Existence and convergence properties of physical measures for certain dynamical systems with holes
aa r X i v : . [ m a t h . D S ] S e p Existence and convergence properties of physicalmeasures for certain dynamical systems with holes.
Henk Bruin ∗ Mark Demers † Ian Melbourne ‡ November 7, 2018
Abstract
We study two classes of dynamical systems with holes: expanding maps of theinterval and Collet-Eckmann maps with singularities. In both cases, we prove thatthere is a natural absolutely continuous conditionally invariant measure µ ( a.c.c.i.m. )with the physical property that strictly positive H¨older continuous functions convergeto the density of µ under the renormalized dynamics of the system. In addition, weconstruct an invariant measure ν , supported on the Cantor set of points that neverescape from the system, that is ergodic and enjoys exponential decay of correlationsfor H¨older observables. We show that ν satisfies an equilibrium principle which impliesthat the escape rate formula, familiar to the thermodynamic formalism, holds outsidethe usual setting. In particular, it holds for Collet-Eckmann maps with holes, whichare not uniformly hyperbolic and do not admit a finite Markov partition.We use a general framework of Young towers with holes and first prove results aboutthe a.c.c.i.m. and the invariant measure on the tower. Then we show how to transferresults to the original dynamical system. This approach can be expected to generalizeto other dynamical systems than the two above classes. Dynamical systems with holes are examples of systems whose domains are not invariant underthe dynamics. Important questions in the study of such open systems include: what is theescape rate from the phase space with respect to a given reference measure? Starting with aninitial probability measure µ and letting µ n denote the distribution at time n conditionedon not having escaped, does µ n converge to some limiting distribution independent of µ ?Such a measure, if it exists, is a conditionally invariant measure . ∗ HB was supported in part by EPSRC grants GR/S91147/01 and EP/F037112/1 † MD was supported in part by EPSRC grant GR/S11862/01 and NSF grant DMS-0801139. ‡ IM was supported in part by EPSRC grant GR/S11862/01MD thanks the University of Surrey for an engaging visit during which this project was started. In addition,HB would like to thank Georgia Tech; MD would like to thank the Scuola Normale Superiore, Pisa; MD andIM would like to thank MSRI, Berkeley, where part of this work was done. R n [PY, CMS1, CMS2];Smale horseshoes [C1, C2]; Anosov diffeomorphisms [CM1, CM2, CMT1, CMT2]; billiardswith convex scatterers satisfying a non-eclipsing condition [LoM, R]; and large parameterlogistic maps whose critical point maps out of the interval [HY].Requirements on Markov partitions have been dropped for expanding maps of the interval[BaK, CV, LiM, D1]; and more recently for piecewise uniformly hyperbolic maps in twodimensions [DL]. Nonuniformly hyperbolic systems have been studied in the form of logisticmaps with generic holes [D2]. Typically a restriction on the size of the hole is introduced inorder to control the dynamics.A central object of study in these open systems is the conditionally invariant measurementioned previously. Given a self-map ˆ T of a measure space ˆ X , we identify a set H ⊂ ˆ X which we call the hole . Once the image of a point has entered H , we do not allow it toreturn. Define X = ˆ X \ H and T = ˆ T | X ∩ ˆ T − X . A probability measure µ is called conditionallyinvariant if it satisfies µ ( A ) = µ ( T − A ) µ ( T − X )for each Borel A ⊆ X . Iterating this relation and setting λ = µ ( T − X ), we see that µ ( T − n A ) = λ n µ ( A ). The number λ is called the eigenvalue of µ and − log λ represents its exponential rate of escape from X .If µ is absolutely continuous with respect to a reference measure m , we call µ an absolutelycontinuous conditionally invariant measure and abbreviate it by a.c.c.i.m. In [D1] and [D2], the author constructed Young towers to study expanding maps ofthe interval and unimodal Misiurewicz maps with small holes. The systems were shown toadmit an a.c.c.i.m. with a density unique in a certain class of densities and converging to theSRB measure of the closed system as the diameter of the hole tends to zero. However, leftopen in these papers was the question of what class of measures converges to the a.c.c.i.m. under the (renormalized) dynamics of T . This question is especially important for opensystems since even for well-behaved hyperbolic systems, many a.c.c.i.m. may exist withoverlapping supports and arbitrary escape rates [DY]. Thus it is essential to distinguisha natural a.c.c.i.m. which attracts a reasonable class of measures, including the referencemeasure.The purpose of this paper is two-fold. First, we prove that for a large class of systemswith holes, including1. C α expanding maps of the interval (see Theorem 2.10), and2. multimodal Collet-Eckmann maps with singularities (see Theorem 2.12),all H¨older continuous densities f which are bounded away from zero converge exponentiallyto the a.c.c.i.m. under the renormalized dynamics of T . To be precise, if L is the transfer op-erator associated with T and | · | the L ( m )-norm, then L n f / |L n f | converges exponentiallyto the density of µ as n → ∞ . Although similar results are known for C expanding mapswith holes [CV, LiM], they are completely new for multimodal maps, and even for unimodalmaps without singularities. In addition, we strengthen the results on the dynamics of thetower which were used in [D1] and [D2]. 2econd, we study the set of nonwandering points of each system: the (measure zero)set of points, X ∞ , which never enter the hole. We construct an ergodic invariant probabil-ity measure ˜ ν supported on X ∞ which enjoys exponential decay of correlations on H¨olderfunctions. The measure ˜ ν is characterized by a physical limit and satisfies an equilibriumprinciple. This implies the generalized escape rate formula for both classes of systems inquestion, log λ = h ˜ ν ( T ) − Z X log J T d ˜ ν where λ represents the exponential rate of escape from X with respect to the referencemeasure ˜ m , h ˜ ν ( T ) is the metric entropy of T with respect to ˜ ν , and J T is the Jacobian of T with respect to ˜ m .This formula is well-known when the usual thermodynamic formalism applies (in thepresence of a finite Markov partition) [Bo, C1, CM1, CMT2, CMS1]. In [BaK], an equi-librium principle was established for piecewise expanding maps with generalized potentialsof bounded variation. The paper [BrK] deals with equilibrium states of the unbounded po-tential − t log | T ′ | , t ≈
1, for Collet-Eckmann unimodal maps T , using a weighted transferoperator, but not allowing any holes. Both [BaK] and [BrK] use canonical Markov extensions(frequently called Hofbauer towers). In Theorem 2.17 we generalize those results to systemswith holes having no Markov structure and nonuniform hyperbolicity by constructing Youngtowers. In contrast to previous results, we do not use bounded variation techniques andso are able to allow potentials which are piecewise H¨older continuous. This answers in theaffirmative a conjecture of Chernov and van dem Bedem regarding expanding maps withholes [CV] and a more general question raised in [DY]. Remark 1.1.
It is important to note that the Young towers must be constructed for eachsystem after the introduction of holes since the presence of holes affects return times ina possibly unbounded way. Thus existing tower constructions for the corresponding closedsystems cannot be used directly.
Throughout the paper, we emphasize the physical properties of the measures involved andtheir characterization as push forward and pull back limits under the renormalized dynamics.In particular, the measures are independent of the Markov extensions used.In Section 2, we formulate our results precisely and include a brief discussion of theissues involved. Section 3 proves the convergence results on the tower while Section 4 appliesthese results to two classes of concrete systems with holes: expanding maps of the intervaland Collet-Eckmann maps with singularities. Section 5 contains proofs of the equilibriumprinciples for both the tower and the underlying dynamical system.
We recall the definition of a Young tower. Let ˆ∆ be a measure space and let Z be acountable measurable partition of ˆ∆ . Given a finite reference measure m on ˆ∆, let R be a3unction on ˆ∆ which is constant on elements of the partition and for which R R dm < ∞ .We define the tower over ˆ∆ asˆ∆ = { ( x, n ) ∈ ∆ × N : n < R ( x ) } , where N = { , , , . . . } . We call ˆ∆ ℓ = ˆ∆ | n = ℓ the ℓ th level of the tower. The action of thetower map ˆ F is characterized byˆ F ( x, n ) = ( x, n + 1) if n + 1 < R ( x )ˆ F R ( x ) ( Z ( x )) = S j ∈ J x Z j for some subset of partition elements of Z indexed by J x where Z ( x ) is the element of Z containing x and ˆ F R ( x ) | Z ( x ) is injective.We will abuse notation slightly and refer to a point ( x, n ) in the tower as simply x andˆ∆ n will be made clear by the context. Also, the partition Z and the action of F inducea natural partition of ˆ∆ which we shall refer to by Z , with elements Z ℓ,j in ˆ∆ ℓ . With thisconvention, it is clear that Z is a Markov partition for ˆ F . The definition of R extends easilyto the entire tower as well: R ( x ) is simply the first time that x is mapped to ˆ∆ under ˆ F .We extend m to each level of the tower by setting m ( A ) = m ( ˆ F − ℓ A ) for every measurableset A ⊂ ˆ∆ ℓ . We define a hole H in ˆ∆ as the union of countably many elements of the partition Z , i.e., H = S H ℓ,j where each H ℓ,j = Z ℓ,k for some k . Also set H ℓ = P j H ℓ,j = H ∩ ˆ∆ ℓ . Thispreserves the Markov structure of the returns to ˆ∆ , but the definition of the return timefunction R needs a slight modification: if x is mapped into H before it reaches ˆ∆ , R ( x )is defined to be the time that x is mapped into H ; otherwise, R ( x ) remains unchanged. If Z ℓ,j ⊂ H , then all the elements of Z directly above Z ℓ,j are deleted since once ˆ F maps apoint into H , it disappears forever.We will be interested in studying the dynamics of the points which have not yet falleninto the hole. To this end, we define ∆ = ˆ∆ \ H and ∆ n = T ni =0 ˆ F − i ∆, so ∆ n is the set ofpoints which have not fallen into the hole by time n . Define the map F = ˆ F | ∆ and itsiterates by F n = ˆ F n | ∆ n . We denote by Z ∗ ℓ,j ⊂ ∆ those elements of Z for which ˆ F ( Z ) ⊂ ˆ∆ .In this paper we will study the map F and the transfer operator associated with it.We consider towers with the following properties. (P1) Exponential returns.
There exist constants
C > θ < m ( ˆ∆ n ) ≤ Cθ n . (P2) Generating partition.
For each x = y ∈ ∆, there exists a separation time s ( x, y ) < ∞ such that s ( x, y ) is the smallest nonnegative integer k such that F k ( x ) and F k ( y ) liein different elements of Z or ˆ F k ( x ), ˆ F k ( y ) ∈ H . (P3) Finite images.
Let Z im be the partition of ∆ generated by the sets { F R Z } Z ∈Z . Werequire that Z im be a finite partition.Due to (P3) we define c := min Z ′ ∈Z im m ( Z ′ ) > d ( x, y ) = β s ( x,y ) for some β ∈ ( θ, θ is as in (P1). (The value of β may be further restricted depending on the underlyingdynamical system to which we wish to apply the tower.)We say that ( F, ∆) is transitive if for each Z ′ , Z ′ ∈ Z im , there exists an n ∈ N such that F n ( Z ′ ) ∩ Z ′ = 0. We say that F is mixing if for each Z ′ ∈ Z im , there is an N such that∆ ⊂ F n ( Z ′ ) for all n ≥ N . Remark 2.1.
We define mixing in this way because the usual requirement, gcd( R | ∆ ) = 1 ,made for towers with a single base (i.e., Z im contains a single element) is not sufficient toeliminate periodicity in towers with multiple bases. Since we may always construct a tower with no holes in the base (by simply choosing areference set in the underlying system which does not intersect the hole), we consider towerswith no holes in ∆ . Define q := X ℓ ≥ m ( H ℓ ) β − ( ℓ − . Our assumption on the size of the hole is, (H1) q < (1 − β ) c C where C is the distortion constant of equation (2.1) below. Remark 2.2.
If one is interested in considering towers with holes in the base, then thedefinition of q is modified to be q := P ℓ ≥ m ( H ℓ ) β − ( ℓ − + c − (1+ C ) m ( H ) P Z ∗ ℓ,j m ( Z ∗ ℓ,j ) β − ℓ .Assumption (H1) remains the same and all the results of this paper apply. In order to study the evolution of densities according to the dynamics of ( F, ∆), we introducethe transfer operator L F defined on L (∆) by L F f ( x ) = X y ∈ F − x f ( y ) g ( y )where g = dmd ( m ◦ F ) . Unless otherwise noted, we will refer to L F as simply L for the rest ofthis paper. Higher iterates of L are given by L n f ( x ) = X y ∈ F − n x f ( y ) g n ( y )where g n = g · g ◦ F · · · g ◦ F n − . For f ∈ L (∆), we define the Lipschitz constant of f to beLip( f ) = sup ℓ,j Lip( f ℓ,j ) and Lip( f ℓ,j ) = sup x = y ∈ Z ℓ,j | f ( x ) − f ( y ) | d ( x, y ) . We will assume that Lip(log g ) < ∞ . This assumption on g implies the following standarddistortion estimate. 5here exists a constant C > n > x, y ∈ ∆ such that s ( x, y ) ≥ n , we have (cid:12)(cid:12)(cid:12)(cid:12) g n ( x ) g n ( y ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C d ( F n x, F n y ) . (2.1)In particular, if E n ( y ) denotes the n -cylinder containing y , then g n ( y ) ≤ ( C + 1) m ( E n ( y )) m ( F n E n ( y )) ≤ ( C + 1) c − m ( E n ( y )) . (2.2)It is easy to see that dµ = ϕ dm is an a.c.c.i.m. with eigenvalue λ if and only if L ϕ = λϕ .Simply write for any measurable set A ⊂ ∆, µ ( F − A ) = Z F − A ϕ dm = Z A L ϕ dm, and λµ ( A ) = λ Z A ϕ dm. Then the two left hand sides are equal if and only if the two right hand sides are equal. Thusthe properties of a.c.c.i.m. for ( F, ∆) are tied to the spectral properties of L . L We begin by proving a spectral decomposition for L corresponding to ( F, ∆) acting on acertain Banach space of functions. The result follows essentially from Proposition 2.3 usingestimates similar to those in [Y2] and [D1]. One important difference in the present settingis that L does not have spectral radius 1, as it does for systems without holes, so carefulestimates are needed to ensure that a discrete spectrum exists outside the disk of radius β < Let V (∆) be the set of functions on ∆ which are Lipschitz continuous on elements of thepartition Z . For each Z ℓ,j and f ∈ V (∆), we set f ℓ,j = f | Z ℓ,j . We denote by | f | ∞ the L ∞ norm of f and define k f ℓ,j k ∞ := | f ℓ,j | ∞ β ℓ , k f ℓ,j k Lip := Lip( f ℓ,j ) β ℓ and || f || = max {k f k ∞ , k f k Lip } where k f k ∞ = sup ℓ,j k f ℓ,j k ∞ and k f k Lip = sup ℓ,j k f ℓ,j k Lip .Our Banach space is then B = { f ∈ V (∆) : k f k < ∞} . The choice β ∈ ( θ,
1) (where θ comes from condition (P1)) guarantees that B ⊂ L (∆) and the unit ball of B is compactlyembedded in L (∆). The proof of this fact is similar to that in [D1, Proposition 2.2]. Let | · | denote the L -norm with respect to m . In Section 3.1, we prove the following.6 roposition 2.3. Let ( F, ∆ , H ) be a tower with holes satisfying properties (P1)-(P3) andassumption (H1). Then there exists C > such that for each n ∈ N and all f ∈ B , kL n f k ≤ Cβ n k f k Lip + C | f | . Proposition 2.3, together with the compactness of the unit ball of B in L ( m ) and thefact that |L n f | ≤ | f | , are enough to conclude that L : B (cid:8) has essential spectral radiusbounded by β and spectral radius bounded by 1 [Ba]. However, since the system is open,we expect the actual spectral radius of L to be a constant λ <
1. We must show that λ > β in order to conclude that there is a spectral gap. This fact is proved in Section 3.2 usingassumption (H1) on the measure of the hole.Once a spectral gap has been established, the next proposition shows that the familiarspectral picture holds true for the open system. This is proved in Section 3.2.
