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THE EXISTENCE OF σ−FINITE INVARIANT MEASURES,

arXiv:math/9201300v1 [math.DS] 15 Jan 1992THE EXISTENCE OF σ−FINITE INVARIANT MEASURES,APPLICATIONS TO REAL1-DIMENSIONAL DYNAMICSMarco MartensIMPA, Estrada Dona Castorina 110, Rio de Janeiro, BrasilAbstract.A general construction for σ−finite absolutely continuous invariantmeasure will be presented. It will be shown that the local bounded distortion of theRadon-Nykodym derivatives of fn∗(λ) will imply the existence of a σ−finite invariantmeasure for the map f which is absolutely continuous with respect to λ, a measureon the phase space describing the sets of measure zero.Furthermore we will discuss sufficient conditions for the existence of σ−finite invari-ant absolutely continuous measures for real 1-dimensional dynamical systems.1.

IntroductionThe statistical study of a dynamical system begins with the question whether ornot the system has an absolutely continuous invariant measure, finite or σ−finite.In 1947 Halmos gave a characterization of the (bijective) dynamical systems whichhave a σ−finite absolutely continuous invariant measure, see [Ha]. During this timethere was some hope that every dynamical system has σ−finite invariant measures.Unfortunately this turned out not to be true.Ornstein gave an example of apiecewise linear bijective map on the interval not having such a measure ([O]).Here we will give a characterization of the ergodic conservative dynamical systemson locally compact spaces having σ−finite absolutely continuous invariant measures.The origin of the characterization presented here can be found in the theory of real1-dimensional dynamics and the theory of Markov processes.

In [H] Harris used1

2MARCO MARTENSlimits of ratios of long term transition probabilities to construct infinite stationarystates for Markov processes on countable state spaces. In section 2 we will use thisidea for constructing σ−finite invariant measures.

The construction gives rise tothe existence theorem A. The distortion of a measure, used in the theorem, willbe defined precisely in section 2.

Furthermore remember that a map f : X →Xis ergodic conservative with respect to a measure λ on X if every set of positivemeasure is hit by the orbits of λ−almost all points (see [P]).Theorem A. Let λ be a Borel probability measure on the σ−compact space X. Theergodic conservative map f : X →X has a σ−finite invariant measure absolutelycontinuous with respect to λ if the Radon-Nykodym derivatives of f n∗λ have locallybounded distortion.As in the general construction of invariant probability measures the construction isdone by pushing forward some initial measure and then considering limits of thesepush-forwards.

It turns out that the procedure only gives rise to σ−finite absolutelycontinuous invariant measures if the initial measures are of some special type. Insection 3 we will construct the initial measures.In section 4 we will study the existence of σ−finite invariant measures for real1-dimensional differentiable dynamics.

As we know from [J] there is no generalexistence theorem for absolutely continuous invariant probability measures: thereexist conservative quadratic maps on the interval not having absolutely continuousinvariant probability measures. Even the existence question for σ−finite absolutelycontinuous invariant measures can not be answered in general.

Katznelson ([Ka])constructed diffeomorphisms of the circle not having σ−finite absolutely continuousinvariant measures.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS3Applying the developed theory we can formulate sufficient conditions implyingthe existence of σ−finite absolutely continuous invariant measures for real 1-dimensional differentiable dynamics.In [HKe2] Hofbauer and Keller gave an existence theorem for some type of conser-vative unimodal maps. Now this theorem can be generalized to multimodal andalso dissipative maps:Theorem B.

Let f be a C3 map on the interval (or the circle) satisfying1) f has only finitely many critical points, points where the derivative vanishes, andthe Schwarzian derivative is everywhere negative except in the critical points;2) there exists a dense orbit;3) the orbits of the critical points stay in a closed invariant set of Lebesgue measurezero.Then f has a σ−finite absolutely continuous invariant measure.In [HKe1] quadratic maps are shown to exist having very strange Bowen-Ruellemeasures. The same techniques can be used to show that there exists a quadraticmap whose critical orbit is in a Cantor set but which doesn’t have an absolutelycontinuous invariant probability measure.

This means that in general the invariantmeasures of Theorem B are really σ−finite.Furthermore we obtain a σ−finiteFolklore theorem.All the results concerns maps whose critical orbits stay in some closed invariantset of Lebesgue measure zero. In the other case, some critical orbits are dense, westate theConjecture.

There exist conservative quadratic maps on the interval not havingσ−finite absolutely continuous invariant measures.

4MARCO MARTENSIn the appendix we will give a short proof of the Chacon-Ornstein Theorem, themain theorem in σ−finite ergodic theory.Remember the following notation: if g : X →X is a Borel measurable function andµ a Borel measure on X then g∗µ is the measure defined by g∗µ(A) = µ(g−1(A)).2. The construction of σ−finite absolutely continuous invariant measuresIn this section we are going to construct σ−finite absolutely continuous invariantmeasures.

