular point. Indeed, if an arc of the boundary between two consecutive hitsby the billiard ball has monotone curvature, then the angle of incidence(reflection)is smaller where the curvature is bigger.

Hence, as an orbit gets closer to the sin-gularity point (the cusp for the cardioid), it is more and more perpendicular to theboundary, and so it cannot accumulate at the singularity.This observation takes care of the Sinai - Chernov Ansatz. We are also guaranteedthat the coefficient σ(DT n) can be made arbitrarily large by increasing n, exceptpossibly for points which end up on the decreasing curve S+ in the future and theincreasing curve S−in the past.

These are the points in S+n ∩S−m, for some n and m,and so there are only countably many such points. (The orbit of such a point ‘dies’both in the future and in the past, and it may fail to pick up enough hyperbolicitybefore then ) We can apply the Main Theorem to all other points and they form a

66CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIconnected set. Hence, the local ergodicity obtained from the Main Theorem impliesergodicity.It remains to check the noncontraction property.

It was pointed out to us byDonnay [D1] that the derivative of T increases |J′|2 on nonzero vectors from thesector. Indeed the interior of the sector is defined byJ′J < −1dso that we have|J′||J| > 1d.If DT(J0, J′0) = (J1, J′1) then we have from (14.4) and (14.5) thatJ1 = −J0 −τJ′0.It follows that|J′1| ≥1d1|J1| = 1d1|J0 + τJ′0| ≥τd1|J′0| −1d1|J0| ≥τ −d0d1|J′0|.In view of (14.6) τ−d0d1> 1.

So indeed |J′|2 gets increased.Moreover, for all vectors in the sector we have the following estimates2( 1r2 ds2 + dϕ2) ≥|J′|2 = |1r ds + dϕ|2 ≥1r2 ds2 + dϕ2.The metric1r2 ds2 + dϕ2 is equivalent to the standard Riemannian metric inthe (s, ϕ) coordinates (ds2 + dϕ2) if only r is bounded away from zero.Thusnoncontraction is established under this additional assumption, which excludes thecardioid.To cover the case of the cardioid, we observe that the noncontraction propertyis used only in the proof of the ‘tail bound’. In that proof some subsets of theneighborhood U are transported back to the neighborhood of the singularity setS−.

We need the property that vectors from the sector C are not contracted toomuch, along the orbits from the vicinity of the singularity set to the neighborhood U,even if the orbit is very long. We obtain readily this property from the observationthat although |J′|2 is, in general, only bigger than the scaled standard Riemannianmetric, it is clearly equivalent to one locally in the neighborhood U.The reader may be worried that the standard Riemannian metric in the (s, ϕ)coordinates does not generate the invariant area element.

However, the Riemannianarea is not smaller than the symplectic area. This is sufficient for the proof of SinaiTheorem.

We could also handle this complication by introducing from the verybeginning coordinates in M in which the symplectic form is standard.We can conclude that T is ergodic, and so Theorem 14.2 is proven.It follows from the results of Katok and Strelcyn [KS] that T is a Bernoullisystem.The framework of this paper allows to cover also the class of billiard systems indomains with more than one smooth piece in the boundary, which are not necessarilyconvex scattering In the recent paper [D2] Donnay introduced a natural condition

ERGODICITY IN HAMILTONIAN SYSTEMS.67(focusing arc) on the convex pieces of the boundary of a billiard table. He provesthat if two focusing arcs are connected by sufficiently long (extremely long maybe required) straight segments, then the billiard system in such a (stadium like)domain has nonvanishing Lyapunov exponents.

This work puts the original stadiumof Bunimovich [B], which had arcs of circles in the boundary, into a large class ofbilliard systems with nonuniform hyperbolic behavior, larger than the class withconvex scattering pieces introduced in [W4].All the properties listed in Section 7 are satisfied for the billiards of Donnay in astraightforward fashion, with the notable exception of the noncontraction property.The problem is that the construction of the bundle of sectors depends heavily on thedynamics, and it is unlikely that there is a geometrically defined Lyapunov metric(like |J′|2 for the convex scattering curves). Instead we use the following two ideas.We have remarked in Section 7 that if the map T is differentiable up to andincluding the boundary of symplectic boxes, and DT is strictly monotone, then thenoncontraction property holds automatically.