Proposition 2.4.
The spectral radius of L on B is λ > β and L is quasi-compact as anoperator on B . In addition,(i) If F is mixing, then λ is a simple eigenvalue and all other eigenvalues have modulusstrictly less than λ . Moreover, there exists δ > such that the unique probability density ϕ corresponding to λ satisfies δλ − ℓ ≤ ϕ ≤ δ − λ − ℓ , on each ∆ ℓ .(ii) If F is transitive and periodic with period p , then the set of eigenvalues of modulus λ consists of simple eigenvalues { λe πik/p } p − k =0 . The unique probability density corre-sponding to λ satisfies the same bounds as in (i).(iii) In general, F has finitely many transitive components, each with its own largest eigen-value λ j . On each component, (ii) applies. Since Proposition 2.4 eliminates the possibility of generalized eigenvectors, the projectionΠ λ onto the eigenspace of eigenvalue λ is characterized for each f ∈ B by the limitΠ λ f = lim n →∞ n n − X k =0 λ − k L k f where convergence is in the k · k -norm. By (iii), the eigenspace V λ := Π λ B has a finite basisof probability densities, each representing an a.c.c.i.m. with escape rate − log λ . Corollary 2.5.
Suppose that F is mixing and let ϕ ∈ V λ denote the unique probabilitydensity given by (i). Then there exists σ ∈ (0 , and C ≥ such that k λ − n L n f − c ( f ) ϕ k ≤ C k f k σ n , for all f ∈ B where c ( f ) is a constant depending on f .Proof. The operator λ − L : B → B has spectral radius 1 and essential spectral radius βλ − .Moreover, there is a simple eigenvalue at 1 with eigenspace V λ spanned by ϕ and no furthereigenvalues on the unit circle. Hence there is an L -invariant closed splitting B = V λ ⊕ W λ and L : W λ → W λ has spectral radius ρ ∈ ( βλ − , σ ∈ ( ρ, L n f |L n f | as n → ∞ where the densityis renormalized at each step. Proposition 2.6.
Let ( F, ∆) be mixing and satisfy the hypotheses of Proposition 2.3, andlet f ∈ B . Then c ( f ) > if and only if lim n →∞ L n f |L n f | = ϕ where convergence is in the k · k -norm. Moreover convergence is at the rate σ n where σ is asin Corollary 2.5.Proof. Note that λ − n |L n f | → | c ( f ) | by Corollary 2.5 so that if c ( f ) >
0, we may writelim n →∞ L n f |L n f | = lim n →∞ L n fλ n λ n |L n f | = ϕ. The converse follows from the linear structure of L . We write B = V λ ⊕ W λ as in the proofof Corollary 2.5. Then W λ = { g ∈ B : c ( g ) = 0 } . Remark 2.7.
In what follows, we will be interested in establishing which functions satisfy c ( f ) > , first on the tower and then for the concrete systems for which towers are con-structed. Proposition 3.3 guarantees that in particular c (1) > so that the reference measureon ∆ converges to the a.c.c.i.m. ( F, ∆) The characterization of ϕ in terms of the physical limit L n f / |L n f | allows us to construct aninvariant measure ν singular with respect to m and supported on ∆ ∞ = ∩ ∞ n =0 ∆ n , the set ofpoints which never enter the hole. Although ν is supported on a zero m -measure Cantor-likeset, the results of this section indicate that it is physically relevant to the system.To state our results, we first introduce a new Banach space B consisting of functionsthat are uniformly bounded and uniformly locally Lipschitz. More precisely, let | f | ∞ denotethe standard sup-norm. Then define | f | Lip = sup ℓ,j
Lip( f ℓ,j ) and k f k = max {| f | ∞ , | f | Lip } .Note that contrary to k f k Lip , the seminorm | f | Lip doesn’t have the weights β ℓ . Finally, let B := { f ∈ B : k f k < ∞} . (2.3)The following proposition is proved in Section 3.3. Proposition 2.8.
Suppose ( F, ∆) satisfies properties (P1)-(P3) and (H1) of Section 2.1.1and is mixing. Then ( F, ∆) admits an invariant probability measure ν supported on ∆ ∞ ,which satisfies ν ( f ) = lim n →∞ λ − n Z ∆ n f dµ or all f ∈ B . In addition, ν is ergodic and (cid:12)(cid:12)(cid:12)Z ∆ ∞ f f ◦ F n dν − ν ( f ) ν ( f ) (cid:12)(cid:12)(cid:12) ≤ C k f k | f | ∞ σ n for all f , f ∈ B , n ≥ . In addition to its characterization as a limit, the invariant measure ν is natural to thesystem in the sense that it satisfies the below equilibrium principle.Let ν := ν (∆ ) ν | ∆ and note that ν is an invariant measure for F R on ∆ ∞ ∩ ∆ .Proposition 5.1 shows that in fact ν is a Gibbs measure for F R .We call a measure η nonsingular provided η ( F ( A )) = 0 if and only if η ( A ) = 0. Thefollowing theorem is proved in Section 5. Theorem 2.9.
Let ( F, ∆) satisfy the hypotheses of Proposition 2.8. Let M F be the set of F -invariant Borel probability measures on ∆ . Then log λ = sup η ∈M F (cid:26) h η ( F ) − Z ∆ log J F dη (cid:27) where h η ( F ) is the metric entropy of η with respect to F and J F is the Jacobian of F withrespect to m . In addition, ν is the unique nonsingular measure in M F which attains thesupremum. We apply the results about abstract towers with holes to two specific classes of dynamicalsystems with holes: C α piecewise expanding maps of the interval and locally C multimodalCollet-Eckmann maps with singularities. By a piecewise expanding map of the unit interval ˆ I , we mean a map ˆ T : ˆ I (cid:9) satisfying thefollowing properties. There exists a partition of ˆ I into finitely many intervals, ˆ I j , such that(a) ˆ T is C α and monotonic on each ˆ I j for some α >
0; and (b) | ˆ T ′ | ≥ τ >
2. Note that wecan always satisfy (b) if | ˆ T ′ | ≥ ε by considering a higher iterate of ˆ T .Let ˆ I nj denote the intervals of monotonicity for ˆ T n . The uniform expansion of ˆ T impliesthe following familiar distortion bound: there exists a constant C > n ,if x and y belong to the same ˆ I nj , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ˆ T n ) ′ ( x )( ˆ T n ) ′ ( y ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ˆ T n ( x ) − ˆ T n ( y ) | α . Introduction of Holes.
A hole ˜ H in [0 ,
1] is a finite union of open intervals ˜ H j . (Weuse the ˜ to distinguish from the hole on the tower.) Let I = ˆ I \ ˜ H and for n ≥
0, define I n = ∩ ni =0 ˆ T − i I . We are interested in studying the dynamics of T n := ˆ T n | I n .Let γ be the length of the shortest interval of monotonicity of T . Our sole condition onthe hole is 9 H2) ˜ m ( ˜ H ) ≤ γ (1 − β )( τ − β − )1 + C where ˜ m is Lebesgue measure on ˆ I and β > max { τ − , τ − α } .The following theorem is proved in [D1]. Theorem A. ([D1]) Let T be a C α piecewise expanding map of the interval and let ˜ H bea hole satisfying the bound given in (H2). Then ( T, I ) admits a tower ( F, ∆) which satisfiesproperties (P1)-(P3) and (H1) of Section 2.1.1 as well as (A1) of Section 4.1 with θ = τ , C = C , c = γ and C = 1 .If in addition, a transitivity condition is satisfied, then there is a unique conditionallyinvariant density ϕ ∈ B with eigenvalue λ . In order to eliminate periodicity and ensure transitivity for the map T and for the tower,we can impose the following transitivity condition. (T1) Let J be an interval of monotonicity for T . There exists an n > T n J covers I up to finitely many points.Property (T1) is analogous to the covering property for piecewise expanding maps of theinterval without holes which is a necessary and sufficient condition for the existence of aunique absolutely continuous invariant measure whose density is bounded away from zero(see [Li]).Fix ¯ α ≥ − log β/ log τ and if necessary, choose β closer to 1 so that ¯ α ≤ α . Let I ∞ denotethe set of points which never escape from I and define G = { ˜ f ∈ C ¯ α ( I ) : ˜ f > I ∞ } .Denote by L T the transfer operator of T with respect to ˜ m and let | · | denote the L ( ˜ m )-norm. We prove the following theorem in Section 4.2. Theorem 2.10.
Let T satisfy the hypotheses of Theorem A in addition to condition (T1).There exists λ > such that for all ˜ f ∈ G , the escape rate with respect to ˜ η = ˜ f ˜ m iswell-defined and equal to − log λ , i.e., lim n →∞ n log ˜ η ( I n ) = log λ. There exists a unique a.c.c.i.m. ˜ µ with density ˜ ϕ and eigenvalue λ such that for all ˜ f ∈ G ,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L nT ˜ f |L nT ˜ f | − ˜ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C T | ˜ f | C ¯ α σ n for some σ ∈ ( β, and C T depends only on the smoothness and distortion of the map T . Remark 2.11.
The results of Theorems A and 2.10 generalize those obtained in [CV] and[LiM] for C expanding maps using bounded variation techniques. We make no assumptionson the position of the holes, only their measure. .4.2 Collet-Eckmann maps with singularities Collet-Eckmann maps are interval maps with critical points such that the derivatives D ˆ T n atthe critical values increase exponentially. We will follow the approach of [DHL] which allowsfor discontinuities and points with infinite derivative as in Lorenz maps. (We try to use thesame notation as in [DHL], but adding a ∗ if there is a clash with our own notation.) Themap ˆ T : ˆ I → ˆ I is locally C and has a critical set Crit = Crit c ∪ Crit s consisting respectivelyof genuine critical points c with critical order 1 < ℓ c < ∞ and singularities with critical order0 < ℓ c ≤
1. At each of these points ˆ T is allowed to have a discontinuity as well, so c ∈ Crithas a left and right critical order which need not be the same. Furthermore, ˆ T satisfiesthe following conditions for all δ > B δ (Crit) = ∪ c ∈ Crit B δ ( c ) is a δ -neighborhood ofCrit): (C1) Expansion outside B δ ( Crit ) : There exist λ ∗ > κ > x and n ≥ x = x, . . . , x n − = ˆ T n − ( x ) / ∈ B δ (Crit), we have | D ˆ T n ( x ) | ≥ κδ ℓ max − e λ ∗ n , where ℓ max = max { ℓ c : c ∈ Crit c } . Moreover, if x ∈ ˆ T ( B δ (Crit)) or x n ∈ B δ (Crit),then we have | D ˆ T n ( x ) | ≥ κe λ ∗ n . (C2) Slow recurrence and derivative growth along critical orbit:
There exists Λ ∗ > c ∈ Crit c there is α ∗ c ∈ (0 , Λ ∗ / (5 ℓ c )) such that | D ˆ T k ( ˆ T ( c )) | ≥ e Λ ∗ k and dist( ˆ T k ( c ) , Crit) > δe − α ∗ c k for all k ≥ . (C3) Density of preimages:
There exists c ∗ ∈ Crit whose preimages are dense in ˆ I , and noother critical point is among these preimages.Condition (C1) follows for piecewise C maps from Ma˜n´e’s Theorem, see [MS, Chapter III.5].The first half of condition (C2) is the actual Collet-Eckmann condition, and the second half isa slow recurrence condition. Condition (C3) excludes the existence of non-repelling periodicpoints.In [DHL], α ∗ c is assumed to be small relative to λ ∗ and Λ ∗ . We keep the same restrictionon α ∗ c and do not need to shrink it further after the introduction of holes. Introduction of Holes.
In order to apply the tower construction of [DHL] to our setting,we place several conditions on the placement of the holes in the interval ˆ I . We adopt notationsimilar to that in Section 2.4.1.A hole ˜ H in ˆ I is a finite union of open intervals ˜ H j , j = 1 , . . . , L . Let I = ˆ I \ ˜ H and set I n = T ni =0 ˆ T − i I . Define T = ˆ T | I and let ˜ m denote Lebesgue measure on ˆ I . The addition of exponent ℓ max − δ in this formula is a correction to [DHL, Condition (H1)], whichaffects the proofs only in the sense that some constants will be different. The formula in [DHL] cannot berealized for any x at distance δ to any critical point c ∈ Crit s with ℓ c > The fact that α ∗ c depends on c in this way is a correction to [DHL, Condition (H2)], where this is notstated, but used in the proof of Lemma 2 of [DHL]. B1)
Let α ∗ c > c ∈ Crit c and k ≥ T k ( c ) , ∂ ˜ H ) > δe − α ∗ c k . Our second condition on ˜ H is that the positions of its connected components are genericwith respect to one another. This condition will also double as a transitivity condition on theconstructed tower which ensures our conditionally invariant density will be bounded awayfrom zero. In order to formulate this condition, we need the following fact about C nonflatnonrenormalizable maps satisfying (C1)-(C2).For all δ > n = n ( δ ) such that for all intervals ω ⊆ ˆ I with | ω | ≥ δ , ( i ) ˆ T n ω ⊇ ˆ I, and( ii ) there is a subinterval ω ′ ⊂ ω such that ˆ T n ′ maps ω ′ diffeomorphicallyonto ( c ∗ − δ, c ∗ + 3 δ ) for some 0 < n ′ ≤ n. (2.4)We also need some genericity conditions on the placement of the components of the hole.Within each component ˜ H j , we place an artificial critical point b j , so Crit hole = { b , . . . , b L } .The points b j are positioned so that the following holds: (B2) (a) orb( b j ) ∩ c = ∅ for all 1 ≤ j ≤ L and c ∈ Crit hole ∪ Crit c .(b) Let ˆ T − ( ˆ T b j ) = ∪ K j i =1 g j,i . For all j, k ∈ { , . . . , L } , there exists i ∈ { , . . . , K j } such that ˆ T ℓ b k = g j,i for 1 ≤ ℓ ≤ n ( δ ).Here n ( δ ) is the integer corresponding to δ in (2.4) and δ is chosen so small that: (i) allpoints in Crit c ∪ Crit s ∪ Crit hole are at least δ apart, and (ii) for each j = 1 , . . . , L , there is τ = τ ( j ) ≥ | D ˆ T τ ( x ) | ≥ max { κe λ ∗ τ , } for all x ∈ B δ ( b j ) . (2.5)Condition (C1) implies that | D ˆ T τ ( x ) | ≥ κe λ ∗ τ whenever x / ∈ B δ (Crit c ) and T τ ( x ) ∈ B δ (Crit c ),so by taking δ small, and using assumption (B2)(a), we can indeed find τ such that also | D ˆ T τ ( x ) | ≥ I ∞ denote the set of points which never escapes from I and let G = { ˜ f ∈ C ¯ α ( I ) : ˜ f > I ∞ } . We prove the following theorem in Section 4.3. Theorem 2.12.
Let ˆ T be a nonrenormalizable map satisfying conditions (C1)-(C3) and let ˜ H be a sufficiently small hole satisfying (B1)-(B2).There exists λ > such that for all ˜ f ∈ G , the escape rate with respect to ˜ η = ˜ f ˜ m iswell-defined and equal to − log λ , i.e., lim n →∞ n log ˜ η ( I n ) = log λ. Moreover, there exists a unique a.c.c.i.m. ˜ µ with density ˜ ϕ and eigenvalue λ such that forall ˜ f ∈ G , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L nT ˜ f |L nT ˜ f | − ˜ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C T | ˜ f | C ¯ α ˜ σ n for some ˜ σ ∈ ( β, and C T depends only on the smoothness and distortion of the map T . emark 2.13. When ˆ T is a Misiurewicz map (i.e., all critical points are nonrecurrent andall periodic points are non-repelling), it is possible to give a constructive bound on the sizeof the hole in terms of explicit constants. In this case we assume that the critical point doesnot fall into the hole, but there is no need for condition (B2)(a), see [D2, Section 2.2]. Small Hole Limit.