Let λ be the Borel measure describing the sets of measure zero. A start-ing point for constructing absolutely continuous invariant measures is consideringlimits of the Birkhoffsums { 1nΣn−1i=0 f i∗λ}n≥0.

Indeed, if for all sets A with λ(A) > 0the measures f n∗λ(A) stay away from 0 we can construct an absolutely continu-ous invariant probability measure, simply by taking a converging subsequence ofBirkhoffsums. In case our system doesn’t have an invariant probability measure,that is there exist closed sets A with f n∗λ(A) converging to zero, we have to considerother limits.In [H] Harris used limits of ratios of long term transition probabilities to constructstationary states for Markov processes.

In our construction we will choose some setI0 of positive measure and consider limits of the normalized sequencePn−1i=0 f i∗λPn−1i=0 f i∗λ(I0).Our construction is strongly related to the Chacon-Ornstein Theorem or, whichis in some sense equivalent to it, the Doeblin-Ratio-Limit Theorem for Markovprocesses (see resp. [K],[P] and [F]).First let us remember some general notions.

Let X be a σ−compact topologicalspace, it can be written as a countable union of compact sets, and B(X) be the

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS5set of Borel measures on X. The set Bσ(X) consists of all measures µ ∈B(X) forwhich there exists a collection {Xn|n ∈N} of pairwise disjoint measurable sets inX such thatµ(Xn) is finite for all n ∈N;µ(X −([n∈NXn)) = 0.The measures in Bσ(X) are called σ−finite measures on X.For discussing the notion of absolute continuity we fix a measure λ ∈Bσ(X) deter-mining the null sets, the sets which have to have measure zero.

We may assumewithout restricting generality that λ is a probability measure. Remember a mea-sure µ ∈Bσ(X) is called absolutely continuous with respect to λ, λ ≫µ, iffthesets of λ−measure zero also have µ−measure zero.

These absolutely continuousmeasures can be expressed by an integral: for all µ with λ ≫µ there exists anessentially unique integrable non-negative function ρ such that µ(A) =RA ρdλ forall measurable sets A ⊂X. This function is called the Radon-Nykodym derivative(or just derivative) of µ with respect to λ.The main objects to be studied here will be absolutely continuous measures whoseRadon-Nykodym derivatives has locally bounded distortion: we say that the deriv-ative of µ ∈Bσ(X), λ ≫µ, on I ⊂X with µ(I) finite, has distortion bounded byK ifffor all measurable sets A ⊂I1Kλ(A)λ(I) ≤µ(A)µ(I) ≤K λ(A)λ(I) .Observe that the constant K can be taken to be equal to 1 iffµ has constantderivative on I.

If you consider a Radon-Nykodym derivative as an object deforming

6MARCO MARTENSthe original measure λ this notion of distortion is related to the concept of distortionfor differentiable maps of the interval (see [GuJ], [LB], [MMS] or [Sw]).In this section we fix a probability measure λ ∈Bσ(X) and a measurable mapf : X →X on the σ−compact space X. Assume that λ is quasi-invariant for f:λ ≫f∗λ.The next step is to give some definitions which enables us to deal with σ−finiteabsolutely continuous invariant measures, shortly acim (if we want to emphasizethat some acim is a probability measure we call it acip: absolutely continuousinvariant probability measure).A λ−partition G of X is a countable collection of pairwise disjoint Borel sets of X,say G = {In|n ∈N}, such that for all n ∈N1) In is σ−compact;2) 0 < λ(In) < ∞;3) λ(X −(SI∈G I)) = 0.If the λ−partition G of X has the additional property4) for all pairs I1, I2 ∈G there exists n ≥0 such that λ(f −n(I1) ∩I2) > 0we will say f is G−irreducible.The role of the λ−partition is the following: its elements will turn out to be sets offinite measure for the acim of the G−irreducible map f.In the sequel we fix a λ−partition G and assume that f : X →X is G−irreducible.To define the measure spaces M(G, f), Ms(G, f) and M∞(G, f), in which theconstruction of the acims will take place, we need some properties of measuresµ ∈Bσ(X).

