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Distribution of Periodic Points

arXiv:math/9301220v1 [math.DS] 23 Jan 1993Distribution of Periodic Pointsof Polynomial Diffeomorphisms of C2E. Bedford, M. Lyubich and J. Smillie§1.

Introduction.This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms ofC2: the polynomial automorphisms. This family of maps has been studied by a number ofauthors.

We refer to [BLS] for a general introduction to this class of dynamical systems.An interesting object from the point of view of potential theory is the equilibrium measureµ of the set K of points with bounded orbits. In [BLS] µ is also characterized dynamicallyas the unique measure of maximal entropy.

Thus µ is also an equilibrium measure fromthe point of view of the thermodynamical formalism. In the present paper we give anotherdynamical interpretation of µ as the limit distribution of the periodic points of f.Fix a polynomial automorphism f. A point p is periodic if f np = p for some n > 0,and the smallest positive n for which this equation holds is the period of p. We let Fixndenote the set of fixed points of f n, and Pern denote the set of points of period exactly n.Thus Fixn = S Perk, where the union is taken over all k which divide n. The map f hasa dynamical degree d which we assume to be larger than 1.

By [FM] f n has exactly dnfixed points counted with multiplicities. Since the multiplicity of each fixed point of f n ispositive we conclude that #Fixn ≤dn.A periodic point p is called a saddle point if the eigenvalues λs(p) and λu(p) of Df n(p)satisfy |λs(p)| < 1 < |λu(p)|.

We let SPern denote the saddle points with period exactlyn, so thatSPern ⊂Pern ⊂Fixn(1)#SPern ≤#Pern ≤#Fixn ≤dn. (2)Theorem 1.

If Pn denotes any of the three sets in (1), thenlimn→∞1dnXa∈Pnδa = µ.There seems to be a general tendency in many classes of dynamical systems for periodicpoints to be equidistributed with respect to the measure of maximal entropy. Indeed, thisis true for subshifts of finite type and their smooth counterparts, axiom A diffeomorphisms(Bowen [B1]).

This is also true for rational endomorphisms of the Riemann sphere [L]. Inthe special case of polynomial maps of the complex plane this result can also be derived byBrolin’s methods [Br] (see Sibony [Si] and Tortrat [T].) For polynomial diffeomorphisms ofC2 the equidistribution property was conjectured by N. Sibony.

In [BS1], this was provenwith the additional hypothesis f is hyperbolic (but without the a priori assumption thatthe periodic points are dense in the Julia set).Brolin uses potential theory to analyze polynomials of one variable, where the har-monic measure is the unique measure of maximal entropy. In the case of (non-polynomial)rational maps there is a unique measure of maximal entropy but it is not in general re-lated to the harmonic measure.

Lyubich’s proof [L] in the case of rational maps uses the1

balanced property of the unique measure of maximal entropy together with a ShadowingLemma. For polynomial automorphisms of C2 the analogue of the balanced property isthe product structure as described in [BLS].

In this paper we use the product structureand the Shadowing Lemma. This argument is of quite a general nature and we expect it tobe useful outside the holomorphic setting.

From our point of view the product structure ofthe measure of maximal entropy is the underlying reason for the equidistribution propertyin the known cases.According to Newhouse there exist polynomial automorphisms with infinitely manysink orbits. The next result shows that the majority of orbits are saddle orbits.Corollary 1.

Most periodic points are saddle points in the sense thatlimn→∞1dn #SPern = 1.A weaker asymptotic formulalim supn→∞(#SPern)1/n = dfollows from a theorem of Katok [K] and the entropy formula h(f) = log d ([FM] and [S]).The following corollary answers a question of [FM]:Corollary 2. f has points of all but finitely many periods.Given a point x, let us define the Lyapunov exponent at x asχ(x) =limn→+∞1n log ∥Df n(x)∥,provided the limit exists. For example, if p is a saddle point of period n, then χ(p) =1n log λu(p).