Fix L distinct points b , . . . , b L ∈ ˆ I which we consider to be infinitesimalholes satisfying (B1) and (B2). We call this hole of measure zero ˜ H (0) with components˜ H (0) j = b j , j = 1 , . . . , L . For each h >
0, we then define a family of holes H ( h ) such that˜ H ∈ H ( h ) if and only if1. b j ∈ ˜ H j and ˜ m ( ˜ H j ) ≤ h for each 1 ≤ j ≤ L ;2. ˜ H satisfies (B1).When we shrink a hole in H ( h ), we keep b , . . . , b L fixed and simply choose a smaller h . Thefollowing theorem is proved in Section 4.3.1. Theorem 2.14.
Let ˆ T satisfy the hypotheses of Theorem 2.12 and let ˜ H ( h ) ∈ H ( h ) be afamily of holes. Let d ˜ µ h = ˜ ϕ h d ˜ m be the a.c.c.i.m. given by Theorem 2.12 with eigenvalue λ h .Then λ h → and ˜ µ h converges weakly to the unique SRB measure for ˆ T as h → . Remark 2.15.
A similar theorem was proved for piecewise expanding maps in [D1] and forMisiurewicz maps in [D2]. ( T , X ) In this section, T is either an expanding map satisfying the assumptions of Theorem 2.10 ora Collet-Eckmann map with singularities satisfying the assumptions of Theorem 2.12.Recall the invariant measure ν supported on ∆ ∞ introduced in Proposition 2.8. Themeasure ˜ ν := π ∗ ν is T -invariant and is supported on X ∞ = π (∆ ∞ ). We show that ˜ ν isphysically relevant to the system ( T, X ) in two ways.
Theorem 2.16.
The invariant measure ˜ ν is characterized by ˜ ν ( ˜ f ) = lim n →∞ λ − n Z X n ˜ f d ˜ µ for all functions ˜ f ∈ C ¯ α ( X ) . In addition, ˜ ν is ergodic and enjoys exponential decay ofcorrelations on H¨older observables. Although ˜ ν is defined simply as π ∗ ν , the preceding theorem gives a characterization of ˜ ν which is independent of the tower construction. This is important for two reasons: first, itimplies that two different tower constructions will yield the same invariant measure; second,it eliminates the need to construct a tower in order to compute ˜ ν .The second theorem is a consequence of Theorem 2.9.13 heorem 2.17. Let M ′ T = π ∗ M F = { π ∗ η : η ∈ M F } be the set of T -invariant Borelprobability measures on X whose lift to ∆ is well-defined. Then log λ = sup η ∈M ′ T (cid:26) h η ( T ) − Z X log J T dη (cid:27) where
J T is the Jacobian of T with respect to ˜ m . The invariant measure ˜ ν is the uniquenonsingular measure ˜ η in M ′ T which attains the supremum. Remark 2.18.
As stated in Theorem 2.17, the equilibrium principle applies to the collectionof invariant measures M ′ T whose lift to ∆ is well-defined. So a priori, ˜ ν need not be theglobal equilibrium state in M T , the set of T -invariant measures supported on Y ∞ = { x ∈ X : ˆ T n ( x ) / ∈ ˜ H for all n ≥ } . (As X ∞ = π (∆ ∞ ) , Y ∞ ⊃ X ∞ .)For C Collet-Eckmann maps without singularities, however, the equilibrium state ˜ ν isindeed global. This is due to [Y1, Theorem 4] which in our setting guarantees log λ ≥ sup η ∈M T (cid:8) h η ( T ) − R X log J T dη (cid:9) . Since ˜ ν ∈ M ′ T ⊆ M T attains the supremum, the inequal-ity is in fact an equality. Theorem 2.16 is proved in Section 5.1 and Theorem 2.17 is proved in Section 5.3. L In this section we prove Proposition 2.3 by deriving Lasota-Yorke type inequalities for k · k ∞ and k · k Lip , kL n f k ∞ ≤ Cβ n k f k Lip + C | f | , kL n f k Lip ≤ Cβ n k f k Lip + C | f | . Proof of Proposition 2.3.
We fix n ∈ N and separate the estimates into two parts: those for Z ℓ,j with ℓ ≥ n and those with ℓ < n . Estimate . For any x ∈ Z ℓ,j with ℓ ≥ n and f ∈ B , note that L n f ( x ) = f ( F − n x ) since g n ( F − n x ) = 1. This allows us to estimate, kL n f ℓ,j k ∞ := | f ( F − n ) ℓ,j | ∞ β ℓ = ( | f ℓ − n,j | ∞ β ℓ − n ) β n = k f ℓ − n,j k ∞ β n . Estimate . Again choose any Z ℓ,j with ℓ ≥ n . Then kL n f ℓ,j k Lip := sup x,y ∈ Z ℓ,j | f ( F − n x ) − f ( F − n y ) | d ( x, y ) β ℓ = β n sup x,y ∈ Z ℓ,j | f ( F − n x ) − f ( F − n y ) | d ( F − n x, F − n y ) β − n β ℓ − n = β n k f ℓ − n,j k Lip since s ( x, y ) = s ( F − n x, F − n y ) − n . 14 stimate . Let x ∈ Z ,j be a point in ∆ . We denote by E n the cylinder sets of length n with respect to the partition Z and let E n ( y ) denote the element of E n containing y . |L n f ( x ) | ≤ X y ∈ F − n x | f ( y ) | g n ( y ) ≤ X y ∈ F − n x | f (¯ a ) | g n ( y ) + g n ( y ) | f ( y ) − f (¯ a ) | d ( y, ¯ a ) d ( y, ¯ a )where ¯ a ∈ E n ( y ) is any point satisfying | f (¯ a ) | ≤ m ( E n ( y )) R E n ( y ) | f | dm . By (2.2), g n ( y ) ≤ ( C + 1) c − m ( E n ( y )) . (3.1)Finally, note that F n y, F n ¯ a ∈ Z ,j so that n is a return time for y and ¯ a . If y ∈ ∆ ℓ ( y ) , thenthe definition of d from Section 2.1.1 implies d ( y, ¯ a ) ≤ β n .Putting this together with (3.1), we estimate |L n f ( x ) | ≤ X y ∈ F − n x g n ( y ) m ( E n ( y )) Z E n ( y ) | f | dm + g n ( y ) β − ℓ ( y ) k f k Lip β n ≤ X y ∈ F − n x (1 + C ) c − Z E n ( y ) | f | dm + (1 + C ) c − m ( E n ( y )) β n − ℓ ( y ) k f k Lip (3.2) ≤ (1 + C ) c − Z ∆ n | f | dm + Cβ n k f k Lip where the second sum is finite since β > θ . Estimate . Let x, y ∈ Z ,j , and let x ′ ∈ F − n x , y ′ ∈ F − n y , denote preimages taken alongthe same branch of F − n . Then summing over all inverse branches gives |L n f ( x ) − L n f ( y ) | d ( x, y ) ≤ X x ′ ∈ F − n x | g n ( x ′ ) f ( x ′ ) − g n ( y ′ ) f ( y ′ ) | d ( x, y ) ≤ X x ′ ∈ F − n x g n ( x ′ ) | f ( x ′ ) − f ( y ′ ) | d ( x, y ) + | f ( y ′ ) | | g n ( x ′ ) − g n ( y ′ ) | d ( x, y ) (3.3) ≤ X x ′ ∈ F − n x (1 + C ) c − m ( E n ( x ′ )) k f k Lip β n − ℓ ( x ′ ) + | f ( y ′ ) | C g n ( y ′ )where we have used (2.1) and (3.1) for g n in the last line as well as the fact that d ( x ′ , y ′ ) = β n d ( x, y ) for x ′ ∈ ∆ ℓ ( x ′ ) . The second sum is identical to that in Estimate |L n f ( x ) − L n f ( y ) | d ( x, y ) ≤ Cβ n k f k Lip + C | f | . Now on Z ℓ,j with ℓ < n , we can combine Estimates kL n f ℓ,j k ∞ ≤ β ℓ kL n − ℓ f ,j k ∞ ≤ β ℓ ( Cβ n − ℓ k f k Lip + C | f | )which implies the estimate for the k · k ∞ -norm.Similarly, we can combine Estimates kL n f ℓ,j k Lip ≤ β ℓ kL n − ℓ f ,j k Lip ≤ β ℓ ( Cβ n − ℓ k f k Lip + C | f | )which completes the estimate for the k · k Lip -norm.15 .2 Spectral Gap
Although Proposition 2.3 implies that the essential spectral radius of L on B is less than orequal to β , we must still ensure that there is a spectral gap, i.e., that there is an eigenvalue β < λ < B .This fact follows from the bound on the measure of the hole H given by (H1). Toprove it, it will be convenient to recall some results from [D1] which concern the nonlinearoperator L f := L f / |L f | . Thus L f represents the normalized push-forward density whichis conditioned on non-absorption by the hole. In [D1] it was shown that for small holes, L preserves a convex subset of B defined by B M = { f ∈ B : f ≥ , | f | = 1 , k f k ∞ ≤ M, k f k log ≤ M } where k f k log = sup ℓ,j Lip(log f ℓ,j ) . We include the proof of the proposition here for clarity and also to formulate the boundon H in terms of the present notation. Proposition 3.1.
Let M ∈ (cid:0) (1 + C ) c − , (1 − β ) q − (cid:1) . Then(i) L n maps B M into itself for n sufficiently large.(ii) There exists β ′ > β such that |L f | ≥ β ′ for all f ∈ B M .Proof. Note that Lip(log f ℓ,j ) is equivalent to sup x,y ∈ Z ℓ,j | f ( x ) − f ( y ) | f ( x ) d ( x, y ) .We will work with this expression in the following estimates. For f ∈ B M , we prove theanalogue of Estimates k · k log . Estimates |L n f ( x ) | ≤ X y ∈ F − n x f (¯ a ) g n ( y ) + g n ( y ) | f ( y ) − f (¯ a ) | f (¯ a ) d ( y, ¯ a ) d ( y, ¯ a ) f (¯ a ) ≤ X y ∈ F − n x g n ( y ) m ( E n ( y )) Z E n ( y ) f dm + g n ( y ) m ( E n ( y )) k f k log β n Z E n ( y ) f dm ≤ (1 + C ) c − (1 + β n k f k log ) Z ∆ n f dm (3.4)where in the last line we have used the fact that P y ∈ F − n x R E n ( y ) f dm ≤ R ∆ n f dm .To modify Estimate P i a i and P i b i are two series of16ositive terms, then P i a i P i b i ≤ sup i a i b i . Equation (3.3) becomes |L n f ( x ) − L n f ( y ) | d ( x, y ) L n f ( x ) ≤ P x ′ ∈ F − n x g n ( y ′ ) | f ( x ′ ) − f ( y ′ ) | d ( x,y ) P x ′ ∈ F − n x g n ( x ′ ) f ( x ′ ) + P x ′ ∈ F − n x f ( x ′ ) | g n ( x ′ ) − g n ( y ′ ) | d ( x,y ) P x ′ ∈ F − n x g n ( x ′ ) f ( x ′ ) ≤ sup x ′ ∈ F − n x g n ( y ′ ) g n ( x ′ ) | f ( x ′ ) − f ( y ′ ) | d ( x, y ) f ( x ′ ) + sup x ′ ∈ F − n x (cid:12)(cid:12)(cid:12) − g n ( y ′ ) g n ( x ′ ) (cid:12)(cid:12)(cid:12) d ( x, y ) ≤ (1 + C ) β n k f k log + C . (3.5)Since k · k log is scale invariant, (3.5) implies for all n ≥ kL n f k log = kL n f k log ≤ (1 + C ) β n k f k log + C . (3.6)Using (3.4), |L n f ( x ) ||L n f | ≤ (1 + C ) c − (1 + β n k f k log ) R ∆ n f dm R ∆ n f dm ≤ (1 + C ) c − (1 + β n k f k log )so that the k · k ∞ -norm stays bounded on the base of the tower. In order for this norm toremain bounded on successive levels, we need to ensure that |L f | ≥ β for each f ∈ B M .Compute that Z ∆ L f dm = Z ∆ f dm = 1 − X ℓ ≥ Z ˆ F − H ℓ,j f dm ≥ − X ℓ ≥ k f ℓ − ,j k ∞ β − ( ℓ − m ( H ℓ ) ≥ − M X ℓ ≥ β − ( ℓ − m ( H ℓ ) . Recall that q = P ℓ ≥ β − ( ℓ − m ( H ℓ ). Thus |L f | > β if 1 − qM > β and M must bechosen large enough so that L n maps B M back into itself for large enough n . Equations (3.4)and (3.5) require that we choose M ∈ (cid:0) (1 + C ) c − , (1 − β ) q − (cid:1) . Thus q < − βM < (1 − β ) c C isa sufficient condition on the size of H and is precisely assumption (H1). Proof of Proposition 2.4.
The proof divides into several steps.
1. Quasi-compactness of L . Proposition 3.1 implies that there exists N ≥ L N B M ⊂ B M . Since L N is continuous on B M , which is a convex, compact subset of L ( m ),the Schauder-Tychonoff theorem guarantees the existence of a fixed point ϕ ∈ B M , which isa conditionally invariant density for L N with eigenvalue ρ = R ∆ N ϕ dm .Proposition 2.3 implies that the essential spectral radius of L N is bounded by β N andProposition 3.1 guarantees that ρ > β N .Thus L N is quasi-compact with spectral radius at least ρ . We conclude that L is quasi-compact with spectral radius at least λ := ρ /N and essential spectral radius β < λ .Let N ≤ N be the least positive integer such that L N ϕ = λ N ϕ . In the next part ofthe proof, Steps 2–5, we assume that F is mixing and that N = 1. These assumptions areremoved in Steps 6 and 7. 17 . The density ϕ . We claim that there exists δ > δ ≤ ϕ | ∆ ≤ δ − . It is thenimmediate from the conditional invariance condition λ − L ϕ = ϕ that δλ − ℓ ≤ ϕ | ∆ ℓ ≤ δ − λ − ℓ .By conditional invariance, for x ∈ ∆ ℓ , ϕ ( x ) = λ − ℓ ϕ ( F − ℓ x ), so that ϕ ≡ ϕ ≡ . Thus there exists x ∈ ∆ such that ϕ ( x ) >
0. Using conditional invariance once more,we obtain x ′ ∈ F − x such that ϕ ( x ′ ) >
0. Let Z be the partition element containing x ′ . Since ϕ ∈ B M , it follows that ϕ ≥ κ > Z . By construction, F ( Z ) ⊇ Z ′ for some Z ′ ∈ Z im .By conditional invariance, inf Z ′ ϕ ≥ λ − κ inf Z g >
0. By transitivity, conditional invariance,and the property that ϕ ∈ B M , we obtain a similar lower bound for each Z ′ ∈ Z im . Theclaim follows from finiteness of the partition Z im .