In the billiards of Donnay the sectorsare pushed strictly inside at the time of crossing from one convex piece to the other.Hence, we can use this observation on the compact part of the phase space madeup of orbits which cross over from one convex piece to the other. We have thenoncontraction property for the return map to this set, where we measure vectorsin C using the form Q defined by the bundle of sectors uniformly larger than C.The construction of the bundle of sectors C by Donnay and his condition on theseparation of convex pieces allows to introduce immediately these larger sectorswith respect to which the derivative of the return map is monotone.It remains to check the noncontraction property along ‘grazing’ orbits whichreflect many times in one convex piece.This is essentially done in [D2], whereLazutkin coordinates are used to put the map T in the vicinity of the boundaryinto a normal form.These two observations, put together, give us the unconditional noncontractionproperty, and thus our Main Theorem applies.B.

Piecewise linear standard map.Let T : T2 →T2 be defined byT(x1, x2) = (x1 + x2 + Af(x1), x2 + Af(x1))where (x1, x2) are taken modulo 1, f is a periodic functionf(t) = |t| −12,for −12 ≤t ≤12,and A is a real parameter. The mapping T preserves the Lebesgue measure.

ForA = 1 there is a simple invariant domain D in the torus shown in Figure 9. Itwas proven in [W5] that the Lyapunov exponents are different from zero almosteverywhere in D.Theorem 14.8.

T is ergodic in D.As in the previous application it follows that T is a Bernoulli system in D.All the conditions of Section 7 are satisfied here in a very simple fashion. Thereader can find all the necessary details in [W5] and [W6].

In this piecewise lin-ear case one does not have to rely on the general results of Katok and Strelcyn

68CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIFigure 9 The domain DThe existence of stable and unstable leaves can be obtained by the straightforwardapproach of Sections 1-3.There are many other values of A for which nonvanishing of Lyapunov exponentswas established for T in some domains in the torus, [W5],[W6]. The most inter-esting is the sequence of A’s (roughly speaking) going to zero for which there is aninvariant domain, with similar geometry as D, where T has nonvanishing Lyapunovexponents.

It is a piecewise linear model for the unstable layer containing the sep-aratrices of the saddle fixed point (0, 14). One can apply Main Theorem to all thesespecial domains , so that in each case the map T is ergodic and hence Bernoulli.

Thereader should not have any difficulties in recovering the details based on the twopapers cited above (incidentally even the noncontraction property was consideredthere).C. The system of falling balls.One of the original motivations for our work was to prove ergodicity of the systemof falling balls.

This is a monotone system ([W7], [W8], [W3]), and all (semi-infinite)smooth orbits are strictly unbounded. (The unboundedness of all orbits is obtained,under mild assumptions, by the application of Proposition 6.9) It follows that allLyapunov exponents are different from zero, and it looks like a prime candidate forthe application of Main Theorem.

It turns out, however, that in this example thesingularity sets are not properly aligned, if the number of balls is greater than two.We will show this, and briefly discuss the case of two balls.The system of falling balls is the system of point particles moving on a verticalline, which also interact by elastic collisions, and are subjected to a potential ex-ternal field which forces the particles to fall down. To prevent the particles fromfalling into an abyss we introduce the hard floor, and assume that the bottomparticle bounces back upon collision with it.

The masses of the particles are in gen-eral different (the system of equal masses is completely integrable since the elastic

ERGODICITY IN HAMILTONIAN SYSTEMS.69collision of equal masses in one dimension amounts to the exchanging of momenta).The Hamiltonian of the system isH =NXi=1 p2i2mi+ miU (qi)where qi are the positions and pi = mivi the momenta of the particles, qi, pi ∈R, i = 1, . .