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS7Let I ∈G and K > 0:m1(I, K): µ(I) ∈[ 1K , K];m2(I, K): For all n ≥0 the µ−measure of f −n(I) is finite and positive and thederivatives of the measures f n∗µ on I with respect to λ has distortionsbounded by K. This property states that measures f n∗µ have locallyuniformly bounded distortions.m3(I): supn≥0 µ(f −n(I)) < ∞;m4(I):∞Xn=0µ(f −n(I)) = ∞.The measure spaces we need areM(G, f) = {µ ∈Bσ(X)| for all I ∈G there is a K > 0 with m1(I, K) and m2(I, K) };Ms(G, f) = {µ ∈M(G, f)| m3(I) holds for all I ∈G};M∞(G, f) = {µ ∈Ms(G, f)| m4(I0) holds for some I0 ∈G}.Because we are only considering the fixed λ−partition G and the fixed G−irreduciblemap f : X →X we will use the short names M, Ms and M∞. Furthermore observethat M∞⊂Ms ⊂M and that M only contains measures which are absolutelycontinuous with respect to λ.The construction of the acims will be by taking converging subsequences in M.Hence we have to describe which kind of convergence we are going to use.Asequence µn in M = M(G, f) is said to converge to µ ∈Bσ(X) ifffor all I ∈G andfor every compact A ⊂I we have weak convergence of µn|A →µ|A.For two reasons the space M is not compact: the measures are not assumed tobe bounded and the underlying space is not compact; mass can disappear to theboundary of the space.

Let us try to describe some compact subsets of M. A

8MARCO MARTENScollection A ⊂M is called uniform ifffor all I ∈G there exists K(I) > 0 such thatm1(I, K(I)) and m2(I, K(I)) hold for all µ ∈A.The campactness in the weak topology of the set of probability measures on acompact space implies easilyLemma 2.1. Uniform collections in M(G, f) have compact closures in M(G, f).Now we are going to use the above measure spaces to construct the acims.

Letµ ∈M and define the following measures in Bσ(X):Snµ =n−1Xi=0f i∗µ;Qnµ =SnµSnµ(I0).for all n ≥0. I0 is a fixed element of G. If µ ∈M∞then we choose I0 such thatm4(I0) holds.The lemma which will assure the existence of limits isLemma 2.2.

Let µ ∈Ms(G, f) then the collectionAµ = {Qnµ|n ≥0}is uniform.Before proving this lemma we are going to use lemma 2.1 and 2.2 to define thefollowing limit set in M. Let µ ∈Ms thenω(µ) ⊂M(G, f)is the set of all limits of the sequence Aµ. We will look for acims in these setsω(µ).

For the moment we know already that it only contains measures which areequivalent to λ.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS9The next lemma shows how the measures of backward orbits of two elements inG are related. It will be used at several places; it serves for gluing together theinformation given by the local boundedness of the distortions.Lemma 2.3.

Let µ ∈M(G, f). Then for every pair I1, I2 ∈G there exist ǫ > 0and n0 ≥0 such thatµ(f −n−n0(I1)) ≥ǫµ(f −n(I2))for n ≥0.proof.

Because f is G−irreducible there exists n0 ≥0 such that λ(f −n0(I1)∩I2) > 0.Hence for µ ∈M(G, f) we get for all n ≥0µ(f −n−n0(I1)) ≥µ(f −n(f −n0(I1) ∩I2))µ(f −n(I2))µ(f −n(I2))≥1Kλ(f −n0(I1) ∩I2)λ(I2)µ(f −n(I2))= ǫµ(f −n(I2)),where K > 0 is such that m2(I2, K) holds for µ.□This lemma is the place where we use the G-irreducibility of f.Lemma 2.4. Let µ ∈Ms(G, f).

For every pair I1, I2 ∈G there exists K < ∞such that1K ≤Snµ(I1)Snµ(I2) ≤Kfor all n ≥0.proof. Let ǫ, n0 be given by lemma 2.3.

For n ≤n0 we have some bound. Let

10MARCO MARTENSn > n0. ThenSnµ(I1)Snµ(I2) ≥Pn−n0−1i=0µ(f −i−n0(I1))Pn−1i=n−n0 µ(f −i(I2)) + Pn−n0−1i=0µ(f −i(I2))≥ǫPn−n0−1i=0µ(f −i(I2))n0 supi≥0 µ(f −i(I2)) + Pn−n0−1i=0µ(f −i(I2))≥ǫµ(I2)n0 supi≥0 µ(f −i(I2)) + µ(I2)which is a finite positive number.

Remark that we used in the last step that thefunction x →xa+x is increasing.By interchanging the role of I1 and I2 we also get an upper bound.□proof of lemma 2.2. Fix I ∈G.m1.

Let K be the number given by lemma 2.4 applied to I and I0. We get directlyfrom lemma 2.4 that m1(I, K) holds for all Qnµ, n ≥0.m2.

Let K > 0 be such that m2(I, K) holds for µ. Fix m ≥0 and let A ⊂I bemeasurable.