3. Spectral radius.
Now suppose f ∈ B such that L f = αf and | α | > λ . Note that f satisfies f ( x ) = α − ℓ f ( F − ℓ x ) for each x ∈ ∆ ℓ , ℓ ≥
1. Since ϕ ≥ δ , there exists K > Kϕ ≥ | f | on ∆ . But since f grows like α − ℓ and ϕ grows like λ − ℓ on level ℓ , we have Kϕ ≥ | f | on ∆. By the positivity of L , K L n ϕ ≥ L n | f | ≥ |L n f | for each n . But this impliesthat Kλ n ϕ ≥ | α | n | f | for each n . Since λ < | α | , it follows that f ≡
0. Hence L has spectralradius precisely λ .
4. Simplicity of λ . Suppose f ∈ B such that L f = λf . As in Step 3, we can choose K > f + Kϕ >
0. Let ψ = ( f + Kϕ ) /C > C = | f | + K is chosen sothat ψ is a probability density. Define ψ s = sϕ + (1 − s ) ψ and let J = { s ∈ R : inf ∆ ψ s > } .Note that for s ∈ J , L ψ s = λψ s and | ψ s | = 1. Since ψ s is Lipschitz and bounded awayfrom zero, k ψ s k log < ∞ . In fact, (3.6) implies that k ψ s k log = lim n →∞ kL n ψ s k log ≤ M , so that ψ s ∈ B M for all s ∈ J .Since ψ s is conditionally invariant, the identity ψ s | ∆ ℓ = λ − ℓ ψ s | ∆ implies that inf ∆ ψ s =inf ∆ ψ s , so that J is open. Now let t ∈ ∂J . Since ψ s ∈ B M for all s ∈ J and B M is closed, wehave ψ t ∈ B M . If ψ t vanishes on ∆ , then ψ t vanishes on an entire element Z ′ ∈ Z im . Since ψ t ≥
0, this implies that ψ t ≡ Z im which map to Z and by transitivity ψ t is zero on all of ∆. Thus ψ t has strictly positive infimum on ∆ and since it is conditionallyinvariant, it must have the same infimum on ∆. Thus t ∈ J , so J is closed. Since J isnonempty, J = R , which is only possible if f = cϕ for some c ∈ R .It remains to eliminate generalized eigenvectors. Suppose f ∈ B such that L f = λ ( f + ϕ ).Then L n f = λ n f + nλ n ϕ = λ n f + L n ( nϕ ) so that L n ( f − nϕ ) = λ n f . This implies that for x ∈ ∆ ℓ , f ( x ) = λ − ℓ ( f − ℓϕ ) ◦ F − ℓ ( x ) . Since for ℓ large enough, f − ℓϕ < , we have f < ∪ ℓ ≥ L ∆ ℓ for some L >
K > ψ := f − Kϕ < ∪ ℓ
5. Absence of peripheral spectrum.
Suppose f ∈ B , | f | = 1, such that L f = αf , where α = λe iω , ω ∈ (0 , π ). We follow an approach similar to Step 4, modified to take into accountthe fact that f is complex and α = λ . Notice that by conditional invariance, f | ∆ ℓ = λ − ℓ e − iωℓ f | ∆ , (3.7)18o that f grows like ϕ plus a rotation up the levels of the tower.Define ψ = (Re( f ) + Kϕ ) /C ′ , where K is chosen large enough that ψ > C ′ normal-izes | ψ | = 1. By replacing f with − f if necessary, we can guarantee that R ∆ Re( f ) dm ≤ C ′ ≤ K . Also notice that since f and ϕ grow at the same rate, there exists δ > δ λ − ℓ ≤ ψ ( x ) ≤ δ − λ − ℓ (3.8)for x ∈ ∆ ℓ .As before, define ψ s = sϕ + (1 − s ) ψ and let J = { s ∈ R : inf ∆ ψ s > } . Due to (3.8), J is open. However, ψ s is not conditionally invariant since α = λ so the second part of theargument needs some modification.Notice that λ − n L n ψ = (Re( e iωn f ) + Kϕ ) /C ′ (3.9)so we may choose a sequence n k such that λ − n k L n k ψ → ψ as k → ∞ . This implies alsothat λ − n k L n k ψ s → ψ s along the same sequence and by (3.6) we have k ψ s k log ≤ M so that ψ s ∈ B M for s ∈ J . Now let t > J . Since B M is closed, we have ψ t ∈ B M . The rest of Step 5 relies on the following lemma. Lemma 3.2. ψ t is bounded away from . It is easy to see that the lemma completes the proof of Step 5 since then t ∈ J and weconclude that J ⊃ R + . Now ψ s > s >
0, implies ϕ > ψ . Thus ϕ > (Re( f ) + Kϕ ) /C ′ ⇒ ( C ′ − K ) ϕ > Re( f ) ⇒ > Re( f )since C ′ ≤ K . But Re( f ) must change sign on ∆ due to the rotation as we move up thelevels of the tower given by (3.7). This contradicts the existence of α . Proof of Lemma 3.2.
Since | ψ t | = 1 and ψ t ≥
0, there exists ℓ ≥ x ∈ ∆ ℓ such that ψ t ( x ) >
0. Since λ − n k L n k ψ t → ψ t , there exists k with n k > ℓ such that λ − n k L n k ψ t ( x ) > x ′ ∈ F − n k ( x ) such that ψ t ( x ) >
0. Let Z ∈ Z be the partitionelement containing x ′ . By construction, Z does not iterate into a hole before reaching ∆ (in m = n k − ℓ iterates). In particular, F m Z covers an element of Z im . Since F is mixing,there exists an N > n ≥ N , F n Z ⊃ ∆ .Since ψ t ∈ B M , it follows that inf Z ψ t =: κ >
0. Note that for any n ≥
0, the definitionof ψ t and equation (3.9) imply that λ − n L n ψ t = tϕ + (1 − t )[Re( e iωn f ) + Kϕ ] /C ′ = ψ t + (1 − t )Re(( e iωn − f ) /C ′ . (3.10)Choose ε < κC ′ | − t |k f k ∞ and define Q ε = { n ∈ N : | e iωn − | < ε } . Notice that Q ε has boundedgaps, i.e., there exists a K = K ( ε ) such that for any n ≥ K , there is a k ≤ K such that n − k ∈ Q ε .It is clear from (3.10) that λ − n L n ψ t ( x ) ≥ ψ t ( x ) − | − t |k f k ∞ /C ′ ε ≥ κ/ n ∈ Q ε and x ∈ Z . 19ix n ≥ N + K and choose k such that N ≤ k ≤ N + K and n − k ∈ Q ε . Note that forany ℓ , inf ∆ ℓ g may be 0 if there are infinitely many Z ⊂ ∆ ℓ with R ( Z ) = 1. However, sincewe only require returns to finitely many Z ′ ∈ Z im for finitely many times, N ≤ k ≤ N + K ,we may choose a set W ⊂ S ℓ ≤ N + K ∆ ℓ containing only finitely many Z such that for each x ∈ ∆ there is a point y ∈ Z such that F k y = x and F i y ∈ W for 0 ≤ i ≤ k − n − k ∈ Q ε , we estimate λ − n L n ψ t ( x ) = λ − k L k ( λ k − n L n − k ψ t )( x ) = λ − k X F k y = x λ k − n L n − k ψ t ( y ) g k ( y ) ≥ λ − k ( λ k − n L n − k ψ t )( y ) g k ( y ) ≥ λ − N κ inf W g N K =: κ ′ > . Thus inf ∆ λ − n L n ψ t ≥ κ ′ for all n ≥ N + K .Now on ∆ ℓ , for n ≥ ℓ + N + K , λ − n L n ψ t ( x ) = λ − ℓ λ ℓ − n L n − ℓ ψ t ( F − ℓ x ) ≥ λ − ℓ κ ′ . Thereforefor large n , inf ∆ ℓ λ − n L n ψ t ≥ κ ′ for all ℓ ≤ n − N − K . Since ψ t = lim k λ − n k L n k ψ t , we haveinf ∆ ψ t ≥ κ ′ .
6. Mixing implies N = 1 . Suppose that L N ϕ = λ N ϕ . The proofs of Steps 2 and 4go through with L replaced by L N , implying that λ N is a simple eigenvalue for L N . (Theproofs are modified in the obvious way. For example, ∆ is replaced by ∆ ∪ · · · ∪ ∆ N − andmixing is used instead of transitivity.) But L N ( L ϕ ) = λ N L ϕ , so we deduce that L ϕ = cϕ for some c ∈ R , with c N = λ N . Positivity of L implies that c >
0, so c = λ . Hence L ϕ = λϕ , that is N = 1.
7. Nonmixing case.
First suppose that F is transitive with period p . Then F p has p distinct components in ∆ and is mixing on each of them. Applying (i) to L p implies that λ p is an eigenvalue of algebraic and geometric multiplicity p and there are no further eigenvalueson or outside the circle of radius λ p . The corresponding eigenvalues for L lie at p th roots of λ p , and it follows easily from transitivity that all p th roots are realized by simple eigenvalues,proving (ii) .Finally, since L is quasi-compact, there are only finitely many transitive components of∆. Restricting to a single component, (iii) reduces to the transitive case (ii) . ∆ ∞ Proof of Proposition 2.8.
We assume that F is mixing and as usual denote by ϕ the uniqueeigenvector with eigenvalue λ . We divide the proof into three parts. (i) Existence of ν . Let f ∈ B . Since ϕ | ∆ ℓ ∼ λ − ℓ where λ > β , it follows from the definitionsof B and B that ϕf ∈ B . By Corollary 2.5, F ( f ) := lim n →∞ λ − n ϕ − L n ( ϕf ) = c ( ϕf ) . (3.12)Hence (3.12) defines a linear functional F : B → R . We also have |L n ( ϕf ) | ≤ | f | ∞ L n ϕ = | f | ∞ λ n ϕ , so that |F ( f ) | ≤ | f | ∞ .Since F is a bounded linear functional on B , there exists a measure ν such that F ( f ) = R f dν for each f in B . Since F (1) = 1, ν is a probability measure. Notice also that we can20rite λ − n L n ( ϕf ) → ϕν ( f ) where convergence takes place in B and hence in L ( m ). Since R ∆ ϕ dm = 1, it follows that ν ( f ) = lim n →∞ λ − n Z ∆ L n ( ϕf ) dm = lim n →∞ λ − n Z ∆ n f ϕ dm = lim n →∞ λ − n Z ∆ n f dµ so that ν is supported on ∆ ∞ . Also, from (3.12) it follows that c ( f ) = ν ( ϕ − f ) for each f ∈ B .Note that L ( ϕ f ◦ F ) = f L ϕ = λϕf and so F ( f ◦ F ) = lim n →∞ λ − n ϕ − L n ( ϕ f ◦ F ) = lim n →∞ λ − n ϕ − L n − ( λϕf )= lim n →∞ λ − ( n − ϕ − L n − ( ϕf ) = F ( f ) . Hence ν is an invariant measure for F (and ˆ F , since F = ˆ F on ∆ ∞ ). (ii) ν is ergodic. Since F is transitive on Z im , given Z ′ , Z ′ ∈ Z im , we may choose n ∈ N suchthat F n ( Z ′ ) ⊇ Z ′ . Since ∆ ∞ is an F -invariant set, this implies that F n ( Z ′ ∩ ∆ ∞ ) ⊇ Z ′ ∩ ∆ ∞ .So F | ∆ ∞ is transitive.Let Z ni ⊂ ∆ ∞ denote a cylinder set of length n with respect to the partition Z ∩ ∆ ∞ .Now suppose A = S i,n Z ni is a countable union of such cylinder sets with F − A = A and ν ( A ) >
0. Since A is a countable union, we must have ν ( Z ni ) > i and n . Thisimplies that F n ( Z ni ) = Z ∩ ∆ ∞ for some Z ∈ Z , and F n + R ( Z ) ( Z ni ) ⊇ Z ′ ∩ ∆ ∞ for some Z ′ ∈ Z im . In particular, Z ′ ∩ ∆ ∞ ⊂ A . Since F is transitive on ∆ ∞ , ∪ k ≥ F k ( Z ′ ∩ ∆ ∞ ) = ∆ ∞ .Thus A = ∆ ∞ so ν ( A ) = 1.Since Z is a generating partition on ∆ ∞ , we conclude that ν is ergodic. (iii) Exponential decay of correlations. Let f , f ∈ B . Recall that ν ( f ) = c ( f ϕ ). Bydefinition of ν , Z ∆ ∞ f f ◦ F n dν − ν ( f ) ν ( f ) = lim k →∞ λ − k Z ∆ k f f ◦ F n ϕ dm − Z ∆ ∞ ν ( f ) f dν = lim k →∞ λ − k Z ∆ k − n L n ( f ϕ ) f dm − lim k →∞ λ n − k Z ∆ k − n ν ( f ) f ϕ dm = lim k →∞ λ n − k Z ∆ k − n [ λ − n L n ( f ϕ ) − c ( f ϕ ) ϕ ] f dm = lim k →∞ λ − k X ℓ ≥ Z ∆ k ∩ ∆ ℓ [ λ − n L n ( f ϕ ) − c ( f ϕ ) ϕ ] f dm. Recall that f ϕ ∈ B . For F mixing, it follows from Corollary 2.5 that (cid:12)(cid:12)(cid:12)Z ∆ k ∩ ∆ ℓ [ λ − n L n ( f ϕ ) − c ( f ϕ ) ϕ ] f dm (cid:12)(cid:12)(cid:12) ≤ | ∆ ℓ ( λ − n L n ( f ϕ ) − c ( f ϕ ) ϕ ) | ∞ | f | ∞ m (∆ k ∩ ∆ ℓ ) ≤ k λ − n L n ( f ϕ ) − c ( f ϕ ) ϕ k β − ℓ | f | ∞ m (∆ k ∩ ∆ ℓ ) ≤ C k f ϕ k| f | ∞ σ n β − ℓ m (∆ k ∩ ∆ ℓ ) . (cid:12)(cid:12)(cid:12)Z ∆ ∞ f f ◦ F n dν − ν ( f ) ν ( f ) (cid:12)(cid:12)(cid:12) ≤ lim k →∞ λ − k X ℓ ≥ C k f ϕ k| f | ∞ σ n β − ℓ m (∆ k ∩ ∆ ℓ )= C k f ϕ k| f | ∞ σ n lim k →∞ λ − k Z ∆ k f β dm, where f β | ∆ ℓ := β − ℓ . In particular, f β ∈ B . By Corollary 2.5, λ − k L k f β converges to c ( f β ) ϕ in B , and hence in L ( m ) so that lim k →∞ λ − k R ∆ k f β dm = c ( f β ), completing the proof. ∆ Notice that the functional analytic approach adopted thus far only tells us that λ representsthe slowest rate of escape from ∆ for elements of B , but in general there are functions whichescape at faster rates. The estimates on the functional F in Section 3.3 and the existenceof the invariant measure ν allow us to establish the uniformity of escape rates for certainfunctions in B . Since the indicator functions of elements of the partition Z are in thisspace, we also obtain uniform escape rates of mass from certain sets and in particular forthe reference measure m on the tower. Proposition 3.3.
Let F be mixing and satisfy properties (P1)-(P3) and (H1) of Section 2.1.1.For each f ∈ B with f ≥ , we have ν ( f ) > if and only if lim n →∞ L n f |L n f | = ϕ (3.13) where as usual, the convergence is in the k · k -norm. In particular, the reference measureconverges to the a.c.c.i.m. Proof.