., N, and U (q) is the potential of the external field . The differentialequations of the system are˙qi = pimi˙pi = −miU ′ (qi) ,i = 1, .

. ., N.The description of the dynamics is completed by the assumptions that the par-ticles are impenetrable, and that they collide elastically with each other and withthe floor q = 0.We choose the following Lagrangian subspacesV1 = {dp1 = · · · = dpN = 0}andV2 = {dh1 = · · · = dhN = 0},where hi =p2i2mi + miU (qi) , i = 1, .

. ., N, are individual energies of the particles.We havedhi = pidpimi+ miU ′ (qi) dqi,i = 1, .

. ., N, so that V1 and V2 are indeed transversal if only U ′ ̸= 0, i.e., if theexternal field is actually present.The form Q is equal toQ =NXi=1dqidpi +pim2i U ′ (qi) (dpi)2.It was shown in the papers cited above that the system is strictly monotone,provided thatU ′ (q) > 0andU ′′ (q) < 0,andm1 > m2 > · · · > mN.The symplectic map T that naturally arises in this system is the map “fromcollision to collision”.

Our dynamical system is a suspension of the map. So thatthe system is ergodic if and only if the map T is ergodic.

As usual, the actualcomputations are easier done in the full phase space of the flow.Singularity set S−corresponds to triple collisions: simultaneous collisions ofthree particles and the collision of two particles with the floor. Part of the firstsingularity set are not properly aligned.

The second set is. So the methods of thispaper apply only to the system of two particles.Let us show that indeed the triple collision of three particles produces the sin-gularity set which is not properly aligned.

We consider the manifold{(q p)|qqq }

70CARLANGELO LIVERANI, MACIEJ WOJTKOWSKIIts tangent subspace is described by the equationsdq1 = dq2 = dq3Its skew orthogonal complement is the two dimensional subspace given by equations(14.9)dq = 0,dp1 + dp2 + dp3 = 0,dpi = 0fori ≥4.Restricting the form Q to this plane we get(14.10)3Xi=1pim2i U ′ (dpi)2.We should assume that the particles emerge from collisions which means thatp1m1< p2m2< p3m3.But the momenta may, as well, be all negative which makes the quadratic form(14.10) negative definite. The actual characteristic line is obtained by intersectingthe plane (14.9) by the tangent to the constant energy manifold.

If all the momentaare negative, it is guaranteed to be outside of the sector. It is not hard to computethat the precise condition for the characteristic line to be contained in the sector isv1m1(v2 −v3)2 + v2m2(v3 −v1)2 + v3m3(v1 −v2)2 ≥0where vi = pimi , i ≥1 are the velocities.We close with the discussion of the system of two balls.

For clarity, we restrictourselves to the case of constant acceleration, U(q) = q. It was established in [W7],that also in this case all orbits are strictly monotone, if there are only two or threeballs and their masses decrease.

(For more than three balls technical problems arise,and it is an open problem to prove strict monotonicity almost everywhere. )Let us fix the value of the total energy of the system, H = 12.

In this manifold weconsider the two dimensional section M of the flow, corresponding to the bottomparticle emerging from the collision with the floor; the surface M is given by {H =12, q1 = 0, v1 ≥0}. The state of the system in M is completely described by thevelocities of the particles (v1, v2); and we use the velocities as coordinates in M.Hence, our phase space M is the domain bounded by the half-ellipsem1v21 + m2v22 ≤1,, v1 ≥0.Let us calculate the symplectic form in these coordinates.

We haveω = dp1 ∧dq1 + dp2 ∧dq2.On the surface of section Mdq1 ≡0anddq2 = −m1 v1dv1 −v2dv2.

