Pointwise normality and Fourier decay for self-conformal measures
aa r X i v : . [ m a t h . D S ] D ec On the decay of the Fourier transform of self-conformal measures
Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang
Abstract
Let Φ be a C γ smooth IFS on an interval J ⊂ R , where γ >
0. We provide mild conditionson the derivative cocycle that ensure that all non-atomic self conformal measures are Rajchmanmeasures, that is, their Fourier transform decays to 0 at infinity. This allows us to give many newexamples of self conformal Rajchman measures, and also provides a unified proof to several pre-existing results. For example, we show that if Φ is C ω and admits a non-atomic non-Rajchmanself conformal measure, then it is C ω conjugate to a self similar IFS satisfying:There exist t, r such that its contractions { r , ..., r n } ⊂ { t · r k : k ∈ Z } This is closely related to the work of Bourgain-Dyatlov [4]. We also prove that if Φ is self similarand does have not this form, then any C γ smooth image of a non atomic self similar measureis Rajchman, assuming the derivative never vanishes. This complements a classical Theorem ofKaufman [20] about homogeneous IFS’s, and extends in many cases recent results of Li-Sahlsten[26] about the Rajchman property in the presence of independent r i , r j .The proof relies on a version of the local limit Theorem for the derivative cocycle, that isadapted from the work of Benoist-Quint [2]. Let ν be a Borel probability measure on R . For every q ∈ R the Fourier transform of ν at q isdefined by F q ( ν ) := Z exp(2 πiqx ) dν ( x )The measure ν is called a Rajchman measure if lim | q |→∞ F q ( ν ) = 0. It is a consequence of theRiemann-Lebesgue Lemma that if ν is absolutely continuous then it is Rajchman. On the otherhand, it follows from Wiener’s Lemma that if ν has an atom then it is not Rajchman. For measuresthat are both continuous (no atoms) and singular, determining whether or not ν is a Rajchmanmeasure may be a challenging problem.Information about the asymptotic behaviour of ν , i.e. whether it is a Rajchamn measure or not- and if it is, about the rate of decay of F q ( ν ), has various geometric implications . To illustrate this,recall that a set F ⊂ [0 ,
1] is called a set of uniqueness if the only trigonometric series P n ≥ a n e πinx that converges to 0 on [0 , \ F is the one satisfying a n = 0 for all n . A set F ⊂ R is called a setof uniquness if its projection to R / Z ≃ [0 ,
1) is a set of uniquness. If F is not a set of uniqueness,it is called a set of multiplicity . Research on sets of uniqueness goes back to the classical works ofRiemann and Cantor. It is a subtle problem: For example, it is known that sets of uniqueness alwayshave zero Lebesgue measure [21, Proposition I.3.1]. However, as Menshov [28] proved, having zeroLebesgue measure does not ensure that F is a set of uniqueness. Later, Salem [33] showed that if F supports a Rajchman measure then F is a set of multiplicity. Thus, the Rajchman property is1ermane to the uniqueness problem. If, moreover, one obtains a sufficiently fast rate of decay, thenthis can be used to prove the existence of normal numbers in the support of ν [11], and may beuseful to estimate its Fourier dimension. See the survey [27] for more details.In this paper we will focus on studying the Rajchman property for self-conformal measures.These are defined as follows: Let Φ = { f , ..., f n } be a finite set of strict contractions of an interval J ⊆ R (an IFS ), such that every f i is differentiable. It is well known that there exists a uniquecompact set ∅ 6 = K = K Φ ⊆ J such that K = n [ i =1 f i ( K )The set K is called a self-conformal set , and the attractor of the IFS { f , ..., f n } . In the specialcase where each f i is affine, i.e. f i ( x ) = r i · x + t i and 0 < | r i | <
1, we call K a self-similar set .Next, let p = ( p , ..., p n ) be a strictly positive probability vector, that is, p i > i and P i p i = 1. It is well known that there exists a unique Borel probability measure ν such that ν = n X i =1 p i · f i ν The measure ν is called a self-conformal measure , and is supported on K . If every f i is affine then ν is called a self-similar measure . We remark that when we write f i ν we mean the pushforward of ν via f i .The following Theorem provides mild conditions on a self conformal IFS that ensure that theRajchman property holds for all non-atomic self-conformal measures. Theorem 1.1.
Let Φ be an IFS on an interval J ⊂ R such that:1. Every f ∈ Φ is at least C γ ( J ) smooth for γ > .2. We have < inf {| f ′ ( x ) | : f ∈ Φ , x ∈ J } ≤ sup {| f ′ ( x ) | : f ∈ Φ , x ∈ J } < .3. For every t, r ∈ R , the set (cid:8) − log (cid:12)(cid:12) f ′ ( y ) (cid:12)(cid:12) : where f ( y ) = y, f ∈ Φ (cid:9) is not included in the set t + r Z .Then any non-atomic self conformal ν is a Rajchman measure, that is, lim | q |→∞ F q ( ν ) = 0In Theorem 1.1, as well as in our other results, we do not assume any separation conditions.Now, assumptions (1) and (2) are standard. To see that the third assumption can be verified for awide class of IFS’s, we introduce a new definition: Let Φ be a self similar IFS with correspondingcontraction ratios { r , ..., r n } . We say that Φ is periodic if there are some r, t ∈ R such that { r , ..., r n } ⊂ { t · r k : k ∈ Z } Otherwise, we say that Φ is aperiodic . It is clear that a self similar aperiodic IFS always satisfiesthe conditions of Theorem 1.1, including the third one. Thus, we can show that any C γ imageof an aperiodic self similar measure is also Rajchman:2 orollary 1.2. Let Φ be an aperiodic self similar IFS and let g : J → R be a C γ smooth map,where γ > and K Φ ⊆ J is an interval. Let ν be a non atomic self similar measure. If g ′ does notvanish on J then gν is a Rajchman measure. For the deduction of Corollary 1.2 from Theorem 1.1 see Section 5. It follows that if K isthe attractor of a non-trivial aperiodic IFS then for any g ∈ C γ ( J ) the set g ( K ) is a set ofmultiplicity, as long g ′ never vanishes. These results should be compared with a recent Theoremof Li and Sahlsten [25] (that was later also recovered by Br´emont [5]): They proved that for anorientation preseving self similar IFS that admits contractions such that log r i log r j Q , any non-atomicself-similar measure is a Rajchman measure. Note that an aperiodic self similar IFS always admitssuch r i , r j . So, for aperiodic IFS’s Theorem 1.2 extends the Li-Sahlsten Theorem to all C γ smooth images. On the other hand, not every IFS considered by Li and Sahlsten is aperiodic,which follows e.g. by noting that if Φ has at most two contractions { r , r } it is never aperiodic.See Section 5.3 for a discussion on the general case when log r i log r j Q . Corollary 1.2 also complementsa classical Theorem of Kaufman [20] (that was later extended by Mosquera and Shmerkin [29], seealso [8]) about the Rajchman property for C images of homogeneous (i.e. r i = r j for all i, j ) selfsimilar measures. Thus, Theorem 1.2 partially answers an open question (see e.g. [32]) about theexistence of a Kaufman Theorem in the non homogeneous setting . We will return to these resultsand put them into context when we survey the relevant literature in Section 1.2.Our next result shows that the only C ω smooth IFS’s that admit a non-Rajchman measure arethose that are C ω conjugate to a periodic self similar IFS: Theorem 1.3.