For any ergodic invariant measure ν the function χ(x) is constant ν almosteverywhere. This common value is called the Lyapunov exponent of f with respect to ν.The harmonic measure µ is ergodic, and we denote by Λ the Lyapunov exponent of f withrespect to µ.

We have an alternate description of Λ as:Λ =limn→+∞1nZlog ||Df n||µ.The following result allows us to compute Λ by averaging the Lyapunov exponents ofthe saddle points.Theorem 2. For any polynomial automorphism f we have:Λ = limn→∞1ndnXp∈SPernχ(p).The statement is equally true with the set SPern replaced by Fixn.

We note (from[BS3]) that Λ ≥log d, so the previous theorem gives a lower bound for the average exponentfor periodic points.2

For quadratic maps in one dimension the Lyapunov exponent with respect to theharmonic measure is closely related to the Green function of the Mandelbrot set.Inparticular the dynamical behavior of f is reflected in the behavior of this function. It isinteresting to investigate the relation between the behavior of Λ = Λ(f) and the dynamicsof f for automorphisms of C2.

Consider a holomorphic parameterization c 7→fc. It isshown in [BS3] that Λ(fc) is subharmonic for all values of c. The following result showsthat non-harmonicity of Λ(fc) corresponds to the creation of sinks (in the dissipative case)or Siegel balls (in the volume-preserving case) for nearby maps.

(A Siegel ball is theanalogue for volume-preserving maps of a Siegel disk, see [BS2], [FS]. )Theorem 3.

Consider a family of {fc} depending holomorpically on a parameter c in thedisk. Assume that c 7→Λ(fc) is not harmonic at c = c0.

If the maps fc are dissipative,then there is a sequence ci →c0 defined for i ≥N so that fci has a sink of period i. If themaps fc are volume preserving, then there is a sequence ci defined for i ≥N so that fcihas a Siegel ball of period i.It is shown in [BS3] that c 7→Λ(fc) is a harmonic function for the values of theparameter for which fc is hyperbolic.In this context hyperbolicity implies structuralstability which implies that the topological conjugacy class of the map is locally constant.The proof of Theorem 3 shows that structural stability directly implies the harmonicity ofΛ:Theorem 4.

If the maps fc are topologically conjugate to one another in a neighborhoodof c = c0, then the function c 7→Λ(fc) is harmonic at c0.§2. Lyapunov charts, Pesin boxes and the Shadowing Lemma.Topological bidisks.

Let us take a pair Ds and Du of standard disks {z : |z| ≤1}, andconsider the standard bidisk D = Du ×Ds. We refer to the sets Ds(b) = Ds ×{b} as stablecross sections and Du(a) = {a} × Du as unstable cross sections.

Also, the boundary ofD = Ds × Du will be partitioned into ∂Ds ∪∂Du, where we set ∂sD := Ds × ∂Du and∂uD := ∂Ds × Du.Let us define a topological bidisk B as a compact set homeomorphic to D togetherwith a homeomorphism h : B →D2. Then B inherits the structure of the stable/unstable(s/u) cross sections Bs(x) and Bu(x), and the partition of the boundary into ∂sB and∂uB, which is induced by this homeomorphism.

We note that the stable/unstable crosssections are not going to be stable or unstable manifolds, but rather an approximation tothem. If h is affine we can talk about the affine bidisk.Let us say that a two dimensional manifold Γ u-overflows a bidisk (B, h) if h(Γ∩B) ⊂D is the graph of a funcion φ : Du →Ds.

If additionally Γ ⊂B we also say that Γ isu-inscribed in B. By the slope of Γ in B we mean max ∥Dφ(z)∥.