ERGODICITY IN HAMILTONIAN SYSTEMS.71Hence, we getω = m1v1dv1 ∧dv2.The map T : M →M is defined by the first return of the flow to M. Oursymplectic box M is split into two symplectic boxes by S+, which is the arc ofthe ellipse {m1v21 + m2(v2 −2v1)2 = 1} contained in M. The symplectic box M+f ,above S+, contains all the initial states for which the bottom particle returns tothe floor without colliding with the top particle. The map T in M+f is linearT(v1, v2) = (v1, v2 −2v1).The symplectic box M+c , below S+, contains all the initial states for which thereis a collision of the two particles before the bottom particle returns to the floor.The map T in M+c is nonlinear and is best described in a coordinate system (h, z)whereh =12m1v21z =v2 −v1.The symplectic form ω = dh ∧dz.

(This coordinate system is derived from thecanonical system of coordinates in the full phase space furnished by the individualenergies and velocities of the particles. The exceptional role of these coordinates iswell documented in [W7], [CW].

)Note that both the energy of the bottom particle and the difference of velocitieschange only in collisions. Now T = F2 ◦F1, whereF1(h, z) = (−h −az2 + b, −z),a = m1m2(m1 −m2)(m1 + m2)2andb =m1m1 + m2,describes the collision of the two particles, andF2(h, z) = (h, z + c√h),c =r8m1,describes the collision of the bottom particle with the floor.To find the image symplectic boxes M−f and M−c we can use the reversibility ofour system.

Namely, if we put S(v1, v2) = (v1, −v2) then T ◦S = S ◦T −1, and soM−f = SM+f , M−c = SM+c .Our bundle of unstable sectors is constant in the coordinates (h, z) and equal tothe positive (and negative) quadrant; the form Q = dhdz. It is immediate that S+and S−= SS+ are properly aligned.We can now check that T is monotone in M+f and strictly monotone in M+c(both F1 and F2 are monotone).

Indeed, in the (h, z) coordinates we haveDF1 =−1−2az0−1and DF2 = 10c2√h1.Moreover the map T in M+f is equal in the coordinates (h, z) to F2.Since the collision of the two particles must eventually occur, we obtain strictmonotonicity of all nondegenerate orbits. Unboundedness of all nondegenerate or-bits follows from Proposition 6 9 So the Sinai Chernov Ansatz holds

72CARLANGELO LIVERANI, MACIEJ WOJTKOWSKITo check the noncontraction property, we observe that the standard Riemannianmetric in the coordinates (h, z) does not decrease on vectors from the sector, whenwe apply one of the above matrices.Finally, we are guaranteed that the coefficient σ(DT n) can be made arbitrarilylarge by increasing n, except for points which end up on the singularity set S+ in thefuture and the singularity set S−in the past. There are only countably many suchpoints in view of the proper alignment of singularity sets, and the Main Theoremapplies to all other points.

It follows that T is ergodic and consequently, by theresults of Katok and Strelcyn, it is a Bernoulli system.The case of variable acceleration (U ′′ < 0) can be treated in a similar fashion. Itis not possible to write down the formulas for the return map T but its derivativein the coordinatesδh = p1m1δp1δz =1m2U ′(q2)δp2 −1m1U ′(q1)δp1,was essentially calculated in [W8].

It is again a product of triangular matrices.Afterword.This paper was greatly improved thanks to many insightful comments and cor-rections by the anonymous referees of the paper.While we were writing this paper, several authors pursued similar goals. Thereare the papers by Chernov [Ch1], [Ch2], the new version of his old preprint byKatok, in collaboration with Burns [K2], by Markarian [M], by Vaienti [Va], andthe papers by Sim´anyi [S1], [S2].References[AW]R.L.

Adler, B. Weiss, Entropy is a complete metric invariant for automorphisms of thetorus, Proc. Natl.

Acad. Sci.

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74CARLANGELO LIVERANI, MACIEJ WOJTKOWSKILiverani Carlangelo, Mathematics Department, University of Rome II, Tor Ver-gata, Rome, Italy.E-mail address: liverani@vaxtvm.infn.itInstitute for Mathematical Sciences, SUNY at Stony Brook, Stony Brook, NY11794, USA.E-mail address: liverani@math.sunysb.eduMaciej Wojtkowki, Mathematics Department, University of Arizona, Tucson, AZ85721, USA.E-mail address: maciejw@math.arizona.edu

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