Let Φ be a C p smooth IFS on an interval J , where either p = 1 + γ for γ > or p = ω , such that:1. We have < inf {| f ′ ( x ) | : f ∈ Φ , x ∈ J } ≤ sup {| f ′ ( x ) | : f ∈ Φ , x ∈ J } < .2. Either p = ω or K Φ is an interval.3. There exists a non-atomic self conformal measure ν that is not a Rajchman measure.Then Φ is C p conjugate to a periodic self similar IFS.That is, there are t, r ∈ R and a C p diffeomorphism h : J → h ( J ) so that every x ∈ h ( J ) hasan open neighbourhood N x such that for every y ∈ N x and f ∈ Φ h ◦ f ◦ h − ( y ) = t · r n f,x y + t f,x for some n f,x ∈ Z and t f,x ∈ R . As before, no separation conditions are assumed. Theorem 1.3 gives many new examples ofself conformal Rajchman measures. It also provides a unified proof to several pre-existing resultsregarding the Rajchman property for C ω IFS’s that are not conjugate to a self similar IFS. Theseinclude those of Bourgain and Dyatlov [4] and of Sahlsten and Stevens [32], and in some cases thoseof Li [23],[22] (the latter papers also extend [4] but in a different direction - see Section 1.2). Forexample, Bourgain and Dyatlov study Patterson-Sullivan measures for convex cocompact Fuchsiangroups. These are known to be self conformal measures with respect to an IFS comprised of M¨obiustransformations that satisfy conditions (1) and (2) of Theorem 1.3. Furthermore, this IFS can beshown to be not conjugate via a M¨obius transformation to a self similar one, outside of trivialcases. On the other hand, for IFS’s that comprise only of M¨obius transformations, it is not hard tosee that if they are C ω conjugate to a self similar IFS, then the conjugating map is also a M¨obiustransformation. Therefore, Theorem 1.3 gives a new proof of results of Bourgain and Dyatlov about3he Rajchman nature of these measures. However, we do not recover the polynomial decay rateproved by Bourgain and Dyatlov, as well as by Li [23] and Sahlsten and Stevens [32].We proceed to survey the rich literature about the Rajchman property for dynamically definedmeasures, including the aforementioned results. Before doing so, we note that all the results in thisSection follow from one unified statement, but we postpone a discussion of this to Section 1.3. A fundamental family of examples is given by Bernoulli convolutions { ν r } r ∈ (0 , : For every 0 < r < { r · x − , r · x + 1 } with the probability vector p = ( , ). A basicquestion in fractal geometry asks: for which r ∈ ( ,
1) is ν r is absolutely continuous?A celebrated result of Erd˝os [12] says that if r − is a Pisot number then ν r is not a Rajchmanmeasure and consequently is not absolutely continuous. Recall that a Pisot number is a realalgebraic integer greater than one whose Galois conjugates all lie inside the unit disc. Later, Salem[33] completed the picture in terms of the Rajchman property, by showing that ν r is not Rajchmanonly if r − is Pisot (see also the related works of Piatetski-Shapiro [43] and Salem and Zygmund[34]) . We remark that through some ground breaking recent papers (e.g. [15], [35], [6], [41] toname a few) the geometric properties of ν r are now far better understood. However, the questionof absolute continuity remains open. More general self similar measures were studied by Strichartz[37], [38]: He proved that their Fourier transforms decay on average, with a recent large deviationsestimate on this decay given by Tsujii [40] (see also [39] for a related paper about self conformalmeasures). However, these papers do not establish the Rajchman property, since they excludecertain frequencies.Very recently there have been major breakthroughs regarding the Rajchman property for selfsimilar measures: As we have already mentioned, Li and Sahlsten [25] proved that if the IFS containstwo maps with contractions r i , r j such that log r i log r j Q , then any non-atmoic self-similar measureis Rajchman. In the complementary case, when all contractions are powers of some r ∈ (0 , r − is Pisot (in fact, Br´emont also reproved the Li-Sahlsten Theorem,and carried out a refined analysis giving some cases when r − is Pisot yet the measure is Rajchman).Another proof of this fact was given by Varj´u and Yu [42]. We remark that while Corollary 1.2covers many IFS’s as in Li and Sahlsten’s paper (and can be adapted to cover all of them), it doesnot cover IFS’s as in the works of Br´emont and of Varj´u and Yu, since these are always periodic.Finally, we note that Li and Sahlsten [26] also generalized their results to self affine measures inhigher dimensions.In the self conformal setting recent results have been obtained by either additive combnatorial orrenewal theortic methods. Via additive combinatorics, Bourgain and Dyatlov [4] established Fourierdecay for Patterson-Sullivan measures for convex cocompact Fuchsian groups, with a polynomialrate (see also [24]). Generlizing this, Li [23] used renewal theory and additive combinatorics toprove polynomial decay for the stationary measure for a random walk on SL (2 , R ), assuming thedriving measure generates a Zariski dense subgroup (thus also generalizing his previous results [22]that such measures are Rajchman). We remark that through the work of Yoccoz [44] and laterAvila, Bochi, and Yoccoz [1], there are known conditions that ensure that such measures are selfconformal with respect to a unifromly contracting C ω IFS. Very recently, Sahlsten and Stevens [32]were able to prove polynomial decay for a class of self conformal measures that do not necessarilyarise from a projective action. Instead, they make a number of other assumptions, including thatthe union K = S f i ( K ) is sufficiently separated, the maps are in the IFS are C ω smooth, and theIFS is totally non-linear - which follows e.g. if the IFS is not C conjugated to a self similar IFS.4or IFS’s that are C conjugate to homogeneous self similar IFS’s, Shmerkin and Mosquera [29],improving a classical Theorem of Kaufman [20], proved polynomial decay for all self conformalmeasures (see also [8]). Finally, Sahlsten and Jordan [18] and later Sahlsten and Stevens [31]proved polynomial Fourier decay for certain Gibbs measures for the Gauss map x x mod 1on the interval. These can be considered as self conformal measures with respect to an IFS withcountably many maps.We end this Section with a quick word about rates in the self-similar case. For Bernoulli con-volutions, it follows from the classical papers of Erd˝os [13] and Kahane [19] that ν r has polynomial(power) decay outside a set of zero Hausdorff dimension (see also [9], [7] [10] for rates in someexplicit examples of r ). Li and Sahlsten [25] prove a logarithmic rate of decay assuming a certainDiophantine condition holds on a pair of independent contractions. A similar rate of decay is ob-tained in the work of Varj´u and Yu [42] assuming the underlying contraction is not Pisot or Salem.Finally, in another very recent paper, by generalizing the Erd˝os-Kahane arguement, Solomyak [36]established polynomial Fourier decay for all non-atomic self-similar measures except for a zeroHausdorff dimensional exceptional set of contractions. Let { f , ..., f n } be an IFS such that each f i is differentiable. From this point forward, we willassume without the loss of generality that the IFS is acting on the interval J = [0 , K = K Φ ⊆ [0 , ω ∈ { , ..., n } N and m ∈ N let f ω | m = f ω ◦ ◦ ◦ f ω m Then we have a surjective coding map { , ..., n } N → K defined by ω ∈ { , ..., n } N x ω := lim m →∞ f ω | m (0)We also choose ρ ∈ (0 ,
1) in a manner dependent only on the IFS (see Claim 2.11), and define ametric on { , ..., n } N via d ρ ( ω, ω ′ ) := ρ min { n : ω n = ω ′ n } Let p be a strictly positive probability vector on { , ..., n } , and let ν be the corresponding selfconformal measure. Let P = p N be the product measure on { , ..., n } N . Then ν is the push-forwardof P via ω x ω . Also, for every 1 ≤ a ≤ n let ι a : { , ..., n } N → { , ..., n } N be the map ι a ( ω , ω , · · · ) = ( a, ω , ω , · · · )Let G to be the free semigroup generated by the family { ι a : 1 ≤ a ≤ n } . We define the derivativecocycle c : G × { , ..., n } N → R via c ( a, ω ) = − log | f ′ a ( x ω ) | Choose some κ ∈ (0 ,
1] and let H κ denote the space of κ -H¨older continuous maps { , ...., n } N → C ,and define Λ c ⊆ R viaΛ c = { θ : There exists φ θ ∈ H κ with | φ θ | = 1 and u θ ∈ S such that (1) φ θ ( ι a ( ω )) = u θ · e − iθ · c ( a,ω ) · φ θ ( ω ) , for all ( a, ω ) ∈ { , ..., n } × { , ..., n } N } It is clear that 0 ∈ Λ c . If Λ c = { } then the cocycle c is aperiodic in the sense of Benoist-Quint [2,Equation (15.8)], and this will be used in an essential way to prove the following Theorem:5 heorem 1.4. Let { f , ..., f n } be a self conformal IFS on an interval J such that:1. Every f i is at least C γ ( J ) smooth for γ > .2. We have < inf {| f ′ ( x ) | : f ∈ Φ , x ∈ J } ≤ sup {| f ′ ( x ) | : f ∈ Φ , x ∈ J } < .3. We have Λ c = { } .Then any non-atomic self conformal ν is a Rajchman measure, that is, lim | q |→∞ F q ( ν ) = 0Theorem 1.4 implies both Theorem 1.1 and Theorem 1.3: For Theorem 1.1, we will show thatcondition (3) therein implies that Λ c = { } . Under the assumptions of Theorem 1.3, we will showthat Λ c = { } only if the IFS can be conjugated to a periodic self similar IFS. For more details,see Section 5.We end this introduction with a road map to the proof of Theorem 1.4, and a comparison withthe method of Li-Sahlsten [25] for the self similar case. For every ω we define a random variablevia X ( ω ) := c ( ω , σ ( ω )) = − log | f ′ ω ( x σ ( ω ) ) | and for any n let X n ( ω ) = X ◦ σ n − ( ω ), where σ : { , ...., n } N → { , ...., n } N is the left shift map.We define a random walk via S n ( ω ) = X ( ω ) + ... + X n ( ω ) = − log | f ′ ω | n (cid:0) x σ n ( ω ) (cid:1) | For every k ∈ N and ω ∈ { , ..., n } N we define the ”stopping time” τ k ( ω ) := min { m : S m ( ω ) ≥ k } By condition (2) of Theorem 1.4 we may assume that S τ k ( ω ) ( ω ) ∈ [ k, k + 1] for every k ∈ N and ω .Notice that τ k is not a stopping time, since the map ω x σ n ( ω ) is not measurable with respect tothe first n digits. On the other hand, τ k is always at a uniformly bounded distance from a bona-fidestopping time, defined similarly only differentiating f ω | n always at 0. This will play an importantrole in our analysis. To complete the setup, for every k we define a partition A k of { , ..., n } N asfollows: ω ∼ A k η if and only if( ω τ k ( ω ) , ..., ω τ k ( ω )+ τ √ k ( σ τk ( ω ) ( ω ) )) = ( η τ k ( η ) , ..., η τ k ( η )+ τ √ k ( σ τk ( η ) ( η ) ))Recall that we want to prove the decay of F q ( ν ) at infinity. For q ∈ R large we fix k ≈ log | q | .The precise choice of k given q is a subtle matter (we will come back to this later). Our proof thengoes through three main steps:1. Linearization: For every s ∈ R let M s : R → R be the multiplication map M s ( t ) = s · t . Thenfor all k large enough, we use the self-conformality of ν to show that |F q ( ν ) | ≤ Z E A k ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ g A k ( ξ ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + o ( q, k ) (2)where by A k ( ξ ) we mean the partition element containing ξ , and g A k ( ξ ) is a smooth map ofthe interval whose derivative we can control: | g ′ A k ( ξ ) ( x ) | ∼ e −√ k for all x ∈ [0 , In practice we will also use the Lyapunov exponent in these definitions, but for clarity we refrain from doing soin the introduction. In practice we will prove such an inequality but with possibly more than one such integral on the right handside, corresponding to suitable maps that only depend on A k ( ξ ). This collection is uniformly bounded, so the effecton the proof is not substantial.