By Proposition 2.6, equation (3.13) holds if and only if c ( f ) >
0. Thus it suffices toprove ν ( f ) > c ( f ) > ν ( f ) = c ( ϕf ) ≥ δc ( f ) since ϕ ≥ δ . So c ( f ) > ν ( f ) > f ∈ B and suppose ν ( f ) >
0. Let ∆ nℓ = ∆ ℓ ∩ ∆ n be the subset of ∆ ℓ which hasnot escaped by time n . Set ∆ n ( K ) = ∪ Kℓ =0 ∆ nℓ and ∆ n + = ∆ n \ ∆ n ( K ) .For ε ∈ (0 , K such that ν (∆ n + ) | f | ∞ < εν ( f ). Then ν ( f ) = lim n →∞ λ − n Z ∆ n ( K ) f dµ + λ − n Z ∆ n + f dµ ! ≤ lim n →∞ λ − n λ − K δ − Z ∆ n ( K ) f dm + λ − n | f | ∞ Z ∆ n + dµ ! ≤ λ − K δ − c ( f ) + | f | ∞ ν (∆ n + ) ≤ λ − K δ − c ( f ) + εν ( f ) . Since ε ∈ (0 , c ( f ) > ν (1) = 1, the normalized push forward of the reference measure m converges to µ as n → ∞ . 22 orollary 3.4. Let A = ∪ ( ℓ,j ) ∈ J Z ℓ,j be a union of partition elements such that ν ( A ) > .Then there exists C > such that C − λ n ≤ m (∆ n ∩ A ) ≤ Cλ n (3.14) for each n ∈ N so that mass with respect to m escapes from A at a uniform rate matchingthat of the conditionally invariant measure.Proof. First note that for any f ∈ B , we have |L n f | ≤ kL n f k X ℓ ≥ β − ℓ m (∆ ℓ ) ≤ C kL n kk f k ≤ Cλ n k f k , so that the upper bound in (3.14) is trivial.Let χ A be the indicator function for A and notice that χ A ∈ B . Integrating the limit inCorollary 2.5, we get c ( χ A ) = lim n →∞ λ − n Z ∆ L n χ A dm = lim n →∞ λ − n Z ∆ n χ A dm. Since c ( χ A ) > m (∆ n ∩ A ) forms a decreasing sequence, there mustexist a C > λ − n m (∆ n ∩ A ) ≥ C − for all n . Corollary 3.5.
Let Z = Z ℓ,j be a cylinder set and let n > R ( Z ) . There exists a constant C > , independent of Z , such that if ∆ n ∩ Z = ∅ , then C − λ n − R m ( Z ) ≤ m (∆ n ∩ Z ) ≤ Cλ n − R m ( Z ) . Proof.
By bounded distortion, we have m (∆ n ∩ Z ) | ( F R ) ′ ( y ) | = m (∆ n − R ∩ F R Z ) for some y ∈ Z . Since F R Z = Z ′ for some Z ′ ∈ Z ′ and Z ′ is finite, by Corollary 3.4, we can find C independent of Z ′ such that C − λ n ≤ m (∆ n − R ∩ Z ′ ) ≤ Cλ n . We complete the proof by noting that | ( F R ) ′ ( y ) | ≈ c /m ( Z ). Corollary 3.6.
Let f ∈ B , f ≥ , such that ν ( x ∈ ∆ : f ( x ) > > . Then lim n →∞ L n f |L n f | = ϕ. Proof.
Let h = min { f, } and note that h ∈ B . Also ν ( h ) > f since h and f share the same support. Thus c ( h ) > c ( f ) = lim n →∞ λ − n Z ∆ n f dm ≥ lim n →∞ λ − n Z ∆ n h dm = c ( h ) > f holds by Proposition 2.6. 23 Applications
We set up our notation as follows. Let ˆ T be a piecewise C α self-map of a metric space( ˆ X, ˜ d ) with open hole ˜ H . Let ˜ m be a probability measure on ˆ X and let ˜ g = d ˜ md ( ˜ m ◦ ˆ T ) . Supposethat a tower ( ˆ F , ˆ∆) with hole H and the properties of Section 2.1.1 can be constructed overa reference set Λ. This implies that there exists a countable partition Z of Λ, a coarserpartition Z im , also of Λ, and a return time function R which is constant on elements of Z and for which ˆ T R ( Z ) ∈ Z im or ˆ T R ( Z ) ⊂ ˜ H for each Z ∈ Z . The set Λ is identified with∆ and each level ∆ ℓ is associated with ∪ R ( Z ) >ℓ T ℓ ( Z ). This defines a natural projection π : ˆ∆ → ˆ X so that π ◦ ˆ F n = ˆ T n ◦ π for each n . In general, we may choose Λ so thatΛ ∩ H = ∅ .Following our previous notation, we define X = ˆ X \ ˜ H and X n = ∩ ni =0 ˆ T − i X . The re-stricted maps are then F n = ˆ F n | ∆ n on the tower and T n = ˆ T n | X n on the underlying space.We use the reference measure ˜ m on ˆ X to define a reference measure m on ∆ by letting m | ∆ = ˜ m | Λ and then simply defining m on subsequent levels by m ( A ) = m ( ˆ F − ℓ A ) formeasurable A ⊂ ˆ∆ ℓ . As before, we let g = dmd ( m ◦ ˆ F ) .Given a measure µ on ˆ∆, we define its projection ˜ µ onto ˆ X , by ˜ µ = π ∗ µ . In terms ofdensities, this implies that if dµ = f dm , then for almost every u ∈ ˆ X , the density ˜ f of ˜ µ isgiven by P π f ( u ) = X x ∈ π − u f ( x ) /J π ( x )where J π = d ( ˜ m ◦ π ) dm . Note that |P π f | L ( X, ˜ m ) = | f | L (∆ ,m ) . Since Radon-Nikodym derivativesmultiply, we have ˜ g n ( πy ) /J π ( y ) = g n ( y ) /J π ( ˆ F n y ) (4.1)for almost every y ∈ ˆ∆ and each n ≥
0. This in turn implies that P π ( L nF f ) = L nT ( P π f ) (4.2)for f ∈ L (∆). The importance of these relations lies in the fact that if ϕ satisfies L F ϕ = λϕ and ˜ f = P π f , then L nF f |L nF f | → ϕ in L ( m ) implies L nT ˜ f |L nT ˜ f | → P π ϕ =: ˜ ϕ in L ( ˜ m ) (4.3)and ˜ ϕ satisfies L T ˜ ϕ = P π ( L F ϕ ) = λ ˜ ϕ so that ˜ ϕ defines a conditionally invariant measurefor T with the same eigenvalue as ϕ .However, the space P π B is not well understood and functions in P π B are a priori no betterthan L . It is not even clear that the constant function corresponding to the original referencemeasure ˜ m is in P π B . Getting a handle on a nice class of functions in P π B is necessary forshowing in particular applications that, for example, Lebesgue measure converges to the a.c.c.i.m. according to the results of the previous section.In what follows, we identify two properties, (A1) and (A2), that guarantee C ¯ α ( X ) ⊂P π B where ¯ α depends on the smoothness and average expansion of T . (A1) is standard in24onstructions of Young towers and (A2) can be achieved with no added restrictions on themap or types of holes allowed. In Sections 4.2 and 4.3, we prove that the towers we constructhave these properties.Let R n ( x ) = R n − ( T R ( x ) ( x )) be the n th good return of x to Λ, for n ≥ (A1) There exist constants τ > C , C > x ∈ Λ, n ≥ k < R n ( x ), | DT R n ( x ) − k ( T k x ) | > C τ R n ( x ) − k .(b) Let x, y ∈ Z ,j and R = R ( Z ,j ). Then (cid:12)(cid:12)(cid:12) ˜ g ℓ ( πx )˜ g ℓ ( πy ) (cid:12)(cid:12)(cid:12) ≤ C for ℓ ≤ R . If T R ( Z ,j ) ⊆ Λ, then (cid:12)(cid:12)(cid:12) ˜ g R ( πx )˜ g R ( πy ) − (cid:12)(cid:12)(cid:12) ≤ C d ( T R ( πx ) , T R ( πy )) α .Property (A1)(a) says that although T may not be expanding everywhere in its phasespace, we only count returns to Λ during which average expansion has occurred. Property(A1)(b) is simply bounded distortion. In fact, (A1) implies the distortion bound (2.1) aswell as (P2) in the towers we use. X Recall that ˜ d is the metric on X and d is the symbolic metric on ∆ defined in Section 2.1.1.Under assumption (A1)(a), these two metrics are compatible in the following sense. Lemma 4.1.
For any ¯ α ≥ − log β/ log τ , let ˜ f ∈ C ¯ α ( X ) and define f on ∆ by f ( x ) = ˜ f ( πx ) for each x ∈ ∆ . Then f ∈ B and k f k ≤ C − | ˜ f | C ¯ α .Proof. First we show that Lip( f ) = sup ℓ,j Lip( f ℓ,j ) < ∞ . Let x, y ∈ Z ℓ,j and let ˜ x = πx and˜ y = πy . Then | f ( x ) − f ( y ) | d ( x, y ) = | ˜ f (˜ x ) − ˜ f (˜ y ) | ˜ d (˜ x, ˜ y ) ¯ α · ˜ d (˜ x, ˜ y ) ¯ α d ( x, y ) ≤ C ¯ α, ˜ f ˜ d (˜ x, ˜ y ) ¯ α d ( x, y ) . (4.4)Note that d ( x, y ) = β s ( x,y ) and that s ( x, y ) is a return time for ˜ x and ˜ y so that | DT s ( x,y ) | ≥ C τ s ( x,y ) on Z ℓ,j by Property (A1)(a). Thus˜ d (˜ x, ˜ y ) = ˜ d (˜ x, ˜ y )˜ d ( T s ( x,y ) (˜ x ) , T s ( x,y ) (˜ y )) ˜ d ( T s ( x,y ) (˜ x ) , T s ( x,y ) (˜ y )) ≤ C − τ − s ( x,y ) diam(Λ) . This, together with (4.4), implies that Lip( f ) < ∞ since β ≥ τ − ¯ α . Also | f | ∞ = | ˜ f | ∞ < ∞ ,so f ∈ B .The problem is that in general P π ( ˜ f ◦ π ) = ˜ f , so Lemma 4.1 does not imply that C α ( X ) ⊂ P π B immediately. P π Given ˜ f ∈ C α ( X ), we want to construct f ∈ B so that P π f = ˜ f . To do this, it is sufficientto have the following property on the tower constructed above the reference set Λ.25 A2)
There exists an index set J ⊂ N × N such that(a) ˜ m ( X \ ∪ ( ℓ,j ) ∈ J π ( Z ℓ,j )) = 0;(b) π ( Z ℓ ,j ) ∩ π ( Z ℓ ,j ) = ∅ for all but finitely many ( ℓ , j ), ( ℓ , j ) ∈ J ;(c) Define J π ℓ,j := J π | Z ℓ,j . Then sup ( ℓ,j ) ∈ J | J π ℓ,j | ∞ + Lip( J π ℓ,j ) =
D < ∞ . Proposition 4.2.
Let T be a piecewise C α self-map of a metric space ( X, ˜ d ) with hole ˜ H . Suppose we can construct a Young tower over a reference set Λ for which T satisfiesproperties (A1) and (A2). Then C ¯ α ( X ) ⊂ P π B for every − log β/ log τ ≤ ¯ α ≤ α .Proof. Let ˜ f ∈ C ¯ α ( X ) be given.If π ( Z ℓ,j ) ∩ π ( Z ℓ ′ ,j ′ ) = ∅ for all other ( ℓ ′ , j ′ ) ∈ J , then we can choose a single preimagefor each u ∈ π ( Z ℓ,j ) on which to define f . In fact, inverting the projection operator P π , wesee that defining f ( x ) = ˜ f ( πx ) J π ( x ) for each x ∈ Z ℓ,j yields the correct value for ˜ f ( πx ).Now consider the case in which π ( Z ℓ ,j ) ∩ π ( Z ℓ ,j ) = ∅ . We may choose a partition ofunity { ρ , ρ } for E = π ( Z ℓ ,j ∪ Z ℓ ,j ) such that ρ i ∈ C α ( E ). Then we define f by f ℓ i ,j i ( x i ) = ˜ f ( πx i ) J π ( x i ) ρ i ( πx i )for x i ∈ Z ℓ i ,j i and i = 1 ,
2. Then for u ∈ E , we set f = 0 on preimages of u which are not in Z ℓ ,j ∪ Z ℓ ,j . It is clear that P π f ( u ) = ˜ f ( u ) for u ∈ E .This construction can be generalized to accommodate finitely many overlaps in the pro-jections π ( Z ℓ,j ) while maintaining a uniform bound on the C α -norm of the ρ i .Let Z J = ∪ ( ℓ,j ) ∈ J Z ℓ,j . Lemma 4.1 tells us that ˜ f ◦ π ∈ B (where B is defined in (2.3))and (A2)(c) implies that J π | Z J ∈ B . Since f ≡ Z J , it follows immediately that f ∈ B . Proof of Theorem 2.10.
Theorem A guarantees that T admits a tower ( F, ∆) satisfying prop-erties (P1)-(P3) and (H1). Property (A1) is automatic for expanding maps.It remains to verify that Property (A2) is satisfied. This follows from the tower construc-tion contained in [D1]. For this class of maps, we may choose the reference set Λ to be aninterval of monotonicity of T and the finite partition of images Z im will consist of the singleelement Λ, i.e., we have a tower with full returns to the base. In the inductive constructionof the partition Z on Λ, at each step, new pieces are created only by intersections withdiscontinuities, intersections with the hole, and returns to the base. In this way, only finitelymany distinct pieces are generated by each iterate and therefore we have only finitely manyoverlaps when we project each level. Since I is covered in finitely many iterates of Λ byassumption (T1), it is also covered by the projection of finitely many levels of ∆, say thefirst N . Thus if we take our index set J to be all indices corresponding to elements in thefirst N levels of the tower, it is immediate that (A2)(a) and (A2)(b) are satisfied.To see that (A2)(c) is satisfied, let x ∈ ∆ and notice that by (4.1), J π ( F ℓ x ) = J π ( x ) g ℓ ( x ) / ˜ g ℓ ( πx ). If ℓ < R ( x ), then J π ( x ) = g ℓ ( x ) = 1 so that J π ( F ℓ x ) = 1 / ˜ g ℓ ( πx ) = | ( T ℓ ) ′ ( πx ) | . (4.5)26ince T is C α , so is T ℓ for each ℓ . Since we are only concerned with ℓ ≤ N and ¯ α ≤ α , byLemma 4.1, J π | Z J ∈ B so (A2)(c) is satisfied. By Proposition 4.2, we have C α ( X ) ⊂ P π B .Property (T1) also implies that we can construct ( F, ∆) to be mixing, since if T n ( Z ′ ) ⊇ I ,then T n +1 ( Z ′ ) ⊃ I so we can avoid periodicity in the return time R by simply delaying areturn by 1 step. Applying Proposition 2.4, we see that L F admits a unique probabilitydensity ϕ for the eigenvalue λ of maximum modulus. Defining ˜ ϕ = P π ϕ , we have L nT ˜ f |L nT ˜ f | → ˜ ϕ at an exponential rate for every ˜ f ∈ P π B for which c ( ˜ f ) > Convergence property.
Let ˜ f ∈ G . Since ( F, ∆) satisfies (A1) and (A2) and ˜ f ∈ C α ( I ),by Proposition 4.2 we can find f ∈ B , supported entirely in elements corresponding to theindex set J , such that P π f = ˜ f . By Corollary 3.6, it suffices to show that ν ( f ) >
0, for thenthe convergence of f to ϕ will imply the convergence of ˜ f to ˜ ϕ := P π ϕ .Since ˜ f ∈ G , we have ˜ f > I ∞ ∩ Λ which implies f > ∞ ∩ ∆ . Since ν is aninvariant measure on ∆, it must be that ν (∆ ) > ν ( f ) > Unified escape rate.
Finally we prove that all functions in G have the same escape rate givenby − log λ . First note that given ˜ f ∈ G and f ∈ B such that P π f = ˜ f , we havelim n →∞ λ − n L nT ˜ f = lim n →∞ λ − n |L nT ˜ f | L nT ˜ f |L nT ˜ f | = lim n →∞ λ − n |L nF f | L nT ˜ f |L nT ˜ f | = c ( f ) ˜ ϕ by Corollary 2.5 and the proof of convergence above. Since ν ( f ) >
0, we also have c ( f ) > η = ˜ f ˜ m , we havelim n →∞ n log ˜ η ( I n ) = lim n →∞ n log |L nT ˜ f | = log λ. Proof of Theorem 2.12.