Then we get for n ≥0Qmµ(f −n(A))Qmµ(f −n(I)) = Smµ(f −n(A))Smµ(f −n(I))=Pm+n−1i=nµ(f −i(A))Pm+n−1i=nµ(f −i(I)).Because m2(I, K) holds for µ we easily get that this last number is in the interval[ 1Kλ(A)λ(I) , K λ(A)λ(I) ]. This proves the lemma.□The following lemma tells under which conditions on µ, ω(µ) will contain invariantmeasures.Lemma 2.5.

Let µ ∈M∞(G, f). Then ω(µ) contains only invariant measures.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS11proof. Let ν ∈ω(µ), say ν = lim Qnµ (lim means : the limit of a certain convergingsubsequence).

Because λ(X −∪G) = ν(X −∪G) = 0 and λ is quasi-invariant for fwe only have to consider A ⊂I, I ∈G. Let A ⊂I be compact.

Thenν(f −1(A)) = lim Snµ(f −1(A))Snµ(I0)= limPn−1i=0 µ(f −i(A)) −µ(A) + µ(f −n(A))Pn−1i=0 µ(f −n(I0))= ν(A) + lim µ(f −n(A)) −µ(A)Snµ(I0).Now we use that m3(I) and m4(I) hold for µ and we getν(f −1(A)) = ν(A).The measure is invariant.□Proposition 2.6. Let λ ∈Bσ(X) be a Borel measure on the σ−compact spaceX and f : X →X a measurable map.

The map f has a λ−equivalent σ−finiteinvariant measure if it is G−irreducible for some λ−partition G of X withM∞(G, f) ̸= ∅.The elements of G will be pieces of X with bounded measure.proof. If µ ∈M∞(G, f) then from lemma 2.2, 2.1 we get ω(µ) ̸= ∅.

FurthermoreLemma 2.5 tells us that ω(µ) only contains invariant measures.□In fact we also want the reverse statement: f has an acim iffit is G−irreducible forsome λ−partition G of X with M∞(G, f) ̸= ∅.It is not hard to prove that for every map f : X →X which has an acim there existsa λ−partition G such that M∞(G, f) ̸= ∅. A problem arises when we want to get

12MARCO MARTENSit such that f becomes G−irreducible. Probably it is possible to get this property.This technical problem can be illustrated by the question: does the feigenbaummap have an acim?We can overcome this technical problem by assuming that f is ergodic and conser-vative with respect to λ: every set of positive λ−measure will intersect every otherset of positive λ−measure after some time.

If f is ergodic and conservative it willbe G−irreducible for every λ−partition G.So we get the following: an ergodic conservative map f has an acim iffthere existsa λ−partition G such that M∞(G, f) ̸= ∅.Using the following lemma we even can state a stronger existence theorem.Lemma 2.7. Let µ ∈Bσ(X) with λ ≫µ.

If f is ergodic and conservative withrespect to λ then for every set A with λ(A) > 0∞Xi=0µ(f −i(A)) = ∞.proof. The ergodicity and conservatively tells us that almost every point in A willreturn to A infinitely many times.

Now use the Borel-Cantelli Lemma.□Corollary 2.8. An ergodic conservative map f has a σ−finite absolutely continu-ous invariant measure iffthere exists a partition G withMs(G, f) ̸= ∅.The precise formulation of Theorem A in the introduction goes as follows.Theorem 2.9.

An ergodic conservative map f has a σ−finite absolutely continuousmeasure if there exists a λ−partition G withλ ∈M(G, f).

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS13This means: once the derivatives of f n∗λ have locally bounded distortion the exis-tence of an acim is assured.3. The initial measuresIn this section we are going to study a condition which will implyM∞(G, f) ̸= ∅, that is, it will imply the existence of acims.Fix λ ∈Bσ(X) and a λ−partition G of X.We say that the measurable mapf : X →X is finite-to-1 with respect toG ifffor all I ∈G f −1(I) is up to anullset contained in a finite subcollection of G.Furthermore PLλ(G) is the setof all distortion free measures: these measures are equivalent to λ with densitieswhich are constant on the element of G.proposition 3.1.

Let λ ∈Bσ(X) be a Borel measure on X and G a λ−partitionsuch that the measurable map f : X →X is G−irreducible.If there exists a λ−partition G0 with1) G is a refinement of G0;2) PLλ(G0) ⊂M(G, f)thenM∞(G, f) ̸= ∅.proof. The condition PLλ(G0) ⊂M(G, f) is a strong condition.

It implies that fis finite-to-1 with respect to G0: if f is not finite-to-1 it is easy to find I ∈G0 andµ ∈PLλ(G0) such that µ(f −1(I)) = ∞.We are going to define a measure µ ∈PLλ(G0) satisfying m3(I0) and m4(I0), whereI0 ∈G is fixed. Using PLλ(G0) ⊂M(G, f) and lemma 2.4 we get µ ∈M∞(G, f).