6. Local equidistribution: We will produce a set X = X k such that P ( X ) = 1 − o k (1), and forevery ξ ∈ X there exists a probability measure Γ A k ( ξ ) on [ k, k + 1] such that:(a) Γ A k ( ξ ) ≪ λ [ k,k +1] , with a uniformly bounded density (the bound only depends on theIFS). Here and throughout the paper, λ is the Lebesgue measure on R .(b) E A k ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ g A k ( ξ ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) = R k +1 k (cid:12)(cid:12) F q (cid:0) M e − z ◦ g A k ( ξ ) ν (cid:1)(cid:12)(cid:12) d Γ A k ( ξ ) ( z ) up to anerror of o ( q, k ), that is independent of ξ .The set X is produced via the central limit theorem, and the other parts follow from a locallimit theorem with target, both proved by Benoist-Quint [2, Theorem 12.1 and Theorem16.15]. We remark that the error o ( q, k ) is not explicit, and this is the main reason thatprevents us from obtaining a rate in Theorem 1.4. Also, this is the only part of the proofwhere the assumption Λ c = { } is used.3. Treating averages of scaled measures: Combining the previous two steps, for some C > |F q ( ν ) | ≤ C · Z X (cid:18)Z k +1 k (cid:12)(cid:12) F q (cid:0) M e − z ◦ g A k ( ξ ) ν (cid:1)(cid:12)(cid:12) dz (cid:19) d P ( ξ ) + o ( q, k ) + o ( q, k ) + o k (1) (3)To handle the terms in the last inequality, we make use of the continuity of ν via a Lemmaof Hochman [16]: Let θ ∈ P ( R ). Then for any r > Z |F q ( M e − t θ ) | dt ≤ e r · | q | + Z θ ( B r ( y )) dθ ( y )Now, the errors o ( q, k ), o ( q, k ) are, informally, large in q and small in e − k , e −√ k . Thus, to gothrough the three stages of the proof, we need to carefully balance these errors, by a careful choiceof k = k ( q ). Such a choice is possible since, roughly speaking, the linearization in step 1 can beshown to be exponentially fast in k + √ k , and using that both g A k ( ξ ) and g − A k ( ξ ) are uniformlyLipschitz in ξ , with constants ∼ e −√ k and ∼ e √ k , respectively (this choice is made in Lemma 3.4).Finally, we compare our method with that of Li-Sahlsten [25]: Here the IFS is self similar andthe assumption is that there exist independent contractions r i r j . They define a stopping time n k similar to our τ k , but by differentiating at 0. Then an inequality as in (2) follows directlyfrom self-similarity, without conditioning on the partition cells A k and without maps g A k . The keyto their argument is a renewal theorem [25, Proposition 2.1], based on Li’s earlier work [22], forthe residue process S n k − k , which is where they use the arithmetic assumption. This yields aninequality similar to (3) and the proof is concluded in an analogous way to ours. We note that inthe self-similar setting this renewal theorem and our local limit theorem (Theorem 2.9 below) canbe interchanged in step 2 of our argument - see Section 5.3 for more details.It is the treatment of non-linear IFS’s that makes the methods significantly different: We studythe asymptotic distribution of S τ k on [ k, k + 1]. This distribution is closely related to evolution ofthe derivative cocycle. However, our τ k is not a stopping time so we can not use the stationarityof ν directly to get (2), nor is it clear that a suitable local equidistribution phenomenon holdstrue for it. To get past both issues, we introduce the partitions A k . This allows us to obtain theinequality (2), with S τ k ( ω ) ( ω ) being the only random term conditional on ω ∈ A k ( ξ ). Furthermore,the error there decays exponentially fast. Using the aperiodicity of the derivative coycycle via theafformentioned limit Theorems from [2] we conclude that except for a small amount of ξ ’s, thelaw of S τ k ( ω ) ( ω ) conditional on ω ∈ A k ( ξ ) is uniformly close to the absolutely continuous measure7 A k ( ξ ) , with this distance both depending only on k and decaying as k → ∞ . Finally, the controlwe gain over the additional diffeomorphisms g A k ( ξ ) that arise from the linearization process as instep 1 is crucially important for the execution of step 3. For more details, see the proof of Theorem1.4 in Section 4. Organization
Section 2 is dedicated to the formulation and proof of a local limit Theorem forcocycle values at our ”stopping time” τ k (Theorem 2.9). This result, which is proved for intervals, isthen adapted to Fourier modes in Section 3.1. The rest of Section 3 is dedicated to the linearizationas in step 1 above, and to Hochman’s Lemma as in step 3 above. Section 4 then contains the proofof Theorem 1.4. We deduce Theorem 1.1, Corollary 1.2, and Theorem 1.3 in Section 5, that alsocontains a discussion about periodic IFS’s. Acknowledgements
This paper grew out of a reading seminar about Hochman’s recent proof[16] of Host’s equidistribution Theorem [17]. We thank Mike Hochman for providing us with apreprint of his work, and for some illuminating discussions about it. We also thank Tuomas Sahlstenand Connor Stevens for providing us with a preprint of their recent paper, and for interesting anduseful discussions.
Throughout this section we work with an IFS as in Theorem 1.4, and follow the notation introducedin Section 1.3. Notice that since for every ω ∈ A N we have x ω = lim m →∞ f ω | m (0)then f ω (cid:0) x σ ( ω ) (cid:1) = f ω (cid:16) lim m →∞ f σ ( ω ) | m (0) (cid:17) = x ω Also, by assumption (2) of Theorem 1.4 there exists
D, D ′ ∈ R such that D := min {− log | f ′ ( x ) | : f ∈ A, x ∈ [0 , } > , D ′ := max {− log | f ′ ( x ) | : f ∈ A, x ∈ [0 , } < ∞ (4)Equivalently, for every f ∈ A and x ∈ [0 , < e − D ′ ≤ | f ′ ( x ) | ≤ e − D < ω , χ := lim n − log | f ′ ω | n (0) | n > Definition 2.1.
For every k ∈ N and ω ∈ A N we define the random variable τ k ( ω ) := min { m : | f ′ ω | m ( x σ m ( ω ) ) | < e − kχ } Recall that, though it closely resembles one, τ k is not formally a stopping time. Definition 2.2.
Let X : A N → R be the random variable X ( ω ) := c ( ω , σ ( ω )) = − log | f ′ ω ( x σ ( ω ) ) | For every n > we define X n ( ω ) = − log | f ′ ω n (cid:0) x σ n ( x ω ) (cid:1) | = X ◦ σ n − θ be the law of the random variable X . Recall the definition of D and D ′ from (4). Thenfor every n , X n ∼ θ . Moreover, θ ∈ P ([ D, D ′ ]). In particular, the support of θ is bounded awayfrom 0. These are immediate from Definition 2.2 and equation (4). Definition 2.3.
For every n ∈ N let S n : A N → R be the random variable S n ( ω ) = n X i =1 X i ( ω )The following Lemma is a direct consequence of the chain rule: Lemma 2.4.
For every k ∈ N and ω ∈ A N we have τ k ( ω ) = min { n : S n ( ω ) ≥ kχ } and − log | f ′ ω | τk ( ω ) ( x σ τk ( ω ) ( ω ) ) | = S τ k ( ω ) ( ω ) ∈ [ kχ, kχ + D ′ ] Definition 2.5.
Given a finite word η = ( η , · · · , η ℓ ) :1. Denote by A η ⊆ A N the set of infinite words that begin with η , A η := { ω : ( ω , ..., ω ℓ ) = η } .
2. For k ∈ N , define A kη ⊆ A η by A kη := { ω ∈ A η : τ k ( ω ) − ℓ } = { ω : ( ω , · · · , ω τ k ( ω ) − ) = η } .
3. Let k ∈ N , and let η ′ = ( η ′ , · · · , η ′ ℓ ′ ) be another word of length ℓ ′ . We define the event A k,η,η ′ := { ω ∈ A η : σ τ k ( ω ) − ( ω ) ∈ A η ′ } . For k ′ ≥ , we also write A k ′ k,η,η ′ := { ω ∈ A η : σ τ k ( ω ) − ( ω ) ∈ A k ′ η ′ } . We note that given k , ℓ , k ′ , the the family of sets { A k ′ k,η,η ′ : η ∈ { , ..., n } ℓ , η ′ ∈ ∞ [ k =1 { , ..., n } ℓ ′ } forms a measurable partition A k ′ k,ℓ of A N . Thus for P -a.e. ω , the conditional measure P A k ′ k,η,η ′ := P A k ′ k,ℓ ω (5)is well defined on the atom A k ′ k,η,η ′ that contains ω . This atom will be denoted by A k ′ k,ℓ ( ω ) := A k ′ k,η,η ′ .In particular, if ω ∈ A η then there exists a unique η ′ such that A k ′ k,ℓ ( ω ) = A k ′ k,η,η ′ .Similarly, given ℓ , the family of sets A ℓ := { A η : η ∈ { , ..., n } ℓ } is a measurable partition. Wecan define A l ( ω ) = A η for P -a.e. ω where η = ( ω , · · · , ω ℓ ), as well as the conditional measure P A η := P A ℓ ω . 9 efinition 2.6. Let k, k ′ ∈ N and let η ∈ { , ..., n } ℓ . If for a finite word η ′ , A k ′ k,η,η ′ has positivemeasure, then we define a probability measure Γ A k ′ k,η,η ′ on [ kχ, kχ + D ′ ] by Γ A k ′ k,η,η ′ := R A k ′ η ′ λ | [ kχ,kχ + X ( ω ′ )] d P ( ω ′ ) R A k ′ η ′ X ( ω ′ ) d P ( ω ′ ) . Note that Γ A k ′ k,η,η ′ actually doesn’t depend on η . But we will still include η in the subscripts fornotational consistence. Lemma 2.7. If P ( A k ′ k,η,η ′ ) > then Γ A k ′ k,η,η ′ ≪ λ [ k · χ,kχ + D ′ ] with a density that depends only on p ,such that its norm can be bounded above by D independently of all parameters (including k and k ′ ).Proof. We write Γ instead of Γ A k ′ k,η,η ′ . It is clear that Γ ≪ λ [ k · χ,kχ + D ′ ] . Next, assuming x ∈ supp(Γ),we need to bound lim r → Γ( B ( x, r )) λ [ k · χ,kχ + D ′ ] ( B ( x, r ) = lim r → Γ( B ( x, r ))2 r assuming x is not an endpoint (which we may assume). Then for every ω ′ ∈ A k ′ k,η,η ′ λ (cid:16) [ kχ, kχ + X ( ω ′ )] \ B ( x, r ) (cid:17) ≤ r so Γ( B ( x, r ))2 r = R A k ′ η ′ λ ([ kχ, kχ + X ( ω ′ )] T B ( x, r )) d P ( ω )2 r · E A k ′ k,η,η ′ ( X ) ≤ E A k ′ k,η,η ′ ( X )Finally, by equation (4) we know that X ( ω ) ≥ D for every ω . We conclude that the density of Γis bounded by D , independently of all parameters. Notation 2.8.