Similarly we can definethe dual concept related to the stable direction.We will say that a topological bidisk B1 u-correctly intersects a topological bidisk B2 (or“the pair (B1, B2) intersects u-correctly”) if every unstable cross-section Bu1 (x) u-overflowsB2 and every stable cross section Bs2(x) s-overflows B1. If the pair (B1, B2) intersects u-correctly, then the intersection B1 ∩B2 becomes a topological bidisk, if we give it the3

unstable cross sections from B1 and the stable cross sections from B2. (In other words,the straightening homeomeorphism is h : x 7→(hs1(x), hu2(x)) where hi = (hsi, hui ) : Bi →Dare straightening homeomorphisms for Bi).

In the case when B1 ⊂B2 the u-overflowingproperty is equivalent to ∂sB1 ⊂∂sB2. Then we also say that B1 is u-inscribed in B2.Of course, we have the corresponding dual s-concepts.Adapted Finsler metric and Lyapunov charts.

We will use the exposition of Pughand Shub [PS] as the reference to the Pesin theory. Below we will adapt our presentationof the theory to our specific goals.Let us consider a holomorphic diffeomorphism f :C2 →C2.

Let µ be an invariant, ergodic, hyperbolic measure (the latter means that ithas non-zero characteristic exponents, χs < 0 < χu). Let R denote the set of Oseledetsregular points for f. For x ∈R there exists an invariant splitting of the tangent spaceTxC2 = Esx ⊕Eux into a contracting direction Esx and an expanding direction Eux whichdepend measurably on x.

Further, for any 0 < χ < min(|χs|, χu) there is a measurablefunction C(x) = Cχ(x) > 0 such thatDf n|Esx ≤C(x)e−nχandDf −n|Eux ≤C(x)e−nχ(2)for n = 1, 2, 3, . .

. Set QC = {x : C(x) ≤C}.

These sets exhaust a set of full µ measure asC →∞. Let us fix a big C and refer to QC as Q.A key construction of the Pesin theory is a measurable change of the metric whichmakes f uniformly hyperbolic.

Namely, for θ ∈(0, χ) and x ∈R, set|v|∗=Xn≥0enθ|Df −nv|forv ∈Eux,(3u)|v|∗=Xn≥0enθ|Df nv|forv ∈Esx. (3s)For arbitrary tangent vector v ∈Tx set |v|∗= max(|vs|∗, |vu|∗) where vs and vu areits stable and unstable components.

This metric is called adapted Finsler or Lyapunov.Observe that12|v| ≤|v|∗≤B(x)|v|(4)with a measurable function B(x) = C(x)/(1 −eθ−χ). Hence the adapted Finsler metric isequivalent to the Euclidean metric on the set Q.Let r(x) > 0 be a measurable function.

Then taking advantage of natural identificationof C2 with Tx(C2), we can consider a family of affine bidisksL(x) = {x + v : |v|∗< r(x)}. (5)By the inner size of L(x) we mean min{|vs|, |vu| : x+v ∈L(x)}.

The Pesin theory providesus with a choice of the “size-function” r(x) with the following properties. (L1) The inner size of Lyapunov charts is bounded away from 0 on the set Q.

Indeed, by(4) on this set the adapted Finsler metric is equivalent to the Euclidean metric. But thenr(x) stays away from 0 as one can see from its explicit definition in [PS], p.13.4

For a complex one-dimensional manifold Γ ⊂L(x), let us define its cut-offiterate,fxΓ, as fΓ ∩L(fx) . (L2) There is a measurable function κ(x) > 0 with the following property.

Denote by Guxthe family of complex one-dimensional manifolds which are u-inscribed into L(x) and havethe slope less than κ(x). Then the operation of cut-offiteration, fx, transforms Gux intoGufx.

In particular, the topological bidisk fL(x) correctly intersects L(x) (where the bidiskstructure on fL(x) comes from L(x)).This allows us to repeat the cut-offprocess for iterates of f.Let f nx denotes thecomposition of the cut-offiterates at x, fx, . .