14MARCO MARTENSLet G0 = {Jn|n ≥0} and G = {In|n ≥0}.We can assume I0 ⊂J0.DefineLN = ∪Ni=0Ji for N ≥0. We are going to define µ by giving its density δ withrespect to λδ =XN≥0cN1JN.The numbers cN > 0 will be defined inductively satisfying the following inductionhypothesissupn≥0µ|LN(f −n(I0)) = 1 −(12)N+1.Because λ(I0) < ∞we can choose c0 > 0 such that the induction hypothesisholds for N = 0.

Suppose that c0, c1, . .

. , cN are defined satisfying the inductionhypothesis.

This means that the measure µ|LN is well defined. Now we have todefine the value cN+1 of the density on JN+1: let the map c →µN,c ∈PLλ(G0)with c ∈[0, ∞) be defined as followsµN,c = µ|LN + cλ|JN+1.Using the fact that f is G−irreducible, that is there exists an n ≥0 such thatλ(f −n(I0) ∩JN+1) ̸= 0, it is easy to see that the map φ : [0, ∞) →R defined byφ(c) = supn≥0µN,c(f −n(I0))tends continuously to infinity for c →∞.

From the definition of c0, c1, . .

. , cN weget φ(0) = 1 −( 12)N+1.

Hence there exists cN+1 > 0 such thatsupn≥0µ|LN+1(f −n(I0)) = φ(cN+1) = 1 −(12)N+2.We finished the induction step; the measure µ is well defined.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS15proof of m3(I0). Suppose there exists n ≥0 such that µ(f −n(I0)) > 1.

Because G0is an exhausting partition of X there exists N ≥0 such thatµ|LN(f −n(I0)) > 1which contradicts the construction of µ.proof of m4(I0). We are going to construct a sequence nk →∞such thatµ(f −nk(I0)) ≥12.This will imply m4(I0).Suppose we have a finite set {ni|i = 0, 1, .

. ., k} such that for all of themµ(f −ni(I0)) ≥12.

Let us find another one having this property. Because f isfinite-to-1 with respect to G0 there exists N ≥1 such thatk[i=0f −ni(I0) ⊂LNup to a set of measure zero.

Hence µ(f −ni(I0)) ≤1 −( 12)N+1 for i = 0, 1, . .

. , k.From the definition of cN+1 we easily get a number nk+1 such that1 −(12)N+1 < µ(f nk+1(I0))which is obviously not one of the previous ones.□Observe that theorem 2.9 gives a much weaker sufficient condition for the existenceof acims for conservative maps.

The use of proposition 3.1 will be for general maps.Indeed it can be shown that lemma 2.7 gives a characterization for dissipativeunimodal maps: A unimodal map is dissipative iffΣ∞i=0λ(f −i(A)) < ∞for all setsA ⊂∪G where G is some λ−partition.

16MARCO MARTENS4. Applications to 1-dimensional real dynamicsIn this section we will discuss the existence of absolutely continuous invariantmeasures for maps on the interval having negative Schwarzian derivative.Theexistence of invariant probability measures is strongly related to the expansionalong the orbits of the critical points, the points where the derivative vanishes.In [CE] the existence of acips was shown for unimodal maps having exponentialgrowth of the derivative along the critical orbit.

In [NS] this result was obtainedfor a weaker growth of the derivative, for example (non-linear) polynomial growthturns out to be sufficient.Another type of existence theorems is described in [Mi] and [S]: if the orbits of thecritical points are not accumulating at critical points, maps having this propertyare called Misiurewicz maps, then an acip exists. Here we will describe some resultscontinuing in this direction.In general the orbit closures of the critical points of Misiurewicz maps will lie in someclosed invariant set of measure zero which doesn’t contain critical points.

In thesequel we will allow our maps to exhibit some recurrence, critical orbits accumulateat critical points but we continue imposing the critical orbits to be in some closedinvariant set of measure zero. For this type of maps the existence of acims will beshown.

A result in this direction was already obtained in [HKe2]: if the criticalpoint of a conservative unimodal map with negative Schwarzian derivative stays ina Cantor set then an acim exists.The question whether these unimodal maps are always conservative is the mainopen problem in the theory of interval dynamics. A natural candidate for havingan absorbing Cantor set was recently proved to be conservative, see [LM].

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS17The results presented in this section can be formulated shortly as follows: mul-timodal maps whose critical orbits lie in a closed invariant set have a σ−finiteabsolutely continuous invariant measure. In particular, unimodal maps having anabsorbing Cantor set have an acim.In the sequel X denotes the interval [0, 1] or the circle endowed with the Lebesguemeasure λ.Let us first define the main analytical tool we need.