We will use superscripts such as o k →∞ ( · ) in O ( · ) and o ( · ) to such bounds take placeas which variables are being varied. The variables on which the implied constants depend on willbe written in subscripts. The implied constant is absolute when no subscript is present. The following Theorem is one of the main keys to the proof of Theorem 1.4:
Theorem 2.9.
Let ℓ be a fixed parameter, k be large, k ′ = k ′ ( k ) depend on k and satisfy k ′ = o k →∞ ( k ) . Then for all η ∈ A ℓ , there exists a subset e A η ⊆ A η such that:(i) P ( e A η ) ≥ P ( A η ) · (1 − o k →∞ ℓ, p (1)) .(ii) for all ξ ∈ e A η , P ( A k ′ k,ℓ ( ξ )) > .(iii) for all ξ ∈ e A η and for any sub-interval J ⊆ [ kχ, kχ + D ′ ] , P A k ′ k,ℓ ( ξ ) ( S τ k ∈ J ) = Γ A k ′ k,ℓ ( ξ ) ( J ) + o k →∞ ℓ, p (1) . The Theorem is proved in the following Section, with some further preliminaries. We remarkthat to prove Theorem 1.4 we only need the case ℓ = 0 which means that A η is the full symbolicspace, but we prove this more general version for possible future use.10 .1 Proof of Theorem 2.9 The proof of Theorem 2.9 relies on two limit Theorems due to Benoist-Quint [2]. Before statingthem, we need some preliminaries.
Definition 2.10.
For every ρ ∈ (0 , we define a metric on { , ..., n } N via d ρ ( ω, ω ′ ) := ρ min { n : ω n = ω ′ n } , ρ ∈ (0 , A η is a locally constant function.For the following Claim, recall the definition of the maps ι a from Section 1.3: Claim 2.11.
For every a ∈ { , ..., n } the following statements hold true:1. For every ρ ∈ (0 , the map ι a is uniformly contracting: d ( ι a ( ω ) , ι a ( η )) = ρd ( ω, η )
2. Assume ρ > e − D · γ . Then the cocycle c ( a, ω ) is uniformly bounded, Lipschitz in ω , with auniformly bounded Lipschitz constant as a ∈ { , ..., n } varies. This is standard, since all the maps f i are C γ smooth and by equation (4). We are nowready to state a consequence of the central limit Theorem for cocycles with target, proved byBenoist-Quint: Theorem 2.12. [2, Theorem 12.1. part (i)] Let η be our fixed element as in Theorem 2.9. Thereexists a variance r = r ( p ) > such that:For every R ∈ R the function ψ = 1 A η × [ R, ∞ ) satisfies that P (cid:18) ω ∈ A η : | S n ( ω ) − n · χ |√ n ≥ R (cid:19) = P ( A η )( N (0 , r ) > R )(1 + o n →∞ ψ (1)) where ( N (0 , r ) > R ) stands for the probability that a Gaussian random variable with mean andvariance r is larger than R . We remark that [2, Theorem 12.1. part (i)] applies here since by Claim 2.11, the cocycle c ( · , · ) satisfies the bounded moment conditions [2, (11.14),(11.15)] and is not constant. We alsoremark that for Theorem 2.12 we do not need the assumption that Λ c = { } made in Theorem1.4. However, this assumption is crucial for the local limit Theorem for cocycles with target, alsoproved by Benoist-Quint: Theorem 2.13. [2, Theorem 16.15] Let η be our fixed element as in Theorem 2.9. Then for every ω ′ , ǫ > , and w ∈ R + P σ − m ( { ω ′ } ) (cid:0) ω ∈ A η , S m ( ω ) ∈ [ w, w + ǫχ ] (cid:1) = G √ mr ( w − mχ ) · P ( A η ) · ǫχ · (1 + o m →∞ ǫ,ℓ, p (1)) where G s ( · ) stands for the density of the Gaussian law N (0 , s ) , and r = r ( p ) is as in Theorem2.12. The decay rate in o m →∞ ǫ,ℓ, p (1) depends only on ǫχ , ℓ and p , and is uniform in ω ′ . To be precise, a-priori the decay rate in o k →∞ ǫ,ℓ, p (1) depends only on ǫχ , A η and p . Hence it onlydepends on ǫ , ℓ , and p as there are only finitely many η ’s of a given length ℓ . In this case, as inTheorem 2.12, Claim 2.11 implies that our cocycle c ( · , · ) satisfies the bounded moment conditions[2, (11.14),(11.15)]. It is aperiodic in the sense of [2, Equation (15.8)] by the assumption thatΛ c = { } . By aperiodicity, the cocycle ˜ c defined by [2, Equation (16.9)] is equal to c . So [2,Theorem 16.15] applies with, in the notations therein, X = { , ..., n } N , ϕ = A η , the convex set C being [0 , ǫχ ], and x = ω ′ . 11 .1.2 Proof of Theorem 2.9 For every r ∈ R let U r ( x ) = x + r be the translation map. In addition, for every k we define theinterval I k = [ k − p k log k, k + p k log k ] . (6)To begin the proof of Theorem 2.9, we decompose the left hand side in (iii) as P A k ′ k,ℓ ( ξ ) ( S τ k ∈ J ) = X m I k P A k ′ k,ℓ ( ξ ) ( τ k = m + 1 , S τ k ∈ J ) + X m ∈ I k P A k ′ k,ℓ ( ξ ) ( τ k = m + 1 , S τ k ∈ J ) . (7)The two terms are respectively treated by Proposition 2.15 and Proposition 2.14 below, andThe theorem follows. Proposition 2.14.
In the setting of Theorem 2.9, there exists a set e A η such that claims (i) and(ii) hold and for all ξ ∈ e A η , P A k ′ k,l ( ξ ) ( τ k − / ∈ I k ) = o k →∞ ℓ, p (1) . Proof.
We first prove the following claim: P A η ( τ k − / ∈ I k ) = o k →∞ ℓ, p (1) , for every η. (8)For the function b = b ( k ) = √ k log k −
1, suppose that | τ k ( ω ) − k | > b = b ( k ). We also fix asmall ǫ >
0. Without loss of generality, suppose first that τ k ( ω ) − k > b . This implies that S ⌊ k + b ⌋ ( ω ) < kχ and therefore | S ⌊ k + b ⌋ ( ω ) − χ · ⌊ k + b ⌋| ≥ | S ⌊ k + b ⌋ ( ω ) − kχ − bχ − χ | ≥ bχ Let r > R = R ( r, ǫ ) > N (0 , r ) > R ) = ǫ . Thenby Theorem 2.12 applied for the corresponding ψ , we get P (cid:0) ω ∈ A η : (cid:12)(cid:12) S ⌊ k + b ⌋ ( ω ) − ⌊ k + b ⌋ χ (cid:12)(cid:12) ≥ bχ (cid:1) = P ω ∈ A η : (cid:12)(cid:12) S ⌊ k + b ⌋ ( ω ) − ⌊ k + b ⌋ χ (cid:12)(cid:12)p ⌊ k + b ⌋ ≥ bχ p ⌊ k + b ⌋ } ! ≤ P ω ∈ A η : (cid:12)(cid:12) S ⌊ k + b ⌋ ( ω ) − ⌊ k + b ⌋ χ (cid:12)(cid:12)p ⌊ k + b ⌋ ≥ R ! = P ( A η )( N (0 , r ) > R )(1 + o k →∞ ψ (1))= ǫ + o k →∞ ǫ,ℓ, p (1) , Here we used that bχ √ ⌊ k + b ⌋ → ∞ as k → ∞ . Since ǫ arbitrary, it follows that P ( τ k ( ω ) − > b ) ≤ P (cid:0)(cid:12)(cid:12) S ⌊ k + b ⌋ ( ω ) − ⌊ k + b ⌋ χ (cid:12)(cid:12) ≥ bχ (cid:1) = o k →∞ ℓ, p (1)A similar argument shows P ( τ k ( ω ) − k < − b )) ≤ P (cid:0)(cid:12)(cid:12) S ⌈ k − b ⌉ ( ω ) − ⌈ k − b ⌉ χ (cid:12)(cid:12) ≥ bχ (cid:1) = o k →∞ ℓ, p (1) . The claim (8) then follows by combining the two inequalities above.The deduction of the proposition from (8) is standard. Indeed, it suffices to set e A η = n ξ ∈ A η : P ( A k ′ k,ℓ ( ξ )) > , P A k ′ k,ℓ ( ξ ) ( τ k − / ∈ I k ) ≤ q P A η ( τ k − / ∈ I k ) o . Then P A η ( A η \ e A η ) ≤ p P A η ( τ k − / ∈ I k ) = o k →∞ ℓ, p (1).12e now take care of the second term in (7) Proposition 2.15.