The construction in [DHL] fixes δ and finds an interval I ∗ with c ∗ ∈ I ∗ ⊂ ( c ∗ − δ, c ∗ + δ ) as base for the induced map. We choose I ∗ such that orb( ∂I ∗ ) isdisjoint from the interior of I ∗ . This is always possible by choosing ∂I ∗ to be pre-periodic.Now by using I ∗ as the base ˆ∆ of the Young tower ˆ∆ (i.e., without hole), and recalling that Z is the natural partition of the tower we have the following:For any Z, Z ′ ∈ Z , the symmetric difference πZ △ πZ ′ = ∅ . (4.6)To show why this is true, write Z = Z ℓ,j and Z ′ = Z ℓ ′ ,j ′ , so π ( Z ) = T ℓ ( πZ ,j ) and π ( Z ′ ) = T ℓ ′ ( πZ ,j ′ ). Assume without loss of generality that k := R ( Z ) − ℓ ≥ R ( Z ′ ) − ℓ ′ =: k ′ .If (4.6) fails, then there are x ∈ ∂πZ ∩ πZ ′ and x ′ ∈ πZ ∩ ∂πZ ′ . But then ˆ T k ( x ′ ) is an interiorpoint of I ∗ , but at the same time ˆ T k ( x ′ ) = ˆ T k − k ′ ( ˆ T k ′ ( x ′ )) ∈ ˆ T k − k ′ ( ∂I ∗ ). This contradictsthe choice of I ∗ . We record property (4.6) for later use in checking condition (A2)(b).Next we adapt the construction of the inducing for the system without hole from [DHL].By (B2)(a) the artificial critical points b j ∈ ˜ H j satisfy ˆ T k ( b j ) = c ∗ for all k ≥
0. Therefore(C3) still holds with the artificial critical points. We set the binding period of x ∈ B δ ( b j )27see [DHL, Section 2.2]) to p ( x ) = τ ( j ) − τ ( j ) as in (2.5). Recall that the hole ˜ H has L components. When the image ˆ T n ( ω ) = ω n of a partition element ω visits B δ ( b j ) (see [DHL,page 432]), we subdivide ω only if ω n intersects ∂ ˜ H j . If ω n has not escaped to large scale, so | ω n | < δ , this results in at most 3 subintervals ω ′ ⊂ ω such that ˆ T n ( ω ′ ) is either contained in˜ H j or disjoint from ˜ H . By (2.5) and our choice of binding period p | B δ ( b j ) , Lemma 2 of [DHL]is automatically satisfied for θ ∗ := θ = ˆ θ = λ ∗ . Remark 4.3.
In [DHL] close visits to Crit c ∪ Crit s that result in a cut are called essentialreturns , whereas those that do not result in a cut are called inessential returns . Let us callcuts caused by ∂ ˜ H hole returns . The cutting of ω ′ at preimages of ∂ ˜ H j is crucial for ourtower to be compatible with the hole. Note also that by the slow recurrence condition (B1), acutting of ω ′ cannot occur within a binding period after a previous visit to a point in Crit c . With this adaptation, the tower construction of [DHL] yields a tower ˆ∆ and a returntime function ˆ R , constant on elements of the partition Z . According to [DHL, Theorem 1],the tower ( ˆ F , ˆ∆ , ˆ R ) satisfies (P1)-(P3) and (A1) of the present paper.Notice that at this point there is no escape. We have simply introduced new cuts at theboundaries of the hole during the construction of the return time function and partition ofthe interval I ∗ so that the induced tower respects the boundary of the hole in the followingsense: For each Z ∈ Z , either πZ ⊂ ˜ H or πZ ∩ ˜ H = ∅ .A crucial feature of this construction is that the exponential rate θ of the tail behavior isindependent of the size of ˜ H when ˜ H is small. To see this, recall the notation introduced inSection 2.4.2 regarding small holes. We first fix the set of points b , . . . b L , which we regardas infinitesimal holes satisfying (B1) and (B2). Then for each h >
0, the family of holes H ( h ) consists of those holes ˜ H satisfying: (1) b j ∈ ˜ H j and ˜ m ( ˜ H j ) ≤ h for each 1 ≤ j ≤ L ;and (2) ˜ H satisfies (B1).For the infinitesimal hole ˜ H (0) with components ˜ H (0) j = b j , j = 1 , . . . , L , we fix δ > I ∗ ⊂ B δ ( c ∗ ) and construct a tower ∆ (0) incorporating the additionalcuts at ∂ ˜ H (0) as described above.An immediate concern is that the presence of additional cuts when we introduce holes ofpositive size interferes with returns to the extent that all full returns to I ∗ are blocked. Thefollowing lemmas guarantee that this is not the case and in fact several properties such asmixing and the rate of returns persist for small holes. Lemma 4.4.
For sufficiently small h , each ˜ H ∈ H ( h ) induces a tower ˆ∆ ( ˜ H ) and returntime function ˆ R ( ˜ H ) over I ∗ using the construction described above. Moreover, ( ˆ F ( ˜ H ) , ˆ∆ ( ˜ H ) ) is mixing if ( ˆ F (0) , ˆ∆ (0) ) is mixing.Proof. Notice that the thickening of the hole at the points b j cannot affect returns whichhappen before a fixed time n h depending only on h . For suppose ω ⊂ I ∗ satisfies ˆ T n ω = I ∗ where n = ˆ R (0) ( ω ) ≤ n h is the return time corresponding to ˜ H (0) . Then in fact ω is in themiddle third of a larger interval ω ′ such that ˆ T n ω ′ ⊃ I ∗ . Cuts made by ∂ ˜ H j must necessarilybe at the endpoints of ω ′ so for sufficiently small h , the return of ω will still take place attime n . By (B2)(a), we can force n h → ∞ as h →
0, guaranteeing the persistence of returnsup to any finite time for sufficiently small h . 28o show ( ˆ F ( ˜ H ) , ˆ∆ ( ˜ H ) ) is mixing, we need only show that g.c.d.( ˆ R ( ˜ H ) ) = 1 since ˆ∆ ( ˜ H ) has asingle base. Since ( ˆ F (0) , ˆ∆ (0) ) is mixing, there exists N such that g.c.d. { ˆ R (0) : ˆ R (0) ≤ N } = 1.Now take h small enough that n h ≥ N . Then g.c.d. { ˆ R ( ˜ H ) } = 1 as well.Our next lemma shows that the rate of return is uniform for small h . Lemma 4.5.
There exist θ < and C > such that ˜ m ( ˆ R ( ˜ H ) > n ) ≤ Cθ n for all ˜ H ∈ H ( h ) with h sufficiently small.Proof. Let θ be the exponential rate of the tail behavior corresponding to ˜ H (0) . We willshow that by choosing δ and h sufficiently small, we can make θ = θ ( ˜ H ) arbitrarily closeto θ for all ˜ H ∈ H ( h ). We do this by showing that the rates of decay given by a series oflemmas in [DHL] vary little for small h . Lemma 1 of [DHL]:
Choose h small enough that n h from the proof of Lemma 4.4satisfies n h ≥ t ∗ in Lemma 1. Then Lemma 1 holds with the same rate since returns in themiddle of large pieces are not affected by the hole before time n h . Lemma 6 of [DHL]:
The notation E n,S ( ω ) stands for the set of subintervals within aninterval ω of size δ/ < | ω | < δ that have not grown to size δ by time n , and have essentialreturn depths summing to S within these n iterates.This lemma estimates the size of any interval ω ′ ∈ E n,S . Let us denote the number ofhole returns used in the history of ω ′ by S hole . Define E n,S,S hole to be the set of subintervals ω ′ ∈ E n,S such that ω ′ has S hole hole returns in its history up to time n .Every hole return, i.e., a cut at ∂ ˜ H j , is followed by a binding period of length τ ( j ) inwhich derivatives grow by an extra factor of 6 by (2.5). Since Lemma 6 is concerned onlywith derivatives, and not with the actual cutting, the conclusion of Lemma 6 becomes: Forevery n ≥ S ≥ S hole ≥ ω ′ ∈ E n,S,S hole we have | ω ′ | ≤ κ − e − θ ∗ S − S hole . where θ ∗ replaces the θ used in [DHL, Lemma 2]. Lemma 7 of [DHL]:
This lemma relies on combinatorial estimates to obtain an upperbound on the number of pieces which can grow to size δ at specific times. By specifying thenumber of hole returns by S hole and using the fact that intervals are cut into at most 3 piecesduring a hole return, we can adapt the conclusion of Lemma 7 to E n,S,S hole ( ω ) ≤ e ˜ ηS S hole . Combining Lemmas 6 and 7 in this form gives |E n,S ( ω ) | = X S hole ≥ |E n,S,S hole ( ω ) |≤ X S hole ≥ κ − e − θ ∗ S − S hole e ˜ ηS S hole = κ − e − ( θ ∗ − ˜ η ) S (4.7)which is precisely formula (21) in [DHL].The free time of an interval ω ′ are all the iterates not spent in a binding period. Wesuppose that ω ′ escapes to ‘large scale’ at time n (i.e., | ˆ T n ( ω ′ ) | ≥ δ ) and consider its history29ntil time n . If ω ′ is cut very short at a hole return, say ∂ ˆ T m ( ω ′ ) ∩ ∂ ˜ H j = ∅ , then we firsthave a binding period of length p ( x ) = τ ( j ) −
1, and the free period after that lasts untileither: (i) ω ′ reaches large scale, (ii) ω ′ has the next artificial cut near b j ∈ Crit hole , (iii) ω ′ has an inessential return near c ∈ Crit c ∪ Crit s . or (iv) ω ′ has the next essential returnnear c ∈ Crit c ∪ Crit s . In case (iv), ˆ T k ( ω ′ ) covers at least three intervals in the exponentialpartition of B δ ( c ) as in [DHL, page 433].Let us call the time from iterate m + τ ( j ) − extended free period of ω after iterate m + τ ( j ) − B δ ( c ) for c ∈ Crit c ∪ Crit s . These are the returns whereˆ T k ( ω ′ ) is too short to result in a cut. Condition (B1) implies that when such an inessentialreturn occurs, the next cut or inessential return will not occur until after the binding periodassociated to dist( ˆ T k ( ω ′ ) , c ). This binding period will restore the small derivative incurredat time k due to [DHL, Lemma 2]. Hence there is λ hole , depending only on λ ∗ and Λ ∗ fromconditions (C1) and (C2), such that | D ˆ T ℓ ( x ) | ≥ e λ hole ℓ (4.8)for each x ∈ ˆ T m + τ ( j ) − ( ω ′ ) and ℓ is the length of this extended free period. We let n hole denote the sum of extended free periods directly following the binding periods due to holereturns in the history of ω ′ up until time n . With this notation, we make adaptations to theremaining lemmas. Lemma 8 of [DHL]:
This lemma can be changed to: there exists n δ such that for every ω ′ ∈ E n, with S hole = 0, ω ′ = ω and n ≤ n δ . The reason is that intervals of definite sizecannot remain small forever if they are not cut during an essential return or hole return, andin fact the n δ can be taken equal to the n ( δ ) used in condition (B2). Lemma 9 of [DHL]:
This lemma can be restated as: for all n ≥ S ≥ E n,S = ∅ , we have S ≥ ( n − n hole − n δ ) / ˜ θ. In other words, we disregard the hole free time n hole . The proof is basically the same as in[DHL] if we keep in mind that at an essential return to c ∈ Crit c ∪ Crit s at time ℓ , following ahole return, the size of the interval ˆ T ℓ ( ω ′ ) that emerges from the cut at this essential returndepends only on the distance of ˆ T ℓ ( ω ′ ) to c .Now Lemmas 8 and 9 of [DHL] and (4.7) combine to give for E n ( ω ) := ∪ S ≥ E n,S ( ω ): |E n ( ω ) | = X ≤ n hole ≤ n X S ≥ ( n − n hole − n δ ) / ˜ θ |E n,S, ( ω ) |≤ X ≤ n hole ≤ n e − λ hole n hole X S ≥ ( n − n hole − n δ ) / ˜ θ κ − e − ( θ ∗ − ˜ η ) S ≤ ˜ C e ( θ ∗ − ˜ η ) n δ / ˜ θ e − min { ( θ ∗ − ˜ η ) / ˜ θ , λ hole } n , for some ˜ C as in the formula given near the bottom of [DHL, page 444]. Having established Lemmas 6 to 9, the rest of the proof in [DHL] goes through basicallyunchanged since the decay in ˆ R ( ˜ H ) depends only on the rates in these lemmas and distortion with the factor κ − inserted where it is missing in [DHL]. H . We see that θ ( ˜ H ) can be made arbitrarily close to θ for h sufficiently small.We have shown that in the presence of additional cuts introduced by ∂ ˜ H , we retain someuniform control over the induced towers ( ˆ F ( ˜ H ) , ˆ∆ ( ˜ H ) ). We are now ready to lift the holesinto the towers and consider the open systems so defined.We define the hole in the tower and the return time with hole to be H = { Z ∈ Z : πZ ⊆ ˜ H } and R ( x ) = min { ˆ R ( x ) , min { j : ˆ T j ( x ) ∈ ˜ H }} . For any partition element Z ℓ,j = H ℓ,j that is identified as a hole, we delete all levels in thetower above Z ℓ,j since nothing is mapped to those elements once the hole is introduced. Wedenote the remaining tower with holes by ∆ and define F = ˆ F | ∆ ∩ ˆ F − ∆ to be the correspond-ing tower map.In order to invoke the conclusions of Proposition 2.6 for ( F, ∆), we must check that itshypotheses, (P1)-(P3) and (H1), are satisfied. We then check conditions (A1) and (A2) inorder to project the convergence results from the tower to the underlying system.Properties (P1)-(P3) are automatic for ( F, ∆ , R ) since they hold for ( ˆ F , ˆ∆ , ˆ R ). Step 1. Condition (H1).