Let g : I →J be C3 andmapping the interval I ⊂X to the interval J ⊂X. The Schwarzian derivativeSg : I →R of g is defined to beSg(x) = D3g(x)Dg(x) −32(D2g(x)Dg(x) )2.An important and easy to derive property of maps with negative Schwarzian deriv-ative is that the iterates of these maps also have negative Schwarzian derivative.The Schwarzian derivative enables us to formulate the following distortion result.Koebe-Lemma.

For every ǫ > 0 there exists K > 0 with the following property.Let g : I →J be a diffeomorphism mapping the interval I ⊂X to the intervalJ ⊂X. Assume that Sg(x) < 0 for all x ∈I.If M ⊂I is an interval such that the components of I −M, denoted by L and R,satisfyλ(g(L))λ(g(M)) ≥ǫ and λ(g(R))λ(g(M)) ≥ǫthen1K ≤|Dg(x1)||Dg(x2)| ≤K

18MARCO MARTENSfor all x1, x2 ∈M.The proof of this fundamental lemma can be found in different places ([GuJ],[MMS]).The class D(X) of functions which we are going to consider is the class of piecewisediffeomorphic maps on X. These maps are defined as follows.

Let f ∈D(X) thenthere exists a λ−partition P of X consisting of open intervals such that for allI ∈P the restriction f|I is a diffeomorphism with negative Schwarzian derivative.Furthermore we assume f to have a dense orbit.An open interval T is called a branch of f n, f ∈D(X), if T is a maximal intervalon which f n is diffeomorphic.For every map f ∈D(X) we define the following functionsrn : X →R,where n ≥1 andrn(y) = inf{ǫ > 0|Bǫ(y) ⊂f i(Ti) with Ti branch of f i, i ≤n, with y ∈f i(Ti)}.with Bǫ(y) = (y −ǫ, y + ǫ). Furthermore define r : X →R to be r = lim rn.The sufficient condition for the existence of acims will be formulated in terms ofthe setS = {y ∈X|r(y) > 0}.Theorem 4.1.

Every conservative ergodic map in D(X) having λ(S) > 0 exhibitsan acim.proof. The set S is backward invariant.Hence the conservativity of f implies|S| = 1.

Let Sρ = r−1(ρ, 1) with ρ > 0.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS19Claim. For every compact set I ⊂Sρ there exist finitely many pairs {Ui, Vi}, i =1, ..., s, of intervals such that1) I ⊂∪{Ui|i = 1, ..., s};2) for all i = 1, ..., s |Ui| = ρ and Ui ⊂Vi with both components of Vi −Ui havelength 12ρ;3) if T is a branch of f n with f n(T )∩(Ui ∩I) ̸= ∅for some i ≤s then Vi ⊂f n(T ).proof of claim.

For every y ∈I the interval Vy = Bρ(y) = (y −ρ, y + ρ) has thefollowing property: if T is a branch with y ∈f n(T ) then Vy ⊂f n(T ).Consider a branch T which covers a point z ∈Vy ∩I, z ∈f n(T ). Because I ⊂Sρwe get immediately y ∈f n(T ).

Hence Vy ⊂f n(T ). Conclusion: for every y ∈Iand every branch T with f n(T ) ∩(Vy ∩I) ̸= ∅we have Vy ⊂f n(T ).Let Uy = B 12 ρ(y).

By using compactness of I we can cover I by finitely manyintervals of the form Uy. The corresponding pairs {Uy, Vy} will satisfy the claim.The claim implies that every compact set I ⊂Sρ has a natural finite partition insets Ii = I ∩Ui.

Let us use this partitions for constructing λ−partitions which willallow us to apply Theorem 2.9.As we saw |S| = 1. Hence for every set K with |K| > 0 there exist a compact setI ⊂K of positive Lebesgue measure and a ρ > 0 such that I ⊂Sρ.

This observationeasily implies the existence of a λ−partition G0 such that for every I ∈G0 I ⊂SρIfor some ρI > 0.Now partition every I ∈G0 as described above: I = ∪sIi=1Ii. Define G to be thecollection consisting of the sets Ii, i = 1, ..., sI and I ∈G0.For applying Theorem 2.9 we have to bound the distortion of the measures