In the setting of Theorem 2.9, for all ξ in the set e A η from Proposition 2.14, X m ∈ I k P A k ′ k,ℓ ( ξ ) ( τ k = m + 1 , S τ k ∈ J ) = Γ A k ′ k,ℓ ( ξ ) ( J ) + o k →∞ ℓ, p (1) . Proof.
Let η ′ be the finite word such that A k ′ k,ℓ ( ξ ) = A k ′ k,η,η ′ . We first notice that X m ∈ I k P A k ′ k,η,η ′ ( τ k = m + 1 , S τ k ∈ J )= P m ∈ I k P (cid:0) ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J (cid:1) P ( A k ′ k,η,η ′ ) (9)Each summand in the numerator can be written as P (cid:0) ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J (cid:1) = P ( ω ∈ A η ∩ σ − m ( A k ′ η ′ ) , S m < kχ, S m +1 ∈ J )= Z A k ′ η ′ P σ − m ( { ω ′ } ) (cid:0) ω ∈ A η , S m < kχ, S m +1 ∈ J (cid:1) d P ( σ − m ( { ω ′ } ))= Z A k ′ η ′ P σ − m ( { ω ′ } ) (cid:0) ω ∈ A η , S m < kχ, S m + X ( ω ′ ) ∈ J (cid:1) d P ( ω ′ )= Z A k ′ η ′ P σ − m ( { ω ′ } ) (cid:0) ω ∈ A η , S m ∈ J ω ′ (cid:1) d P ( ω ′ ) (10)where the interval J ω ′ is defined by J ω ′ := [ kχ − X ( ω ′ ) , kχ ) ∩ U − X ( ω ′ ) J. (11)Fix an arbitrarily small ǫ >
0. By Theorem 2.13, for all translates W ⊆ J ω ′ of the form[ w, w + ǫχ ), P σ − m ( { ω ′ } ) (cid:0) ω ∈ A η , S m ∈ W (cid:1) = G √ mr ( w − mχ ) · P ( A η ) · ǫχ · (1 + o m →∞ ǫ,ℓ, p (1)) (12)where we recall that G s ( w ) dw stands for the density of N (0 , s ), r > o k →∞ ǫ,ℓ, p (1) in (12) depends only on ǫ , ℓ , and p , and is uniform in ω ′ . Because m ∈ I k , m → ∞ if andonly if k → ∞ , so we know by Lemma 2.16 below that P σ − m ( ω ′ ) ( ω ∈ A η , S m ∈ W ) = G √ kr (( m − k + β ) χ ) P ( A η ) ǫχ · (1 + o k →∞ ǫ,ℓ, p (1))for all ω ′ ∈ A k ′ η ′ , m ∈ I k and β ∈ [0 ,
1) as k → ∞ .Now, since the interval J ω ′ contains ⌊ λ ( J ω ′ ) ǫχ ⌋ disjoint intervals of the form [ w, w + ǫχ ) and iscovered by ⌈ λ ( J ω ′ ) ǫχ ⌉ such intervals, we know that for all m ∈ I k and β ∈ [0 , P σ − m ( ω ′ ) ( ω ∈ A η , S m ∈ J ω ′ )= G √ kr (( m − k + β ) χ ) · P ( A η ) · (cid:16) λ ( J ω ′ ) ǫχ + O (1) (cid:17) · ǫχ · (1 + o k →∞ ǫ,ℓ, p (1))= G √ kr (( m − k + β ) χ ) · P ( A η ) · (cid:0) λ ( J ω ′ ) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) , (13)13here the implied constant in the O ( ǫχ ) is 1: the term represented by O ( ǫχ ) is of absolute valuebounded by ǫχ . The error term o k →∞ ǫ,ℓ,p (1) is uniform in J , ω ′ and β .Integrating (13) inside (10) leads to P ( ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J )= Z A k ′ η ′ G √ kr (( m − k + β ) χ ) · P ( A η ) · (cid:0) λ ( J ω ′ ) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) d P ( ω ′ )= G √ kr (( m − k + β ) χ ) · P ( A η ) · P ( A k ′ η ′ ) · (cid:0) E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) By summing over m ∈ I k and integrating over β ∈ [0 , X m ∈ I k P ( ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J )= Z k + √ k log k +1 k −√ k log k G √ kr (( t − k ) χ ) dt ! · P ( A η ) P ( A k ′ η ′ ) · (cid:0) E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) = Z √ k log k +1 −√ k log k G √ kr ( tχ ) dt ! · P ( A η ) P ( A k ′ η ′ ) · (cid:0) E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) = Z √ log k + √ k −√ log k G r ( tχ ) dt ! · P ( A η ) P ( A k ′ η ′ ) · (cid:0) E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) =( 1 χ − o k →∞ p (1)) · P ( A η ) P ( A k ′ η ′ ) · (cid:0) E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫχ ) + o k →∞ ǫ,ℓ, p (1) (cid:1) = P ( A η ) P ( A k ′ η ′ ) · (cid:18) χ E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + O ( ǫ ) + o k →∞ ǫ,ℓ, p (1) (cid:19) (14)as k → ∞ . The term represented by O ( ǫ ) is uniformly bounded by ǫ in absolute value.As ǫ > J ⊂ [ kχ, kχ + D ′ ), X m ∈ I k P ( ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J )= P ( A η ) P ( A k ′ η ′ ) · (cid:18) χ E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + o k →∞ ℓ, p (1) (cid:19) , (15)where J ω ′ is defined by (11) and the error term o k →∞ ℓ, p (1) is uniform in J , ω ′ and β .Consider the special case of J = [ kχ, kχ + D ′ ), where J ω ′ = [ kχ − X ( ω ′ ) , kχ ). Because theevent { τ k = m + 1 , S τ k ∈ [ kχ, kχ + D ′ ) } coincides with { τ k = m + 1 } , we obtain X m ∈ I k P ( ω ∈ A k ′ k,η,η ′ , τ k = m + 1)= P ( A η ) P ( A k ′ η ′ ) · (cid:18) χ E ω ′ ∈ A η ′ (cid:0) λ ([ kχ − X ( ω ′ ) , kχ )) (cid:1) + o k →∞ ℓ, p (1) (cid:19) = P ( A η ) P ( A k ′ η ′ ) · (cid:18) χ E ω ′ ∈ A η ′ X ( ω ′ ) + o k →∞ ℓ, p (1) (cid:19) . (16)14herefore, by (9), (15) and (16), X m ∈ I k P A k ′ k,η,η ′ ( τ k = m + 1 , S τ k ∈ J )= P m ∈ I k P (cid:0) ω ∈ A k ′ k,η,η ′ , τ k = m + 1 , S m +1 ∈ J (cid:1)P m ∈ I k P (cid:0) ω ∈ A k ′ k,η,η ′ , τ k = m + 1 (cid:1) · P m ∈ I k P (cid:0) ω ∈ A k ′ k,η,η ′ , τ k = m + 1 (cid:1) P ( A k ′ k,η,η ′ )= χ E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + o k →∞ ℓ,p (1) χ E ω ′ ∈ A η ′ X ( ω ′ ) + o k →∞ ℓ, p (1) · P A k ′ k,η,η ′ ( τ k − ∈ I k )= χ E ω ′ ∈ A ′ η ( λ ( J ω ′ )) + o k →∞ ℓ,p (1) χ E ω ′ ∈ A η ′ X ( ω ′ ) + o k →∞ ℓ, p (1) · (cid:0) − o k →∞ ℓ,p (1) (cid:1) (since ξ ∈ e A η and A k ′ k,η,η ′ = A k ′ k,ℓ ( ξ ) )= E ω ′ ∈ A ′ η λ ( J ω ′ ) E ω ′ ∈ A η ′ X ( ω ′ ) + o k →∞ ℓ,p (1) ! · (cid:0) − o k →∞ ℓ, p (1) (cid:1) (since 0 < D ≤ E ω ′ ∈ A η ′ X ( ω ′ ) ≤ D ′ )= E ω ′ ∈ A ′ η λ ( J ω ′ ) E ω ′ ∈ A η ′ X ( ω ′ ) + o k →∞ ℓ, p (1) . (17)To conclude, it suffices to notice that E ω ′∈ A ′ η λ ( J ω ′ ) E ω ′∈ Aη ′ X ( ω ′ ) is exactly Γ A k ′ k,η,η ′ ( J ). Lemma 2.16.