., f n−1x. The hyperbolicity of µ implies:(L3) For any two manifolds Γi ∈Gux, their cut-offiterates are getting close exponentiallyfast:C1−dist(f nx Γ1, f nx Γ2) ≤A(x)e−θnwith a measurable A(x).

Moreover, A(x) is bounded on Q. (L4) Let p = f −nx.Then for any manifold Γ ∈Gup , and for any two points y, z ∈f np Gamma,dist(f −npy, f −npz) ≤A(x)dist(y, z)e−θnwith A(x) as above.

(In the stable direction we of course have the corresponding properties with the re-versed time.) Such a family of affine bidisks L(x) will be called a family of Lyapunovcharts.Stable and unstable manifolds.Let us consider the push-forward of a Lyapunovchart and trim it down by intersecting with the image chart.Since by (L2) the pair(fL(x), L(fx)) intersects u-correctly, the set Lu1(fx) = L(fx) ∩fL(x) a topological bidisku-inscribed into L(fx).

Similarly, Ls1(x) = L(x) ∩f −1L(fx) is a topological bidisk s-inscribed into L(x), and f : Ls1(x) →Lu1(fx) is a bidisk diffeomorphism.Property (L2) allows us continue inductively by noting that the pair (L(f nx), fLun−1)intersects u-correctly, and we can consider topological bidisksLun(f nx) = L(f nx) ∩fLun−1(f n−1(x) =f n{x : x ∈L(x), fx ∈L(fx), ..., f nx ∈L(f nx)},(6a)andLsn(x) = f −nLun(f nx) = {x : x ∈L(x), fx ∈L(fx), ..., f nx ∈L(f nx)}. (6b)Observe that Lsn(x) can also be defined as the connected component of L(x) ∩f −nL(f nx)containing x.Furthemore, Lsn(x) is s-inscribed into L(x), and by the dual property to (L3) the sizesof Lsn(x) in the stable direction shrink down to zero.

Hence the intersectionW sloc(x) =\n≥0Lsn(x) = {x : f nx ∈L(f nx), n = 0, 1, ...}(7)5

is a manifold of the family Gux s-inscribed into L(x).This is exactly the local stablemanifold through x. By the dual to (L4), for any y, z ∈W s(x)dist(f ny, f nz) ≤A(x)e−θnwith a measurable function A(x).Reversing the time we obtain local unstable manifolds as well.Pesin boxes.

Recall that Q = QC ⊂R denotes the set on which C(x) ≤C in (2). Thisset is compact, and it may be shown that Q ∋x 7→Es/uxis continuous.

Further, W s(x)and W u(x) intersect transverally at x, and W s/uloc (x) varies continuously with x ∈Q. Thusif x, y ∈Q are sufficiently close to each other, then there is a unique point of intersection[x, y] := W sloc(x) ∩W uloc(y).

It follows that [ , ] : Q × Q →C2 is defined and continuousnear the diagonal. Given a small α > 0, let us setQ = QC,α = {[x, y] : x, y ∈Q, dist(x, y) < α}.This is a closed subset of R on which C(x) < C/2 in (2) (provided α is sufficiently small),and which is locally closed with respect to [ , ]-operation.

By (L1) the inner size of theLyapunov charts L(x) stays away from 0 for x ∈Q. Let ρ > 0 be a lower bound for thissize.We say that a set P has product structure if P is closed under [ , ].

If in addition P is(topologically) closed and has positive measure, then it will be called a Pesin box. Let uscover Q with countably many closed subsets Xi of positive measure with diameter < α.Taking the [ , ]-closure of Xi, we obtain a covering of Q with a countably many Pesin boxesPi ⊂Q.

Since the µ almost whole space can be exhausted by the sets QC, we concludethat it can be covered by countably many Pesin boxes of arbitrarily small diameter. Ournext goal is to make these boxes disjoint (at expense of losing a set of small measure).Lemma 1.