We split the sum in (H1) into pieces that encounter ˜ H during theirbound period and those that encounter it when they are free, X ℓ m ( H ℓ ) β − ℓ = X bound m ( H ℓ ) β − ℓ + X free m ( H ℓ ) β − ℓ . To estimate the bound pieces, we use the slow recurrence condition given by (B1). If ω ⊂ I ∗ is some partition element, and c ∈ Crit c the last critical point visited by ω before ω falls into the hole, then dist( ˆ T ℓ c, ∂ ˜ H j ) ≥ δe − α ∗ c ℓ for each j . Therefore, if ˆ T ℓ ω ∩ ˜ H j = ∅ , wemust have δe − α ∗ c ℓ < ˜ m ( ˜ H j ) and so ℓ > − (1 /α ∗ c ) log( ˜ m ( ˜ H j ) /δ ). Thus by Property (P1), X bound m ( H ℓ ) β − ℓ ≤ X ℓ> − α ∗ c log( ˜ m ( ˜ H ) /δ ) m ( ˆ∆ ℓ ) β − ℓ ≤ X ℓ> − α ∗ c log( ˜ m ( ˜ H ) /δ ) Cθ ℓ β − ℓ ≤ C ′ δ ˜ m ( ˜ H ) α ∗ c log( θ − β ) . (4.9)To estimate the free pieces, we will need some facts about the tower without holes, ( ˆ F , ˆ∆).It was shown in [Y2] that ˆ F admits a unique absolutely continuous invariant measure η withdensity ρ ∈ B , ρ ≥ a >
0. Moreover, π ∗ η = ˜ η is the unique SRB measure for ˆ T . BySection 4.1, ˜ ρ = P π ρ is the density of ˜ η .Notice that ρ | ˆ∆ is an invariant density for ˆ F ˆ R so that ˜ ρ := P π ( ρ ˆ∆ ) is an invariantdensity for ˆ T ˆ R . Since π ′ ≡ , we have a ≤ ˜ ρ ≤ A . This implies that we can alsoobtain the invariant density ˜ ρ by pushing forward ˜ ρ under iterates of ˆ T .It is clear that pushing forward ˜ ρ will result in spikes above the orbits of the criticalpoints, hence ˜ ρ is not bounded on ˆ I . However, when an interval ω ⊂ π ( ˆ∆ ) is free at time n , condition (C2) and [DHL, Lemma 1] imply that the push forward of the density on ω attime n will be uniformly bounded. 31efine neighborhoods N k ( ˆ T k c ) of radius δe − α ∗ c k for each c ∈ Crit c . These are preciselythe points starting in B δ ( c ) whose orbits are still bound to c at time k . From the aboveconsiderations, it is clear that outside of the set ∪ c ∈ Crit c ∪ k ≥ N k ( ˆ T k c ), the density ˜ ρ isbounded. This is the sum of the push forwards of ˜ ρ on free pieces. Thus, we may define ameasure ˜ η free = X ( ℓ,j ): Z ℓ,j is free π ∗ η ( Z ℓ,j )whose density with respect to Lebesgue, ˜ ρ free , is bounded on ˆ I . Then since ρ ≥ a >
0, wehave X free m ( H ℓ,j ) ≤ X free η ( H ℓ,j ) /a = ˜ η free ( ˜ H ) /a ≤ C ˜ m ( ˜ H ) . Now set P = − log ˜ m ( ˜ H ). We estimate the contribution from free pieces by X free m ( H ℓ,j ) β − ℓ = X free: ℓ>P m ( H ℓ,j ) β − ℓ + X free: ℓ ≤ P m ( H ℓ,j ) β − ℓ ≤ X free: ℓ>P Cθ ℓ β − ℓ + β − P X free: ℓ ≤ P m ( H ℓ,j ) ≤ C ′ ( θβ − ) P + C ′′ β − P ˜ m ( ˜ H ) ≤ C ′ ˜ m ( ˜ H ) log( βθ − ) + C ′′ ˜ m ( ˜ H ) β . (4.10)Putting together (4.9) and (4.10), we see that the left hand side of (H1) is proportionalto ˜ m ( ˜ H ) γ , for some γ >
0. This quantity can be made sufficiently small to satisfy (H1) bychoosing ˜ m ( ˜ H ) small since θ (and hence β ) are independent of ˜ H by Lemma 4.5. Step 2. Property (A1).
The bounded distortion required by (A1)(b) is satisfied by the cuttingof pieces introduced in the construction of ∆ (see [DHL, Proposition 3]). The expansionrequired by (A1)(a) follows from two estimates: property (C1) guarantees that starting atany x / ∈ B δ (Crit), there is exponential expansion upon entry to B δ (Crit); [DHL, Lemma2] guarantees that exponential expansion occurs at the end of a binding period. Since anyreturn must occur at a free entry to B δ ( c ∗ ), we may concatenate these estimates as manytimes as needed in order to obtain (A1)(a) at any return time ˆ R n . However, once the holeis introduced, a partition element may fall into the hole during a bound period and so thereturn time with hole, R , may be declared when there has not been sufficient expansion tosatisfy (A1)(a). Since this property is only needed to prove Lemma 4.1, we give an alternateproof of this lemma which uses (A1)(a) only for ˆ R . Proof of Lemma 4.1 for ( F, ∆) . First note that because (A1) is satisfied by ( ˆ
F , ˆ∆), Lemma 4.1holds for lifts ˜ f ◦ π of ˜ f ∈ C ¯ α ( ˆ I ), with ¯ α ≥ − log β/ log τ . Here τ is the rate of expansionfrom (A1) and β is the constant chosen for the symbolic metric on ˆ∆ (see Section 2.1.1).The separation time ˆ s ( · , · ) is shortened by the introduction of the hole in the tower sothat the new separation time satisfies s ( x, y ) ≤ ˆ s ( x, y ). Thus the separation time metric isalso loosened on ∆: d β ( x, y ) := β s ( x,y ) ≥ β ˆ s ( x,y ) =: ˆ d β ( x, y ) . (4.11)32hus if f is Lipschitz with respect to ˆ d β on ˆ∆, its restriction to ∆ is also Lipschitz withrespect to d β .Now for ˜ f ∈ C ¯ α ( ˆ I ), with ¯ α ≥ − log β/ log τ , we have ˜ f ◦ π ∈ B ( ˆ∆) by Lemma 4.1. Thenby (4.11), ˜ f ◦ π ∈ B (∆) as well. Step 3. Property (A2).
We focus first on finding an index set J ⊂ N × N such that (A2)(a)is satisfied. The following lemma is the analogue of (2.4) for T , the map with holes. (Seealso [D2, Lemma 5.2].) Lemma 4.6.
Let δ be the radius of B δ ( c ∗ ) as above. Let n = n ( δ ) be defined by (2.4) . For h sufficiently small, given any interval ω ⊂ I such that | ω | ≥ δ/ , we have n [ i =0 T i ω ⊃ I mod 0 Proof.
Suppose there exists an interval A such that A ∩ ( ∪ n i =0 T i ω ) = ∅ . Since A ⊆ ˆ T n ω , wemust have A ∩ ˆ T i k ˜ H k = ∅ for some ˜ H k such that ˜ H k ∩ ˆ T i ′ k ω = ∅ for some integers i k , i ′ k with i k + i ′ k = n . In other words, the piece of ω that should have covered part of A fell into ˜ H k before time n .Condition (B2)(b) implies that there exists 1 ≤ j k ≤ k such thatmin ≤ ℓ ≤ n dist( g k,j k , ˆ T ℓ b k ) > . Thus for small h , we have ˆ T ℓ ( ˜ H k ) ∩ B h ( g k,j k ) = ∅ for all 1 ≤ ℓ ≤ n . So B h ( g k,i ) is coveredby time n under T , i.e., B h ( g k,j k ) ⊂ T n ω .Since ˆ T ( b k ) = T ( g k,j k ), condition (B2)(a) says that B h ( g k,j k ) cannot fall into the holebefore time n for small h . Thus T i k B h ( g k,j k ) ⊇ ˆ T i k ˜ H k and we conclude that the part of A which should have been covered by the piece of ω that fell into ˜ H k is at the latest coveredat time n + i k by an interval passing through B h ( g k,j k ).Doing this for each k , we have A ⊂ ∪ Lk =1 T i k B h ( g k,j k ) and so A ⊂ ∪ n i =0 T i ω .Lemma 4.6 implies that I can be covered by the projection of finitely many levels of ∆,say the first N . If π ( Z ℓ,j ) ⊂ π ( Z ℓ ′ ,j ′ ) and both ℓ, ℓ ′ ≤ N , we eliminate ( ℓ, j ) from our indexset, but retain ( ℓ ′ , j ′ ). By (4.6), the remaining index set J ⊂ { , . . . , N − } × N satisfies(A2)(a) and (A2)(b). As before, set Z J = ∪ ( ℓ,j ) ∈ J Z ℓ,j .By (4.5), J π ( F ℓ x ) = ( T ℓ ) ′ ( πx ) so that we are only concerned with the first N iterates of T ℓ . It is clear that if Crit s = ∅ and T is globally C , then J π | Z J ∈ B by Lemma 4.1 and so(A2)(c) is satisfied.In the case when Crit s is nonempty, (A2)(c) does not hold and so Proposition 4.2 mustbe modified. We do this in Step 5 of the proof when we address the convergence propertyfor the a.c.c.i.m. . Step 4. ( F, ∆) is mixing. Since we have constructed a tower over a single base, it sufficesto show that g.c.d.( R ) = 1. The fact that ˆ T is nonrenormalizable guarantees that for theinfinitesimal hole ˜ H (0) , ( ˆ F (0) , ∆ (0) ) can be constructed to be mixing by making g.c.d.( ˆ R (0) ) =33. Indeed, (2.4) implies that as in the case of expanding maps, we can simply wait onetime step on a given return to destroy any periodicity in ˆ R (0) . Once this is accomplished,Lemma 4.4 implies that ( ˆ F ( ˜ H ) , ˆ∆ ( ˜ H ) ) is mixing for ˜ H ∈ H ( h ) with h small enough thatg.c.d.( ˆ R ( ˜ H ) ) is still 1 (by making n h sufficiently large). But since holes cannot affect returnsbefore level n h in ∆, the tower with holes, we have that g.c.d.( R ) = 1 as well. Step 5. Convergence property.
We have already verified in Steps 2 and 3 that ( F, ∆) satisfies(A1) and there is an index set J satisfying (A2)(a) and (A2)(b).By (4.5), the problem spots where ( T ℓ ) ′ (and therefore J π ) are unbounded are neighbor-hoods of T k ( c ) for c ∈ Crit s , k ≥
1. In fact, we only need to address the iterates of c ∈ Crit s up until the time when a neighborhood of T k ( c ) is covered by some other element in thetower on which the derivative is bounded. Since I is covered by the first N levels of ∆, weneed consider at most the first N iterates of c ∈ Crit s .Notice that if a neighborhood A of T k ( c ) can only be reached by an interval ω originatingin a neighborhood of c , then due to the exponential partition of B δ ( c ) which subdivides ω ,there are countably many elements Z ⊂ ∆ ℓ whose projections cover A and in which | π ′ | becomes unbounded the closer that πZ is to T k ( c ).Fix ε > N ε ( c ) denote the ε -neighborhood of those iterates of c ∈ Crit s whichcan only be reached by passing through B δ ( c ). Let N ε = ∪ c ∈ Crit s N ε ( c ) and let J ⊂ J bethe index set of those elements Z such that πZ ⊂ N ε . Denote by 1 ε the indicator functionof the set { y ∈ I : y ∈ πZ ℓ,j , ( ℓ, j ) ∈ J } .Now let ˜ f ∈ G and write ˜ f = ˜ f + ˜ f ε where ˜ f ε := ˜ f · ε and ˜ f = ˜ f − ˜ f ε . We define a liftof ˜ f by f = ˜ f ◦ π · J π on elements of J as in the proof of Proposition 4.2. The lifts f and f ε are defined analogously. Although f / ∈ B , we do have f ∈ B by Proposition 4.2 since˜ f ◦ π ≡ J π becomes unbounded. Using Corollary 3.6 preciselyas in Section 4.2, we have lim n →∞ λ − n L n ˜ f = c ( ˜ f ) ˜ ϕ (4.12)where convergence is in the L -norm and c ( ˜ f ) >
0. Since λ − n L n ˜ f = λ − n L n ˜ f + λ − n L n ˜ f ε , our strategy will be to show that the L -norm of the second term above can be madeuniformly small in n by making ε small. This will imply that λ − n L n ˜ f → c ( ˜ f ) ˜ ϕ in L ( ˜ m )where c ( ˜ f ) = lim ε → c ( ˜ f ) >
0, implying the desired convergence result.Estimating |L n ˜ f ε | L ( ˜ m ) is equivalent to estimating |L n f ε | L ( m ) . λ − n Z L n f ε dm = λ − n X ( ℓ,j ) ∈ J Z ∆ n ∩ Z ℓ,j f ε dm ≤ λ − n X ( ℓ,j ) ∈ J | ˜ f | ∞ | J π ℓ,j | ∞ m (∆ n ∩ Z ℓ,j ) . (4.13)Since we are concerned with finitely many problem spots where the derivative blows up,it suffices to show that the sum in (4.13) over elements in J corresponding to one of theproblem spots is proportional to ε . For simplicity, we fix c ∈ Crit s and denote by A ε theset of elements in ∆ projecting to the ε -neighborhood of ˆ T ( c ). There exists k ε > Z ∈ A ε , then πZ lies in an element of the partition E − k = ˆ T ( c − e − k +1 , c − e − k ) and E + k = ˆ T ( c + e − k , c + e − k +1 ) with k ≥ k ε . 34or Z ℓ,j ∈ A ε , let Z ,j = F − ℓ Z ℓ,j . We split the sum in (4.13) into those elements Z ∈ A ε with R ( Z ) ≥ n and those with R ( Z ) < n . Let 0 < ℓ c < c .We estimate terms with R ( Z ) < n using Corollary 3.5 and the bounded distortion givenby (A1)(b) for J π . λ − n X Z ℓ,j ∈A ε : R ( Z ℓ,j )
We show that the convergence of λ − n L n ˜ f estab-lished in Step 5 occurs at an exponential rate. Since |L n ˜ f | L ( ˜ m ) = |L n f | L ( m ) , it suffices toshow this convergence for the lift on ∆.Let ε = e − tn for some small constant t to be chosen later. Define N ε as above and noticethat outside of N ε , the C norm of ˆ T ℓ for ℓ = 1 , . . . , N is proportional to e − k ε ( ℓ ∗ c − where ℓ min c > c ∈ Crit s . Since k ε is on the order of − log ε ,we have | ˆ T ℓ | ˆ I \N ε | C = O ( e tn (2 − ℓ min c ) ). Let Z J = ∪ ( ℓ,j ) ∈ J Z ℓ,j and let Z J,ε ⊂ Z J denote thoseelements which project into N ε . Then k J π | Z J \Z J,ε k ≤ Ce tn (2 − ℓ min c ) . (4.19)Define ˜ f , f , ˜ f ε , f ε as in Step 5. By Lemma 4.1, (4.19) implies that k f k ≤ Ce tn (2 − ℓ min c ) sothat by Corollary 2.5, | λ − n L n f − c ( f ) ϕ | L ( m ) ≤ k λ − n L n f − c ( f ) ϕ k ≤ Ce tn (2 − ℓ min c ) σ n (4.20)Next, when the holes are sufficiently small, λ − ≥ max { θ γ , θ ( p − /p } , so (4.18) yields, λ − n |L n f ε | ≤ Ce − tn (1 − p (1 − ℓ min c )) /p + C ′ e − tn ( ℓ min c − γ ) ≤ C ′′ e − tnγ ′ (4.21)for some γ ′ >
0. In particular, we see from (4.21) that the constants c ( f ) converge to c ( f )exponentially fast as well. | c ( f ) − c ( f ) | = lim n →∞ λ − n ( |L n f | − |L n f | ) = lim n →∞ λ − n Z L n f ε dm ≤ C ′′ e − tnγ ′ This estimate together with (4.20) and (4.21) imply that λ − n L f → c ( f ) ϕ exponentially fastonce we choose t < − log σ/ (2 − ℓ min c ). Step 7. Unified escape rate.
By Step 5, for each ˜ f ∈ G , we have f ∈ L ( m ) such that P π f = ˜ f and λ − n L nF f = c ( f ) ϕ for some c ( f ) > λ − n L nT ˜ f = c ( f ) ˜ ϕ by (4.3).Letting ˜ η = ˜ f ˜ m , we havelim n →∞ n log ˜ η ( I n ) = lim n →∞ n log |L nT ˜ f | = log λ. Proof of Theorem 2.14.