For m ∈ I k , w ∈ [ kχ − D ′ , kχ ] and β ∈ [0 , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G √ mr ( w − mχ ) G √ kr (( m − k + β ) χ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k − (log k ) as k → ∞ , where the implied constant depends only on p .Proof. Recall G s ( x ) = √ πs exp( − x s ) and log G s ( x ) = − log √ π − log s − x s .So as k → ∞ , (cid:12)(cid:12) log G √ mr ( w − mχ ) − log G √ kr ( w − mχ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) log( √ mr ) − log( √ kr ) | + ( w − mχ ) r (cid:12)(cid:12) m − k (cid:12)(cid:12) ≤ (cid:12)(cid:12) log mk (cid:12)(cid:12) + ( | k − m | χ + D ′ ) mr (cid:12)(cid:12) mk − (cid:12)(cid:12) . (cid:12)(cid:12) mk − (cid:12)(cid:12) + ( √ k log k + 1) k (cid:12)(cid:12) mk − (cid:12)(cid:12) . (log k ) (cid:12)(cid:12) mk − (cid:12)(cid:12) . (log k ) √ log k √ k = k − (log k ) , where the implied constant depends only on χ , D ′ and r , and hence only on p .Moreover, (cid:12)(cid:12) log G √ kr ( w − mχ ) − log G √ kr (( m − k + β ) χ ) (cid:12)(cid:12) = 12 kr (cid:12)(cid:12) ( w − mχ ) − ( kχ − mχ + βχ ) (cid:12)(cid:12) = 12 kr · | w − kχ − βχ | · | k − m − β ) χ + ( w − kχ − βχ ) |≤ kr · ( D ′ + βχ ) · (cid:0) p k log k + 1) χ + ( D ′ + βχ ) (cid:1) . k − (log k ) , p .Combining the two inequalities aboves shows (cid:12)(cid:12) log G √ mr ( w − mχ ) − log G √ kr (( m − k + β ) χ ) (cid:12)(cid:12) . k − (log k ) , which in turn implies the lemma. In this Section we still assume the conditions and setting of Theorem 1.4. First, we recall two usefulresults. The first is the bounded distortion property, which holds in our situation since every f i isat least C γ smooth and strictly contracting in the sense of assumption (2) of Theorem 1.4 (seee.g. [30, The discussion about equation (1.3)]): Theorem 3.1.
There exists some
L > such that: for any k ∈ N and for any word η ∈ { , ..., n } k L − ≤ (cid:12)(cid:12) f ′ η ( x ) (cid:12)(cid:12)(cid:12)(cid:12) f ′ η ( y ) (cid:12)(cid:12) ≤ L, for any x, y ∈ [0 , Lemma 3.2.
Let ν be a self conformal measure with respect to an IFS satisfying condition (2) ofTheorem 1.4. If ν has an atom then ν is a Dirac mass. In particular, if ν is not a Dirac mass, itis continuous. That is, for every ǫ > there is a δ > such that for any y ∈ [0 , , ν ( B δ ( y )) < ǫ where B δ ( y ) is the open ball about y of radius δ > . For a given Borel probability measure ρ ∈ P ( R ), for every q ∈ R we define a function g q,ρ : R → R via g q,ρ ( t ) = |F q ( M e − t ρ ) | where M s ( x ) = s · x for any s, x ∈ R .Next, fixing ℓ = 0 in Theorem 2.9, we define the sequence o k := o k →∞ p . Notice that theassumption ℓ = 0 means that A η is the entire symbolic space (since the only word of length 0 is theempty word). The following Theorem is needed in conjunction with Theorem 2.9, since in practicewe will need a version of Theorem 2.9 for functions rather than intervals. Theorem 3.3.
Let q be large, let C > and let k = k ( q ) be defined implicitly as an integersatisfying | q | = Θ C (cid:18) o − k e ( k + k ′ ) χ (cid:19) (18) where we assume k ′ = [ √ k ] . Let ρ ∈ P ( R ) be a measure such thatdiam ( supp ( ρ )) = O ( e − k ′ χ ) Then for every ξ ∈ e A η ⊂ A η as in Theorem 2.9, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E A k ′ k ( ξ ) (cid:2) g q,ρ ( S τ k ( ω ) ) (cid:3) − Z kχ + D ′ kχ g q,ρ ( x ) d Γ A k ′ k,ℓ ( ξ ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( o k )16 roof. We first claim that the function g q,ρ ( t ) is 4 πqe − χk · diam (supp ( ρ )) Lipschitz, whenever t ∈ [ kχ, kχ + D ′ ]. Indeed, since the complex exponential is a 1-Lipschitz function, for any x, y ∈ supp( ρ )and t, s ∈ [ kχ, kχ + D ′ ] we have | exp(2 πiqe − t ( x − y )) − exp(2 πiqe − s ( x − y )) | ≤ | πq ( x − y ) | · | e − t − e − s | ≤ π diam (supp ( ρ )) · e − kχ Since the L norm is always bounded by the L ∞ norm, and since g q,ρ ( t ) = |F q ( M e − t ρ ) | = Z Z exp(2 πiqe − t ( x − y )) dρ ( x ) dρ ( y )the Claim follows.Recalling that k = k ( q ) satisfies q = Θ C (cid:18) o − k e ( k + k ′ ) χ (cid:19) and that diam (supp ( ρ )) = O ( e − k ′ χ )it follows that the function t ∈ [ kχ, kχ + D ′ ] g q,ρ ( t )is o − k Lipschitz (up to a constant universal multiplicative factor). Therefore, there exists a stepfunction ψ : [ kχ, kχ + D ′ ] → R such that:1. ψ consists of o − k steps (indicators of intervals).2. || ψ − g q,ρ || ∞ ≤ o k on the interval [ kχ, kχ + D ′ ].Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E A k ′ k ( ξ ) (cid:2) g q,ρ ( S τ k ( ω ) ) (cid:3) − Z kχ + D ′ kχ g q,ρ ( x ) d Γ A k ′ k,ℓ ( ξ ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E A k ′ k ( ξ ) (cid:2) g q,ρ ( S τ k ( ω ) ) (cid:3) − E A k ′ k ( ξ ) (cid:2) ψ ( S τ k ( ω ) ) (cid:3)(cid:12)(cid:12)(cid:12) (19)+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E A k ′ k ( ξ ) (cid:2) ψ ( S τ k ( ω ) ) (cid:3) − Z kχ + D ′ kχ ψ ( x ) d Γ A k ′ k,ℓ ( ξ ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20)+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z kχ + D ′ kχ ψ ( x ) d Γ A k ′ k,ℓ ( ξ ) ( x ) − Z kχ + D ′ kχ g q,ρ ( x ) d Γ A k ′ k,ℓ ( ξ ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (21)Now, the terms in (19) and (21) are bounded by o k by point 2 above.The term in (20) is bounded by o − k · o k since by point 1 above there are at most o − k steps in ψ ,and since by Theorem 2.9 each such step introduces an error of at most o k .In the context of Theorem 3.3, it is natural to ask about the existence of integers k that satisfy(18) with respect to some C >
1. This is the content of the following Lemma:17 emma 3.4.
By potentially making o k go to zero slower, we may assume that there exists a constant C > such that for every q large there exists an integer k = k ( q ) such that | q | = Θ C (cid:18) o − k e ( k +[ √ k ]) χ (cid:19) Proof.
We first make the following assumptions on o k :1. o k ≥ k . Otherwise, we move to the sequence a k = max { o k , k } . Then o k ≤ a k and still a k → k define v k = sup { o n : n ≥ k } . Then o k ≤ v k , v k is decreasing, and it is clear that v k → k we have · o k ≤ o k +1 ≤ o k . Otherwise, we move to the recursively definedsequence b k , where b = o , b k = max { o k , b k − } Then b k ≥ o k ≥ k and b k ≤ b k +1 ≤ b k , b k → g : R + → R + be a smooth monotonic decreasing function such that g ( k ) = o k . Let q be large. Find x ∈ R + such that | q | = g ( x ) − · e ( x + √ x ) χ Notice that |
14 log 1 g ([ x ]) + ( x + √ x ) χ −
14 log 1 g ( x ) − ([ x ] + hp [ x ] i ) χ |≤ |
14 log g ( x ) g ([ x ]) | + | ( x + √ x − [ x ] − hp [ x ] i ) · χ |≤ | log g ([ x ] + 1) g ([ x ]) | + 3 χ ≤ | log 4 | + 3 χ It follows that g ( x ) − · e ( x + √ x ) χ g ([ x ]) − · e ([ x ]+ h √ [ x ] i χ = exp (cid:18)
14 log 1 g ([ x ]) + ( x + √ x ) χ −
14 log 1 g ( x ) − ([ x ] + hp [ x ] i ) χ (cid:19) = O (1)We thus choose, for every q large, our k as [ x ]. It follows that there is some C > C − ≤ qo − k e ( k +[ √ k ]) χ ≤ C which is what we claimed. 18 .2 Linearization of compositions of C γ functions Suppose I ⊂ R is a compact interval and Φ is a family of differentiable contractions φ : I → I satisfying λ ′ ≤ k φ ′ k C ≤ λ, k φ k C γ ≤ C, ∀ φ ∈ Φ (22)for some 0 < λ ′ < λ < γ ∈ (0 ,
1) and
C >
0. DefineΦ ∗ n := { φ ◦ · · · ◦ φ n : φ , · · · , φ n ∈ Φ } . Lemma 3.5.
For every β ∈ (0 , γ ) there exists ǫ ∈ (0 , such that for all n ≥ , g ∈ Φ ∗ n and x, y ∈ I satisfying | x − y | < ǫ , (cid:12)(cid:12) g ( x ) − g ( y ) − g ′ ( y )( x − y ) (cid:12)(cid:12) ≤ | g ′ ( y ) | · | x − y | β . What Lemma 3.5 means is that for every y ∈ B ǫ ( x ) the function g may be approximatedexponentially fast on B ǫ ( x ) by an affine map with similarity ratio g ′ ( y ). Proof.