For any ǫ > 0 and η > 0 there is a finite family of disjoint Pesin boxes Pi suchthat diam(Pi) < η and µ(S Pi) > 1 −ǫ.Proof. As we can cover a set of full measure by countably many Pesin boxes, we can covera set X of measure 1 −ǫ by finitely many boxes.

Now let us make them disjoint by thesame procedure as used by Bowen (cf. [B2], Lemma 3.13).

Namely, if two boxes P1 andP2 intersect, subdivide each of them into four sets P ji ⊂Pi with product structure in thefollowing way:P 11 = {x ∈P1 : W s(x) ∩P2 ̸= ∅, W u(x) ∩P2 ̸= ∅},P 21 = {x ∈P1 : W s(x) ∩P2 = ∅, W u(x) ∩P2 ̸= ∅},P 31 = {x ∈P1 : W s(x) ∩P2 ̸= ∅, W u(x) ∩P2 = ∅},P 41 = {x ∈P1 : W s(x) ∩P2 = ∅, W u(x) ∩P2 = ∅}.Repeating this procedure, we can cover the set X by a finite number of disjoint setsQi, i = 1, . .

., N, with product structure. However, Qi need not be closed.

To restore thisproperty, let us consider closed sets Kj ⊂Qj such that µ(Qj −Kj) < ǫ/N. Completingthese sets with respect to the product structure, we obtain a suitable family of Pesin boxes.6

A common chart. Our goal is to have a common Lyapunov chart for all points returningto a Pesin box.Observe that the family of Lyapunov charts can be reduced in size by a factor t ≤1.Then the manifolds of the family Gux (correspondingly Gsx) become almost parallel withinthe scaled charts.

In particular, the stable cross sections of the bidisks Lsn(x), as well asthe local stable manifolds truncated by the scaled family of charts become almost parallel.Let ρ be the lower bound of the size of the Lyapunov charts L(x), x ∈Q. Let us takea Pesin box P of size η < ρ/8.

Let a ∈P, and let Bs(a, r) ⊂Esa, Bu(a, r) ⊂Eua denote theEuclidean disks of radius r centered at a in the corresponding subspaces. Let us consideran affine bidisk B = Bs(a, ρ/2) ×Bu(a, ρ/2).

Since the stable/unstable directions throughthe points x ∈P are almost parallel, we have the inclusions:P ⊂B ⊂\x∈PL(x).For x ∈P ∩f −nP let T = Bsn,x be the component of B ∩f −nB containing x. Letalso y = f nx, R = f nT = Bu−n,y.Lemma 2. The set T/(respectively R) is a topological bidisk which is s/u-correctly in-scribed in B.The s/u-cross sections of T/R belong to the families Gs/u respectively.Moreover, T s-correctly intersects R.Proof.

For z ∈T let T u(z) = T ∩(z+Eua), and similarly for ζ ∈R let Rs(ζ) = R∩(ζ +Esa).The first claim will follow from the following dual statements:(i) f nx T u(z) is a topological disk u-correctly inscribed into B;(ii) f −nyRs(ζ) is a topological disk which is s-correctly inscribed into B.Let us prove (i). Let us consider the Lyapunov chart L(x) ⊃B.

Then the disk K =L(x) ∩(z + Eua) belongs to the family of graphs Gux. Hence its cut-offiterate f nx K ∈Gu(y)overflows L(y).Since it is almost parallel to Eua, f nx T u(z) = f nx K ∩B is u-correctlyinscribed into B.

This proves the first two claims.The last one now follows from the transversality of the families Gu and Gs.Set W u/sB(x) = W u/sloc (x) ∩B.Lemma 3. Under the above circumstances we haveW sB(x) ⊂T(8)andW uloc(x) ∩T ⊂f −nW uloc(y).(9)Proof.

Since f nW sB(x) has an exponentially small size, it is contained in B. So W sB(x) ⊂B ∩f −nB.