20MARCO MARTENSf n∗λ|I, I ∈G.Fix I ∈G, say I ⊂Sρ, and let A ⊂I. The definition of the sets I ∈G allows us tocover f −n(I) by branches T1, ..., Tkn, kn ∈N ∪{∞}, satisfying:1) f n|Ti is diffeomorphic;2) both components of f n(Ti) −{convex −hull(I)} have length bigger than 12ρ.Now the Koebe-Lemma states the existence of K > 0, only depending on ρ, suchthat1K ≤|Df n(x1)||Df n(x2)| ≤Kfor all x1, x2 ∈f −n(I) ∩Ti, i = 1, ..., kn.Nowλ(f −n(A))λ(f −n(I)) =Pkni=1 λ(f −n(A) ∩Ti)λ(f −n(I))=Pkni=1λ(f −n(A)∩Ti)λ(f −n(I)∩Ti) λ(f −n(I) ∩Ti)λ(f −n(I))=Pkni=1λ(I)λ(f−n(I)∩Ti)λ(A)λ(f−n(A)∩Ti)λ(A)λ(I) λ(f −n(I) ∩Ti)λ(f −n(I)).Using the mean value theorem and the distortion result above we get1Kλ(A)λ(I) ≤λ(f −n(A))λ(f −n(I)) ≤K λ(A)λ(I) ,the measures f n∗λ|I have distortion bounded by K. We proved λ ∈M(G, f).

HenceTheorem 2.9 states the existence of an acim.□A map f ∈D(X) is called a Markov map if there exists a λ−partition P consistingof intervals such that for every I ∈P the image f(I) is a union (up to a set of

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS21measure zero) of elements of P. The map f is said to satisfy the Markov propertywith respect to P. Obviously these Markov maps have r(y) > 0 for all y ∈∪P.Corollary 4.2. Every conservative ergodic Markov map has an acim.This statement has to be compared with a theorem of Harris (see [H]) stating theexistence of infinite stationary states for certain Markov processes on countablemany state spaces.

In fact examples are known of Markov process not having astationary state (see [D]). These examples also serve for showing that we cannotomit the conservativity.On the other hand we can weaking this condition byimposing a topological condition.

By doing so we kill the metrical subtilities andget a general existence theorem which is valid as well in the conservative as in thedissipative case.σ−Folklore Theorem. Every finite-to-1 Markov map has an acim.proof.

Suppose f satisfies the Markov property with respect to G0. Let G be aλ−partition refining G0 and consisting of intervals.These intervals I ∈G arechosen in such a way that both components of T −I have length bigger than λ(I),where T ∈G0 with I ⊂T .Once we provedPLλ(G0) ⊂M(G, f)Theorem 3.1 assures the existence of an acim.Fix I ∈G with I ⊂T ∈G0.

Furthermore let A ⊂I. Because f is a finite-to-1 Markov map the set f −n(I), n ≥o, can be covered by finitely many intervalsT1, ..., Tkn satisfying1) f n|Ti is diffeomorphic;

22MARCO MARTENS2) f n(Ti) = T ;3) there exists T ′i ∈G0 with Ti ⊂T ′i.for all i = 1, ..., kn.Take µ ∈PLλ(G0).To prove that µ ∈M(G, f) first we have to show thatµ(f −n(I)) < ∞, n ≥0. However this is a direct consequence of f being finite-to-1.

Secondly we have to study the local distortion of the measures f n∗µ.Again the Koebe-Lemma gives a constant K > 0 such that1K ≤|Df n(x1)||Df n(x2)| ≤Kfor all x1, x2 ∈f −n(I) ∩Ti, i = 1, ..., kn.As beforeµ(f −n(A))µ(f −n(I)) =Pkni=1λ(I)µ(f−n(I)∩Ti)λ(A)µ(f−n(A)∩Ti)λ(A)λ(I) µ(f −n(I) ∩Ti)µ(f −n(I))=Pkni=1λ(I)λ(f−n(I)∩Ti)λ(A)λ(f−n(A)∩Ti)λ(A)λ(I) µ(f −n(I) ∩Ti)µ(f −n(I)).In the last step we used the fact that all measures in PLλ(G) have constant densitieson the elements T ∈G0 and property 3) above. Using the mean value theorem andthe distortion result above we get1Kλ(A)λ(I) ≤µ(f −n(A))µ(f −n(I)) ≤K λ(A)λ(I) .We proved µ ∈M(G, f).□The usual Folklore theorem states that every Markov map having derivative big-ger and bounded away from 1 has an absolutely continuous invariant probabilitymeasure.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS23Another possible σ−Folklore theorem could be formulated by considering Markovmaps whose branches are all mapped onto X.As the main consequence of the σ−Folklore Theorem we get Theorem B of theintroduction.Corollary 4.3. Let f be a C3 map on the interval (or circle) satisfying1) f has only finitely many critical points and the Schwarzian derivative is every-where negative except in the critical points;2) there exists a dense orbit;3) the orbits of the critical points stay in a closed invariant set of Lebesgue measurezero.Then f has a σ−finite absolutely continuous invariant measure.proof.

Let Λ be a closed invariant set which contains the critical orbits. Assume ithas Lebesgue measure zero.

Furthermore consider the λ−partition P consisting ofthe gaps of this set (the connected components of its complement). The map weare considering has only finite critical points.