For all x, y ∈ I and φ ∈ Φ, there is an intermediate value z between x and y such that (cid:12)(cid:12) φ ( x ) − φ ( y ) − φ ′ ( y )( x − y ) (cid:12)(cid:12) = (cid:12)(cid:12) φ ′ ( z )( x − y ) − φ ′ ( y )( x − y ) (cid:12)(cid:12) ≤| φ ′ ( z ) − φ ′ ( y ) || x − y | ≤ C | z − y | γ | x − y |≤ C | x − y | γ . (23)Define a sequence β > β > β > · · · by β = γ , β n = β n − − bλ ( n − γ , where the constant b := (1 − λ γ )( γ − β ) > β n = β . Choose ǫ ∈ (0 , e ) sufficiently small, suchthat 4 Cλ ′ ǫ γ − β < min(1 − e , β . (24)We will prove inductively that: If n ≥ , | x − y | < ǫ and g ∈ Φ n , then (cid:12)(cid:12) g ( x ) − g ( y ) − g ′ ( y )( x − y ) (cid:12)(cid:12) ≤ | g ′ ( y ) | · | x − y | β n . (25)For the n = 0 case, assume | x − y | < ǫ and g ∈ Φ ∗ = { Id } , then | g ( x ) − g ( y ) − g ′ ( y )( x − y ) (cid:12)(cid:12) = | ( x − y ) − ( x − y ) | = 0 and (25) holds.Assume n ≥ n −
1. Suppose | x − y | = δ < ǫ and g ∈ Φ ∗ n . Then g = φ ◦ ˜ g where φ ∈ Φ and ˜ g ∈ Φ ∗ ( n − , and (cid:12)(cid:12) ˜ g ( x ) − ˜ g ( y ) − ˜ g ′ ( y )( x − y ) (cid:12)(cid:12) ≤ | ˜ g ′ ( y ) | δ β n − . (26)In particular, it follows that | ˜ g ( x ) − ˜ g ( y ) | ≤ | ˜ g ′ ( y ) | δ (1 + δ β n − ) . (27)Combining (23) and (27), we get (cid:12)(cid:12) g ( x ) − g ( y ) − g ′ ( y )( x − y ) (cid:12)(cid:12) = (cid:12)(cid:12) φ (˜ g ( x )) − φ (˜ g ( y )) − φ ′ (˜ g ( y ))˜ g ′ ( y )( x − y ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) φ (˜ g ( x )) − φ (˜ g ( y )) − φ ′ (˜ g ( y ))(˜ g ( x ) − ˜ g ( y )) (cid:12)(cid:12) + | φ ′ (˜ g ( y )) | · (cid:12)(cid:12) ˜ g ( x ) − ˜ g ( y ) − ˜ g ′ ( y )( x − y ) (cid:12)(cid:12) ≤ C | ˜ g ( x ) − ˜ g ( y ) | γ + | φ ′ (˜ g ( y )) | · | ˜ g ′ ( y ) | δ β n − ≤ C (cid:0) | ˜ g ′ ( y ) | δ (1 + δ β n − ) (cid:1) γ + | g ′ ( y ) | δ β n − (28)19ecause 0 < δ < ǫ <
1, we know (28) is bounded by: (cid:12)(cid:12) g ( x ) − g ( y ) − ( D y g )( x − y ) (cid:12)(cid:12) ≤ C | ˜ g ′ ( y ) | · | ˜ g ′ ( y ) | γ δ γ + | g ′ ( y ) | δ β n − = (cid:16) C | φ ′ (˜ g ( y )) | − | ˜ g ′ ( y ) | γ δ γ − β n + δ β n − − β n (cid:17) | g ′ ( y ) | δ β n ≤ (cid:16) Cλ ′ λ ( n − γ δ γ − β n + δ β n − − β n (cid:17) | g ′ ( y ) | δ β n . (29)To complete the induction, it suffices to prove4 Cλ ′ λ ( n − γ δ γ − β n + δ β n − − β n ≤ . (30)We distingush between the cases log δ ≥ β n − − β n and log δ < β n − − β n .If log δ ≥ β n − − β n , then δ β n − − β n ≤ e . Moreover, by (24), Cλ ′ λ ( n − γ δ γ − β n ≤ Cλ ′ ǫ γ − β < − e .Thus (30) holds in this case.Assume now log δ < β n − − β n . Since e − t ≤ − t on [0 , λ β n − − β n = e − (log δ )( β n − − β n ) ≤ −
12 (log 1 δ )( β n − − β n ) ≤ −
12 ( β n − − β n ) = 1 − bλ ( n − γ ≤ − Cλ ′ λ ( n − γ ǫ γ − β ≤ − Cλ ′ λ ( n − γ δ γ − β n . Here we used the facts that 0 < δ < ǫ < e , β n ≤ β < γ and assumption (24). Hence (30) holds inthis case as well.We have established the inductive statement (25). As β < β n and | x − y | < ǫ <
1, the lemmathen follows.
The following Lemma is adapted from a recent paper of Hochman:
Lemma 3.6. [16, Lemm 3.2] Let θ ∈ P ( R ) , k ∈ N and χ, D as in the previous Sections. Then forany r > and q = 0 , Z kχ + D ′ k · χ |F q ( M e − t θ ) | dt ≤ D ′ · (cid:18) e r · | q | + Z θ ( B e χk · r ( y )) dθ ( y ) (cid:19) In fact, Hochman’s Lemma states that for any θ ∈ P ( R ), any r >
0, an any m = 0, Z |F m ( M p t θ ) | dt ≤ r · | m | · log p + Z θ ( B r ( y )) dθ ( y )where here p >
1. 20n the context of Lemma 3.6, we apply this result for p = e − and the measure M e − kχ θ betweenthe scales 0 and D . Then the same proof yields Z kχ + D ′ k · χ |F m ( M e − t θ ) | dt = Z D ′ |F m ( M e − t ( M e − kχ θ )) | dt ≤ D ′ · (cid:18) e r · | m | + Z M e − kχ θ ( B r ( y )) dM e − kχ θ ( y ) (cid:19) = D ′ · (cid:18) e r · | m | + Z θ ( B e χk · r ( y )) dθ ( y ) (cid:19) which is Lemma 3.6. Let ν be as in Theorem 1.4. Under the assumptions of Theorem 1.4, our goal is to show thatlim | q |→∞ F q ( ν ) = 0So, let ǫ >
0, let | q | be large, and choose k = k ( q ) ∈ N as in Lemma 3.4. Recall that this meansthat for some C > q = Θ C (cid:18) o − k · e ( k + k ′ ) χ (cid:19) where our standing assumption is that k ′ = [ √ k ]By Lemma 3.4 (and its proof), any requirement that k be large translates to a requirement on q being large. In the subsequent argument, we let k be large, and fix ℓ = 0. Recall that ℓ = 0means that the event A η is the entire symbolic space. We also define a bona-fide stopping time˜ β k : { , ..., n } N → N by ˜ β k ( ω ) = min { m : | f ′ ω | m (0) | < e − ( k + k ′ ) χ } (31) Lemma 4.1.
For every k ∈ N , ν = E ( f ω | ˜ βk ( ω ) ν ) Proof.
This is standard, and follows since ν is self-conformal. See e.g. [3, Lemma 2.2.4].So, by Lemma 4.1 and Jensen’s inequality we obtain |F q ( ν ) | = (cid:12)(cid:12)(cid:12) F q ( E ( f ω | ˜ βk ( ω ) ν )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E (cid:16) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ E (cid:18)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:19) Next, appealing to Theorem 2.9 with our choice of k, ℓ, k ′ , there is a subset ˜ A η ⊆ A η with P ( ˜ A η ) ≥ − o k (1) such that 21 (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) == Z ξ ∈ A η \ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ )Combining this with the previous equation array, and using that (cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) ≤ |F q ( ν ) | ≤ Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + o k (1) (32)Next, we take a closer look at the maps f ω | ˜ βk ( ω ) : Lemma 4.2.
There exists some integer
P > such that:For all k large enough, and for every ω , letting η ′ be such that ω ∈ A k ′ k,η,η ′ , we have (cid:12)(cid:12)(cid:12) ˜ β k ( ω ) − τ k ( ω ) − | η ′ | (cid:12)(cid:12)(cid:12) ≤ P Proof.
We first observe that, by the definition of ˜ β k from (31), f ω | ˜ βk ( ω ) = f ω | τk ( ω ) ◦ f ω | ˜ βk ( ω ) τk ( ω ) , and f ω | τk ( ω )+ | η ′| = f ω | τk ( ω ) ◦ f η ′ So, either ω | ˜ β k ( ω ) τ k ( ω ) is a prefix of η ′ , or vice versa. By the last displayed equation, for any x ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12) f ′ ω | ˜ βk ( ω ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ω | τk ( ω ) f ω | ˜ βk ( ω ) τk ( ω ) ( x ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | f ′ ω | ˜ βk ( ω ) τk ( ω ) ( x ) | Now, it is a consequence of Theorem 3.1 (bounded distortion), the definition of ˜ β k , and of theDefinition 2.1 of τ k , that for some L > y ∈ [0 , L − · e − ( k + k ′ ) χ − D ′ ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ′ ω | ˜ βk ( ω ) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L · e − ( k + k ′ ) χ , L − · e − kχ − D ′ ≤ (cid:12)(cid:12)(cid:12) f ′ ω | τk ( ω ) ( y ) (cid:12)(cid:12)(cid:12) ≤ L · e − kχ (33)combining the last two displayed equations, we see that there some constant C ′ > | f ′ ω | ˜ βk ( ω ) τk ( ω ) ( x ) | = Θ C ′ (cid:16) e − k ′ χ (cid:17) , ∀ x ∈ [0 ,
1] (34)On the other hand, by the definition of the event A k ′ k,η,η ′ e − χk ′ − D ′ ≤ | f ′ η ′ ( x σ τk ( ω ) ( ω ) ) | < e − χk ′ So, applying Theorem 3.1 | f η ′ ( x ) | = Θ L (cid:16) e − k ′ χ (cid:17) , ∀ x ∈ [0 ,
1] (35)Therefore, combining equation (35) with (34) (and noting that the constants C ′ , L are uniform),that either ω | ˜ β k ( ω ) τ k ( ω ) is a prefix of η ′ or vice versa, and equation (4), the Lemma follows.22et P be as in Lemma 4.2. For every η ′ ∈ { , ..., n } ∗ of length | η ′ | > P we define¯ η ′ := η ′ | | η ′ |− P That is, ¯ η ′ is the prefix of η ′ of length | η ′ | − P . It is now a corollary of Lemma 4.2 that for any ω ,if η ′ = η ′ ( ω ) is as in Lemma 4.2, then there is a word ρ ω,k such that f ω | ˜ βk ( ω ) = f ω | τk ( ω ) ◦ f ¯ η ′ ◦ f ρ ω,k and | ρ ω,k | ≤ P .With this information, we revisit equation (32). Recall that M s ( t ) = s · t . Claim 4.3.