Since W sB(x) is connected, (8) follows.In order to get (9), observe that T ⊂Lsn(x). But we know by the construction of thestable/unstable manifolds that W uloc(x) ∩Lsn(x) = f −nW uloc(y).7

Observe that by definition the sets Bun,x are either disjoint or coincide. So, for each nwe can consider the following equivalence relation on the set P ∩f −nP of returning points:x ∼y if Bun,x = Bun,y.Lemma 4.

The equivalence classes have a product structure. Moreover, if x, z ∈P ∩f −nPare equivalent then f n[x, z] = [f nx, f nz].Proof.

Denote q = [x, z]. Clearly, f nq ∈W sloc(f nx).

Further, by (8), q ∈T. Hence by (9)f nq ∈W uloc(f nz), and the property f n[x, z] = [f nx, f nz] follows.

Moreover, it follows thatq ∈T ∩P ∩f −nP, so that it is equivalent to x, z.The Shadowing Lemma. For any x ∈P ∩f −nP, y = f nx, there is a unique saddlepoint α ∈Bsn,x ∩Bu−n,y of period n. Moreover, dist(α, P) ≤Ce−nθ.Proof.Let us consider the family Gu of manifolds u-inscribed into B with the slope≤κ ≤infx∈P κ(x) (with respect to the decomposition Eua ⊕Esa) .

Take a returning pointx ∈P ∩f −nP, and let y = f nx. Then we can consider the cut-offiterate Φu : Γ 7→f nx Γ∩B.Since its image ΦuΓ is close to the unstable manifold W uB(y), it is almost parallel to Eua.It follows that for sufficiently big n, Φu maps Gu into itself.

Moreover, this transformationis contracting according to (L3).Hence there is a unique Φu-invariant manifold Gu ∈Gu. This manifold is u-inscribedinto Bu−n,y, and can be characterized as the set of all points non-escaping Bu−n,y underbackward iterates of f n. Similarly, we can define a transformation Φs corresponding tothe return of y back to P under f −n, and find a unique Φs-invariant manifold Γs ∈Gs,Γs ⊂Bsn,x.

This manifolds intersect transversally at a unique point α which is the desiredperiodic point.There are no other periodic ponts of period n in Bux,n. Indeed, all points escape Bux,neither under forward or backward iterates of f n. Finally since the bidisks Bsn,x and Bu−n,yare exponentially thin, the point α is exponentially close to [x, y] ∈P.§3.

Proofs of the theorems.According to the Shadowing Lemma, to each returning point x ∈P ∩f −nP we can assigna periodic point α = α(x) ∈Bsn,x ∩Bun,x of period n. Given such an α, setT(α) = {x ∈P ∩f −nP : α(x) = α} = P ∩f −nP ∩Bsn,α,where Bsn,α = Bsn,x is the component of B ∩f −nB containing x. The sets T(α) actuallycoincide with the equivalence classes introduced above.In what follows we assume that µ is the measure of maximal entropy of the polynomialdiffeomorphism f. Since P has the product structure, it is homeomorphic to P s × P u,were P s and P u are cross sections.

It was proved in [BLS] that µ is a product measurewith respect to this topological structure, i.e. µP = µ−|P s ⊗µ+|P u.Lemma 5. µ(T(α)) ≤µ(P)d−n.Proof.

By Lemma 4, T is naturally homeomorphic to T s × T u, where T s = T s(x) andT u = T u(x) be the cross sections of T = T(α) through the point x ∈T(α) . By theproduct property of µ,µ(T) = µ−(T s)µ+(T u).

(10)8

Since T s ⊂P s(x),µ−(T s) ≤µ−(P s(x)) = µ−(P s). (11)By (9), f nT u ⊂W uloc(f nx)∩P = P u(f nx).

By the transformation rule for µ+ we concludethatµ+(T u) ≤d−nµ+(P u). (12)Now the result follows from (10)–(12).In the following proof we let SFixn be the set of saddle periodic points of perioddividing n.Lemma 6.