Hence it is finite-to-1. In other wordsthe map is a finite-to-1 Markov map.□In the unimodal case the last corollary can be stated simpler.

In the unimodal casethe measure of the Cantor set containing the critical orbit always has Lebesguemeasure zero ([M]). So, unimodal maps which are only finitely renormalizable andhaving their critical orbit in an invariant Cantor set have acims.Appendix: The Chacon-Ornstein TheoremIn this section we will give a short proof of the main theorem in conservativeσ−finite ergodic theory.

It is based on the classical BirkhoffErgodic Theorem. The

24MARCO MARTENSproof can be summarized by: a map has an acim iffthe return map on some sethas an acip.Theorem A.1. Let f : X →X be ergodic and conservative with respect to µ ∈Bσ(X) which is a σ−finite invariant measure for f. For every pair of Riemannintergrable functions φ, ψ : X →R the equalitylimn→∞Pn−1i=0 φ(f i(x))Pn−1i=0 ψ(f i(x))=RφdµR ψdµholds for µ−almost every point x ∈X.proof.

Let B ⊂X having µ(B) < ∞. Using the fact that f is ergodic and conser-vative we can write the space as a stack: X is up to a set of measure zero equal toa countable union of pairwise disjoint sets Bk, k ≥0, whereBk = {x ∈X|x, f(x), .

. .

, f k−1(x) /∈B and f k(x) ∈B}.The key for the theorem isclaim. Let A ⊂Bk for some k ≥0.

Then for µ−almost every point x ∈Xlimn→∞#n(A)#n(B) = µ(A)µ(B).Here #n(U) = #{i = 0, 1, . .

., n −1|f i(x) ∈U}.proof of claim. The return map on B is denoted by R : B →B.

It has the followingproperties1) R is ergodic;2) for all x ∈B the set {f(x), . .

. , R(x)} contains at most one point of A;3) the measureµµ(B) is an acip for R.

THE EXISTENCE OF σ−FINITE INVARIANT MEASURES, APPLICATIONS TO REAL 1-DIMENSIONAL DYNAMICS25LetXA = {x ∈B|{f(x), . .

. , R(x)} ∩A ̸= ∅}.Then4) µ(XA) = µ(A).The statements 1) and 2) are obvious.

Let us prove 3) and 4). Consider a setA ⊂Bk and define inductively the sets Al with l ≥k:Ak = A;Al+1 = f −1(Al) ∩Bl+1.Define Rl+1 = f −1(Al) ∩B for l ≥k.

Using induction we getµ(A) =nXl=k+1µ(Rl) + µ(An)for all n ≥k. Applying this to A = B and using the fact that almost every pointin B returns to B we get µ(Bk) →0 for k →∞.Consider A ⊂Bk and observe XA = ∪∞l=k+1Rl.

Using µ(An) ≤µ(Bn) →0 we getµ(XA) = µ(∞[l=1Rl) =∞Xl=1µ(Rl) = µ(A).We proved 4). Furthermore observe R−1(A) = XA for every A ⊂B.

We proved 3).The proof of the claim is based on the BirkhoffErgodic theorem. Consider a pointx whose orbit behaves according to the invariant measure of R. Let y = f i0(x) bethe first time when f i(x) ∈B.

Furthermore partition the orbit according to thereturns to B. Then we getlimn→∞#n(A)#n(B) = limn→∞#{i = 0, 1, .

. .

, #n(B) −1|Ri(y) ∈XA}#n(B)= µ(XA)µ(B) = µ(A)µ(B).

26MARCO MARTENSObserve that up to time i0 we hit A at most i0 times. Furthermore the part of theorbit from R#n(B)(x) to R#n(B)+1(x) hit A at most once.

The conservativity andergodicity implies #n(B) →∞. Hence the initial and final part of the orbit arenot influencing the limit.

We proved the claim.For general sets A ⊂X we getlim infn→∞#n(A)#n(B) = lim infn→∞Xk≥0#n(A ∩Bk)#n(B)≥Xk≥0µ(A ∩Bk)µ(B)= µ(A)µ(B).Using the symmetry in A and B we getµ(A)µ(B) ≥1lim supn→∞#n(B)#n(A)= lim infn→∞#n(A)#n(B) ≥µ(A)µ(B).This implies, again using the symmetry, the equalitylimn→∞#n(A)#n(B) = µ(A)µ(B).The Chacon-Ornstein Theorem obviously follows for linear combinations of indica-tor functions. The general statement is a direct consequence of the definition ofRiemann integrability.□References[BL] A.M.Blokh, M.Ju.Lyubich, Non-existence of wandering intervals and structureof topological attractors of one-dimensional smooth dynamics, II.

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