By (4.9) and (4.10), the quantity q := P ℓ ≥ m ( H ℓ ) β − ( ℓ − can bemade arbitrarily small by choosing h to be small. By Proposition 3.1, the escape rate λ iscontrolled by the size of q so that λ → q →
0. Thus λ h → h → µ h is a sequence of probability measures on the compact interval ˆ I , it follows thata subsequence, { ˜ µ k } corresponding to h k , converges weakly to a probability measure ˜ µ ∞ .We show that ˜ µ ∞ is an absolutely continuous invariant measure for ˆ T . Since there is only36ne such measure, this will imply that in fact the entire sequence converges to this sameinvariant measure. Step 1. ˜ µ ∞ is absolutely continuous with respect to Lebesgue. For each ˜ H ( k ) , we have twotowers: ( ˆ F ( k ) , ˆ∆ ( k ) ) which has no holes but is constructed using ∂ ˜ H ( k ) as artificial cuts asdescribed in the proof of Theorem 2.12; and ( F ( k ) , ∆ ( k ) ), the tower with holes obtained fromˆ∆ ( k ) . By Lemma 4.5, there exist uniform constants C > θ < m ( ˆ∆ ( k ) ℓ ) ≤ Cθ ℓ .We have an invariant density ρ k on ˆ∆ ( k ) and a conditionally invariant density ϕ k on ∆ ( k ) .By Proposition 3.1, both ρ k , ϕ k ∈ B M where M is independent of k (to see the results for ρ k , simply apply the proposition to the case H = ∅ ). In addition, by Proposition 2.4(i), ρ k ≥ a > a is independent of i because the uniform decay given byLemma 4.5 implies that ˆ∆ ( k )0 must retain some positive minimum measure for all k .Let ˆ π k be the projection corresponding to ˆ∆ ( k ) and let π k = ˆ π k | ∆ ( k ) . Letting ˜ ρ denote theunique invariant density for ˆ T and J ˆ π k the Jacobian of ˆ π k etc., we have for each k ,˜ ρ ( x ) = P ˆ π k ρ k ( x ) = X y ∈ ˆ π − k x ρ k ( y ) J ˆ π k ( y ) and ˜ ϕ k ( x ) := P π k ϕ k ( x ) X y ∈ π − k x ϕ k ( y ) J π k ( y ) . (4.22)Now for any ε >
0, choose
L > P ℓ>L CM β − ℓ θ ℓ < ε . Next choose k such thatfor all k ≥ k , λ − Lk ≤
2. Now for any Borel A ⊂ ˆ I ,˜ µ k ( A ) = X ℓ ≤ L µ k (∆ ℓ ∩ π − k A ) + X ℓ>L µ k (∆ ℓ ∩ π − k A ) =: ˜ µ k,L ( A ) + ˜ µ k, + ( A ) . By (4.22), the measure ˜ µ k,L has density ˜ ϕ k,L bounded independently of k ≥ k :˜ ϕ k,L ( x ) = X y ∈ π − k x : ℓ ( y ) ≤ L ϕ k ( y ) J π k ( y ) ≤ Ma X y ∈ ˆ π − k x : ℓ ( y ) ≤ L λ − ℓ ( y ) k ρ k ( y ) J ˆ π − k ( y ) ≤ M ˜ ρ ( x ) a . (4.23)where ℓ ( y ) is the level of y in ˆ∆ ( k ) . The remaining measure ˜ µ k, + has small total mass:˜ µ k, + ( ˆ I ) = X ℓ>L ˜ µ k (∆ ℓ ) ≤ X ℓ>L M β − ℓ m (∆ ℓ ) ≤ X ℓ>L CM β − ℓ θ ℓ < ε. (4.24)Putting together (4.23) and (4.24), we see that µ ∞ = µ ∞ ,L + µ ∞ , + where µ ∞ ,L has densitybounded by 2 M ˜ ρ/a while µ ∞ , + is possibly singular with total mass less than ε . Since thisis true for each ε >
0, we conclude that in fact µ ∞ is absolutely continuous with densitybounded by 2 M ˜ ρ/a . Step 2. ˜ µ ∞ is invariant. Let I k = ˆ I \ ˜ H ( k ) and as usual, let I nk = ∩ nj =0 ˆ T − j I k and T k = ˆ T | I k .By Step 1, ˜ µ ∞ has density bounded by 2 M ˜ ρ/a , which is in L ( ˜ m ). Thus ˜ µ ∞ gives 0measure to the singularity points of ˆ T . This fact allows us to write, for any continuousfunction f on ˆ I ,˜ µ ∞ ( f ◦ ˆ T ) = lim k →∞ ˜ µ k ( f ◦ ˆ T ) = lim k →∞ Z I k f ◦ T k d ˜ µ k + Z ˆ I \ I k f ◦ ˆ T d ˜ µ k . (4.25)37ince λ k →
1, the first term in (4.25) is equal tolim k →∞ Z I k f d (( T k ) ∗ ˜ µ k ) = lim k →∞ λ k ˜ µ k ( f ) = ˜ µ ∞ ( f ) . The second term in (4.25) is bounded by | f | ∞ ˜ µ k ( ˆ I \ I k ). This quantity tends to 0 as k → ∞ because of the uniform bounds on the densities of ˜ µ k obtained in Step 1. In this section we consider the invariant measures ν and ˜ ν = π ∗ ν and prove Theorems 2.9,2.16 and 2.17. We assume throughout that F is mixing and satisfies (P1)-(P3) and (H1). ˜ ν Proof of Theorem 2.16.
Let ˜ f ∈ C ¯ α ( X ) and note that ˜ f ◦ π ∈ B . Thus,˜ ν ( ˜ f ) = ν ( ˜ f ◦ π ) = lim n →∞ λ − n Z ∆ n ˜ f ◦ π dµ = lim n →∞ λ − n Z ∆ n ˜ f ◦ πϕ dm = lim n →∞ λ − n Z π (∆ n ) P π ( ˜ f ◦ π ϕ ) d ˜ m = lim n →∞ λ − n Z X n ˜ f ˜ ϕ d ˜ m = lim n →∞ λ − n Z X n ˜ f d ˜ µ where in the first line we have used Proposition 2.8.The ergodicity of ˜ ν follows from that of ν and the relation X ∞ = π (∆ ∞ ). If A ⊂ X is T -invariant, then since F − ◦ π − ( A ) = π − ◦ T − ( A ) = π − ( A ), we conclude that π − ( A ) is F -invariant. This implies that ˜ ν ( A ) is 0 or 1.To prove exponential decay of correlations let ˜ f , ˜ f ∈ C α ( X ). Set f i = ˜ f i ◦ π and notethat R X ˜ f i d ˜ ν = R ∆ f i ◦ π dν for i = 1 ,
2. So Z X ˜ f ˜ f ◦ T n d ˜ ν = Z ∆ ˜ f ◦ π ˜ f ◦ T n ◦ π dν = Z ∆ f ˜ f ◦ π ◦ F n dν = Z ∆ f f ◦ F n dν, from which exponential decay of correlations follows using Proposition 2.8 and the fact that f , f ∈ B . First note that since F is mixing, Property (P3) implies that there exists an n ∈ N suchthat F n ( Z ′ ) ⊇ ∆ , for all n ≥ n and Z ′ ∈ Z im .Let ν := ν (∆ ) ν | ∆ and define S = F R : ∆ ∞ ∩ ∆ (cid:9) . Let R n ( x ) = P n − k =0 R ( S k x ) be the n th return time starting at x and let M S be the set of S -invariant Borel probability measureson ∆ ∞ ∩ ∆ . 38 roposition 5.1. The measure ν is a Gibbs measure for S and S is topologically mixingon ∆ ∞ . Accordingly, sup η ∈M S (cid:26) h η ( S ) + Z ∆ log(( J S ) − λ − R ) dη (cid:27) = 0 . and ν is the only nonsingular measure η ∈ M S which attains the supremum. We first prove the following two lemmas.
Lemma 5.2.
Let χ be the indicator function for ∆ . There exists a k ∈ N such that forall k ≥ k and all x ∈ ∆ , λ − k ϕ − ( x ) L k ( ϕχ )( x ) ≥ ν ( χ ) / . Proof.
Note that χ ∈ B so that λ − k L k ( χ ) → c ( χ ) ϕ in the k · k -norm. This means thatthe functions converge pointwise uniformly on each level of the tower. Thus0 < ν (∆ ) = lim k →∞ λ − k ϕ − ( x ) L k ( χ ϕ )( x )uniformly for x ∈ ∆ . The uniform convergence implies the existence of the desired k .The next lemma establishes the Gibbs property for ν . Lemma 5.3.
Let [ i , i , . . . , i n − ] ⊂ ∆ denote a cylinder set of length n with respect to S .Then there exists a constant C > such that for all n , C − λ − R n ( y ∗ ) ( J S n ( y ∗ )) − ≤ ν ([ i , i , . . . , i n − ]) ≤ Cλ − R n ( y ∗ ) ( J S n ( y ∗ )) − where y ∗ is an arbitrary point in [ i , i , . . . , i n − ] and J S n is the Jacobian of S n with respectto m .Proof. Let χ A be the indicator function of A := [ i , i , . . . , i n − ]. Although χ A / ∈ B , we dohave L k χ A ∈ B for k ≥ n since 1-cylinders are in B . Thus ν ( χ A ) is characterized by thelimit ν ( χ A ) = lim k λ − k ϕ − L k ( ϕχ A ). Since this convergence is in the k · k -norm, it is uniformfor x ∈ ∆ .For x ∈ ∆ and k ≥ R n ( A ), L k ( ϕχ A )( x ) = X F k y = x ϕ ( y ) χ A ( y ) g k ( y )= X y ∈ A,F k y = x ϕ ( y ) g k − R n ( A ) ( F R n y ) g R n ( y )= X z ∈ F Rn ( A ) ,F k − Rn ( A ) z = x ϕ ( y ) g k − R n ( A ) ( z ) g R n ( y ) , (5.1)39here in the last line we have used the fact that F R n ( A ) | A is injective. Note that by (2.1),we may replace g R n ( y ) by g R n ( y ∗ ) where y ∗ ∈ A is an arbitrary point. Also, since both y and F R n y are in ∆ and δ ≤ ϕ ≤ δ − on ∆ , we may estimate (5.1) by L k ( ϕχ A )( x ) ≤ Cg R n ( y ∗ ) X z ∈ F Rn ( A ) ,F k − Rn ( A ) z = x ϕ ( z ) g k − R n ( A ) ( z ) ≤ Cg R n ( y ∗ ) X F k − Rn ( A ) z = x ϕ ( z ) g k − R n ( A ) ( z )= Cg R n ( y ∗ ) L k − R n ( A ) ϕ ( x ) = Cg R n ( y ∗ ) λ k − R n ( A ) ϕ ( x ) . (5.2)Combining this estimate with the definition of ν and noticing that g R n = ( J S n ) − , we havethe upper bound, ν ( A ) ≤ C ( J S n ( y ∗ )) − λ − R n ( A ) . To obtain the lower bound, we again work from equation (5.1) and choose k ≥ R n ( A ) + n + k . L k ( ϕχ A )( x ) = X y ∈ A,F k y = x ϕ ( y ) g k − R n ( A ) − n ( F R n + n y ) g n ( F R n y ) g R n ( y ) ≥ X z ∈ F Rn + n ( A ) ∩ ∆ F k − Rn ( A ) − n z = x ϕ ( y ) g k − R n ( A ) − n ( z ) g n ( F R n y ) g R n ( y ) . (5.3)We again replace g R n ( y ) by g R n ( y ∗ ) using (2.1). Note also that g n | ∆ is bounded belowand that F R n ( A ) y ∈ ∆ . Since we are only considering y, z ∈ ∆ , we know that ϕ ( y ) isproportional to ϕ ( z ). Thus L k ( ϕχ A )( x ) ≥ Cg R n ( y ∗ ) X z ∈ F Rn + n ( A ) ∩ ∆ F k − Rn ( A ) − n z = x ϕ ( z ) g k − R n ( A ) − n ( z )= Cg R n ( y ∗ ) X F k − Rn ( A ) − n z = x χ ( z ) ϕ ( z ) g k − R n ( A ) − n ( z )= Cg R n ( y ∗ ) L k − R n ( A ) − n ( χ ϕ )( x ) . (5.4)where in the second to last line we have used the fact that F R n + n ( A ) ⊇ ∆ . Combiningequation (5.4) with Lemma 5.2, since k − R n ( A ) − n ≥ k , we estimate L k ( ϕχ A )( x ) ≥ Cg R n ( y ∗ ) λ k − R n ( A ) − n ϕ ( x ) ν ( χ )2 . The lower bound follows from the definition of ν . Proof of Proposition 5.1.
Lemma 5.3 implies that ν is a Gibbs measure with potential φ = − log( λ R J S ). We define a topology on ∆ using the cylinder sets with respect to Z as ourbasis. The fact that S | ∆ ∩ ∆ ∞ is topologically mixing follows immediately from the conditionthat F be mixing on elements of Z im together with the finite images condition (P3). Thiscan be seen as in the proof of Proposition 2.8(ii) in Section 3.3.40he formalism of [S] completes the proof of the proposition. Theorem 3 of [S] impliesthat P G ( φ ) = sup η ∈M S (cid:26) h η ( S ) + Z ∆ φ dη (cid:27) (5.5)where P G ( φ ) = sup { P top ( φ | Y ) : Y ⊂ ∆ ∩ ∆ ∞ , top. mixing finite Markov shift } is the Gure-vich pressure of φ for S .Lemma 5.3 of this paper combined with [S, Theorems 7 and 8] implies that P G ( φ ) = 0and that the supremum is obtained by our Gibbs measure ν . In addition, ν is the onlynonsingular S -invariant probability measure which attains the supremum.We now prove an equilibrium principle for F using the one for S . Lemma 5.4.
Let M F be the set of F -invariant Borel probability measures on ∆ . For any η ∈ M F , let η = η (∆ ) η | ∆ . Then Z ∆ log J S dη = Z ∆ log J F dη Z ∆ R dη . Proof.
Notice that η ∈ M S . For x ∈ ∆ , J S ( x ) = J F R ( x ) = Π R ( x ) − i =0 J F ( F i x ). But J F ( F i x ) = 1 for i < R ( x ) −
1, so that
J S ( x ) = J F ( F R − x ). In other words, we have Z ∆ log J S dη = η (∆ ) − Z F − ∆ log J F dη = ν (∆ ) − Z ∆ log J F dη. (5.6)Since the measure of a partition element Z ℓ,j ∈ Z does not change as it moves up thetower, we have 1 = X ( ℓ,j ) η ( Z ℓ,j ) = X j η ( Z ,j ) R ( Z ,j ) = Z ∆ R dη.
So by definition of η , we have Z ∆ R dη = η (∆ ) − Z ∆ R dη = η (∆ ) − . This, together with (5.6), proves the lemma.Since S = F R is a first return map to ∆ , the general formula of Abramov [A] impliesthat h η ( F ) = h η ( S ) η (∆ ) so that h η ( S ) = η (∆ ) − h η ( F ) = h η ( F ) Z ∆ R dη . (5.7)Since R ∆ R dη = η (∆ ) − = 0 and there is a 1-1 correspondence between measures in M S and M F , putting equation (5.7) and Lemma 5.4 together with (5.5), we havelog λ = sup η ∈M F (cid:26) h η ( F ) − Z ∆ log J F dη (cid:27) . (5.8)Moreover, ν is the only nonsingular F -invariant probability measure which attains the supre-mum. This completes the proof of Theorem 2.9.41 .3 An Equilibrium Principle for ( T , X ) The proof of Theorem 2.17 consists simply of projecting (5.8) down to X to get the desiredrelation for T .Note that for any η ∈ M F , we can define ˜ η = π ∗ η ∈ M T . Then given a function ˜ f on X , we have R X ˜ f d ˜ η = R ∆ ˜ f ◦ π dη . From the relation π ◦ F = T ◦ π , we have J π ( F x ) J F ( x ) = J T ( πx ) Dπ ( x )for each x ∈ ∆. Thus, Z X log J T d ˜ η = Z ∆ (log J F + log
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Statistical properties of dynamical systems with some hyperbolicity , Annals ofMath. (1998), 585-650.Department of MathematicsUniversity of SurreyGuildford, Surrey, GU2 7XHUK [email protected]://personal.maths.surrey.ac.uk/st/H.Bruin/
Department of Mathematics and Computer ScienceFairfield UniversityFairfield, CT 06824USA [email protected]://cs.fairfield.edu/ ∼ demers/ Department of MathematicsUniversity of SurreyGuildford, Surrey, GU2 7XHUK [email protected]://personal.maths.surrey.ac.uk/st/I.Melbourne/[email protected]://personal.maths.surrey.ac.uk/st/I.Melbourne/research.html