For any ǫ > 0, there exists C > 0 depending on the Pesin box P such thatlim infn→∞1dn #{α ∈SPern : dist(α, P) < Ce−nθ} ≥µ(P).Proof.In our work above, we have assigned to any returning point x ∈P ∩f −nP aperiodic point α(x) ∈SFixn exponentially close to P. Let An denote the set of periodicpoints obtained in this way. Hence, by the Lemma 5,µ(P)d−n#An ≥Xα∈Anµ(T(α)) = µ(P ∩f −nP)and by the mixing property of µ [BS],#Andn≥µ(P ∩f −nP)µ(P)→µ(P)as n →∞.Thus we have shown thatlim infn→∞1dn #{α ∈SFixn : dist(α, P) < Ce−nθ} ≥µ(P).It remains to show that SFixn can be replaced by SPern.

For this, we note that {k :k|n, k < n} ⊂{k ≤n/2}. Since #Fixn ≤dn, it follows that#Fixn −#Pern ≤n2 dn/2.Since this is o(dn), we see that almost all points of SFixn have period precisely n.Proof of Theorem 1.

Let νn = d−n Pa∈Pern δa. Consider any limit measure ν = lim νn(i)of a subsequence of these measures.

It follows from Lemma 6 that for any Pesin box,ν(P) ≥µ(P). Now let G be any open set.

Consider a compact set K ⊂G such thatµ(G −K) < ǫ. Let η = dist(K, ∂G).

By Lemma 1 we can cover all but a set of measureǫ by disjoint Pesin boxes Pi of size less than η. Let I denote the set of those Pesin boxeswhich intersect K. Thenν(G) ≥Xi∈Iν(Pi) ≥Xi∈Iµ(Pi) ≥µ(K) −ǫ ≥µ(G) −2ǫ.Hence ν ≥µ, and as the both measures are normalized, ν = µ.9

Now we turn to the Lyapunov exponent Λ(f). For x ∈R we havelimn→∞1n log ||Df n(x)|| = limn→∞1n logDf(x)|Eux = limn→∞1nn−1Xj=0logDf(f jx)|Eufj x .Thus with the notationψ(x) = logDf(x)|Eux(13)we haveΛ(f) =Zψ(x)µ(x).

(14)Lemma 7. Let P denote a Pesin box, and let ǫ > 0 be given.Then there exists Nsufficiently large that the periodic saddle point α(x) generated by a returning point x ∈P ∩f −nP with n ≥N satisfies dist(Euα(x), Eu(x)) < ǫ.Proof.

By the construction above, the unstable manifold W u(α(x)) goes across the topo-logical bidisk B, and W u(α(x)) ⊂Bun. Since the stable cross section of Bun is exponentiallysmall in n, we have the desired estimate on the distance of the tangent spaces.Proof of Theorem 2.

The measures d−n Pa∈Pn δa converge to µ. To evaluate the integralin (14), we may work on a countable, disjoint family of Pesin boxes Pj, with the propertythat Df and the tangent spaces Eu, and thus the expression ψ in (13), vary by no morethan ǫ on each Pj.

By Lemma 7, as n →∞, the tangent spaces Euα(a) converge to withinǫ of the tangent spaces Eu given by the Oseledec Theorem. Thus ψ(α(x)) is within ǫ ofthe value of ψ on Pj.

It follows that the expression in Theorem 2 will be within ǫ of theintegral (14).Corollary 3. If Pn denotes any of the sets defined in (1), thenΛ(f) = limn→∞d−n Xp∈Pnψ(p).Proof.

Since K is compact, it follows that ||Dfx|| ≤M for all x ∈K. And since Pn ⊂K,it follows that that ψ(p) ≤M.

Further, the Jacobian of f is a constant a, so even at asink orbit of order n, we have ψ(p) ≥logp|a|. Thus we may use any of the three sets Pnin (1) in defining the limit.Proof of Theorem 3.The conclusion of the theorem is a consequence of the followingstatement: For any neighborhood U of c there is a number NU so that for each n ≥NUthere is a c ∈U so that fc has a sink of period n. We will show that if the previous assertiondoes not hold then Λ(fc) is harmonic.

Thus assume that there exists a neighborhood Uand an infinite sequence ni such that fc has no sink of period ni for any c ∈U and any i.Fix an n = ni. Let V = {(c, p) ∈U ×C2 : f nc (p) = p}.

The set V is a one dimensionalanalytic variety with a projection onto U. We can remove from V a discrete set of points{(c1, p1), (c2, p2), .

. .} corresponding to “bifurcations” where either V is singular or theprojection onto the first coordinate has a singularity.

Let V ′ = V −{(c1, p1), (c2, p2), . .

. }10

and let U ′ = U −{c1, c2, . .

. }.

On V ′ the period of a periodic point is constant on eachcomponent. Remove components for which the period is less than n. Call the resultingset V ′′.Assume first that we are in the dissipative case, and there are no sinks of period n.Since there is no sink, the modulus of the larger eigenvalue must be at least as large as 1,and since it is dissipative, the modulus of the smaller eigenvalue must be no larger thanp|a| < 1.

Thus at each point (c, p) in V ′ there is a unique largest eigenvalue for Df nc (p)call it λ+(c, p). The function λ+ is a continuously chosen root of the characteristic equationof Df nc (p) so it is holomorphic.

In particular the function log |λ+(c, p)| is harmonic on V ′and the functionU ′ ∋c 7→1dnXp∈Pn1n log |λ+(c, p)| = 1dnXp∈Pnψ(p)is harmonic on U ′. This function extends to a harmonic function Λn on U.

Now Theorem 2implies that Λn converges pointwise to Λ. Since each function Λn is harmonic we concludethat Λ is harmonic.

This proves our assertion in the dissipative case.Assume now that we are in the volume preserving case. Note that since the functionc 7→det fc is holomorphic and of constant norm it is actually constant.

Assume that thereare no Siegel balls of period n. By the two-dimensional version of the Siegel linearizationtheorem (see [Z]), this implies that there are no elliptic periodic points of period n forwhich the eigenvalues satisfy certain Diophantine conditions. In particular if at any ellipticperiodic point the eigenvalues are not constant as a function of c then we can vary theparameter so that λ1 varies through an interval on the unit circle.This implies thatthere is some parameter value at which the Diophantine conditions are verified for theeigenvalues λ1 and λ2 = const/λ1.

We conclude that at any elliptic point (c, p) of period nthe eigenvalues are locally independent of c. This implies that the eigenvalues are constanton the component of V ′ which contains the point (c, p).As before we remove from V ′ the components on which the period is less than n. Inaddition we remove components on which both eigenvalues are constant and have modulus1. The remaining variety V ′′ consists of saddles.

Arguing as before we see that Λn isharmonic hence Λ is harmonic. This completes the proof of the theorem.Proof of Theorem 4.

Let J∗denote the support of µ, which is the closure of the saddlepoints (cf. [BS3]).

If the maps fc|J∗are topologically conjugate, then the conjugacy pre-serves the set of periodic points, so that each periodic point p is part of a selection c 7→p(c).As in the proof of Theorem 3, each function Λn is harmonic; hence Λ is harmonic.References[BLS] E. Bedford, M. Lyubich, and J. Smillie, Polynomial diffeomorphisms of C2. IV: Themeasure of maximal entropy and laminar currents.

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), Lecture Notes in Math., vol. 597,Springer-Verlag, New York, 855–866 (1977)Indiana University, Bloomington, IN 47405SUNY at Stony Brook, Stony Brook, NY 11794Cornell University, Ithaca, NY 1485312

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