Fix β ∈ (0 , γ ) . Then for all k large enough, |F q ( ν ) | ≤ Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F q M e − Sτk ( ω )( ω ) ◦ M sign (cid:18) f ′ ω | τk ( x ) ( x στk ( ω )( ω ) ) (cid:19) ◦ f ¯ η ′ ◦ f ρ ω,k ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d P ( ξ )+ O ( q · e − ( k + k ′ ) χ − β · k ′ χ ) + o k (1) where for every ξ ∈ ˜ A η , recalling that A k ′ k,ℓ ( ξ ) = A k ′ k,η,η ′ , η ′ is defined as η ′ = η ′ ( ξ ) .Proof. Fix ξ ∈ ˜ A η . Assuming A k ′ k,ℓ ( ξ ) = A k ′ k,η,η ′ , let η ′ be this η ′ = η ′ ( ξ ). Assume ω ∈ A k ′ k,η,η ′ . Thenwe have seen that there is a word ρ ω,k such that f ω | ˜ βk ( ω ) = f ω | τk ( ω ) ◦ f ¯ η ′ ◦ f ρ ω,k (36)and | ρ ω,k | ≤ P . It follows from the proof of Lemma 4.2 (specifically, equation (34)) that theresome constant C ′ > | (cid:0) f ¯ η ′ ◦ f ρ ω,k (cid:1) ′ ( x ) | = Θ C ′ (cid:16) e − k ′ χ (cid:17) , ∀ x ∈ [0 ,
1] (37)Also, it is a consequence of (36) that there exists some z ∈ [0 ,
1] with f ¯ η ′ ◦ f ρ ω,k ( z ) = x σ τk ( ω ) ( ω ) .Now, plug into Lemma 3.5 the parameters g = f ω | τk ( ω ) , y = f ¯ η ′ ◦ f ρ ω,k ( z ) and for x ∈ [0 ,
1] weplug in f ¯ η ′ ◦ f ρ ω,k ( x ). Then, by (36) and assuming k (and therefore k ′ ) are large enough, (cid:12)(cid:12)(cid:12) f ω | ˜ βk ( ω ) ( x ) − f ω | τk ( x ) (cid:0) f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:1) − f ′ ω | τk ( x ) (cid:0) f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:1) (cid:0) f ¯ η ′ ◦ f ρ ω,k ( x ) − f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ | f ′ ω | τk ( x ) ( f ¯ η ′ ◦ f ρ ω,k ( z )) | · (cid:12)(cid:12) f ¯ η ′ ◦ f ρ ω,k ( x ) − f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:12)(cid:12) β And, by the definition of τ k , since f ¯ η ′ ◦ f ρ ω,k ( z ) = x σ τk ( ω ) ( ω ) , and by (37) | f ′ ω | τk ( x ) (cid:0) f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:1) | ≤ e − kχ , (cid:12)(cid:12) f ¯ η ′ ◦ f ρ ω,k ( x ) − f ¯ η ′ ◦ f ρ ω,k ( z ) (cid:12)(cid:12) ≤ O ( e − k ′ χ )Now, for every ω ∈ A k ′ k,η,η ′ we define a smooth map S ω,k,η ′ : [0 , → R via S ω,k,η ′ ( x ) = (cid:12)(cid:12)(cid:12) f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) (cid:12)(cid:12)(cid:12) · sign (cid:16) f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) (cid:17) · f ¯ η ′ ◦ f ρ ω,k ( x ) (38) − f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) · x σ τk ( ω ) ( ω ) + f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) )23his map is affine in sign (cid:16) f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) (cid:17) · f ¯ η ′ ◦ f ρ ω,k ( x ). Then we have just shown that || f ω | ˜ βk ( ω ) − S ω,k,η ′ || C ([0 , ≤ O ( e − ( k + k ′ ) χ − βk ′ χ )So, since F q ( · ) is a 2 πq -Lipschitz function, (cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17) − F q (cid:0) S ω,k,η ′ ν (cid:1)(cid:12)(cid:12)(cid:12) ≤ O ( q · e − ( k + k ′ ) χ − βk ′ χ )Therefore (cid:12)(cid:12)(cid:12) |F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17) | − |F q (cid:0) S ω,k,η ′ ν (cid:1) | (cid:12)(cid:12)(cid:12) ≤ O ( q · e − ( k + k ′ ) χ − βk ′ χ )and so for every event A k ′ k,η,η ′ we have (cid:12)(cid:12)(cid:12)(cid:12) E A k ′ k,η,η ′ (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | ˜ βk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) − E A k ′ k,η,η ′ h(cid:12)(cid:12) F q (cid:0) S ω,k,η ′ ν (cid:1)(cid:12)(cid:12) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( q · e − ( k + k ′ ) χ − βk ′ χ ) (39)Finally, recall equation (32), and recall the definition of the maps S ω,k,η ′ from (38). Note thatlog | f ′ ω | τk ( ω ) ( x σ τk ( ω ) ( ω ) ) | = − S τ k ( ω ) ( ω )by Lemma 2.4. The Claim follows from (32) and (39), since the translation of S ω,k,η ′ does noteffect the absolute value of F q ( · ), by integrating over all ξ ∈ ˜ A η (using that the bounds we got areuniform in ξ ). Corollary 4.4.
There is some K = K ( ǫ ) such that for all k > K , |F q ( ν ) | ≤ X | ρ |≤ P Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + ǫ (40) where P is the constant from Lemma 4.2. Furthermore, there is some global constant C ′ > suchthat for all ¯ η ′ and ρ as above | (cid:0) f ¯ η ′ ◦ f ρ (cid:1) ′ ( x ) | = Θ C ′ (cid:16) e − k ′ χ (cid:17) , ∀ x ∈ [0 , Remark 4.5.
For notational convenience, in this Corollary and the subsequent argument, we makethe assumption that we always have f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) > . Otherwise, we simply make the sumon the right hand side of (40) larger, by including the possibility that it is negative. Since thereare uniformly finitely many such options, still the sum above is over uniformly finitely many terms,and the proof follows through.Proof. For every k large enough, for every ξ ∈ ˜ A η and every ω ∈ A k ′ k,ℓ ( ξ ), as | ρ ω,k | ≤ P (cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ω,k ν (cid:17)(cid:12)(cid:12)(cid:12) ≤ X | ρ |≤ P (cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ν (cid:17)(cid:12)(cid:12)(cid:12) and so, by Claim 4.3, assuming f ′ ω | τk ( x ) ( x σ τk ( ω ) ( ω ) ) > |F q ( ν ) | ≤ Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ω,k ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + O ( q · e − ( k + k ′ ) χ − β · k ′ χ ) + o k (1)24 X | ρ |≤ P Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) + O ( q · e − ( k + k ′ ) χ − β · k ′ χ ) + o k (1)Recalling the choice of k = k ( q ) (which is as in Lemma 3.4), O ( q · e − ( k + k ′ ) χ − βk ′ χ ) = O ( o − k · e − βk ′ χ )So, since o k decays in at most a polynomial rate (by e.g. Lemma 3.4), there is some K = K ( ǫ ) aswe claimed. The last assertion is a consequence of equation (35), Theorem 3.1 (bounded distortion),that | ρ | ≤ P , and of equation (4).Now, fix some ρ with | ρ | ≤ P and consider the corresponding term in (40) Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ )We next appeal to Theorem 3.3 for every event A k ′ k,ℓ ( ξ ) separately. To do this, we notice that byCorollary 4.4, for every f ¯ η ′ ◦ f ρ involveddiam (cid:0) supp (cid:0) f ¯ η ′ ◦ f ρ ν (cid:1)(cid:1) = O ( e − k ′ χ )Notice that the the error term in Theorem 3.3 is O ( o k ) independently of the event A k ′ k,ℓ ( ξ ). So, Z ξ ∈ ˜ A η E A k ′ k,ℓ ( ξ ) (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω )( ω ) ◦ f ¯ η ′ ◦ f ρ ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) d P ( ξ ) ≤ Z ξ ∈ ˜ A η Z kχ + D ′ kχ (cid:12)(cid:12) F q (cid:0) M e − x ◦ f ¯ η ′ ◦ f ρ ν (cid:1)(cid:12)(cid:12) d Γ A k ′ k,ℓ ( ξ ) ( x ) d P ( ξ ) + O ( o k )Let K = K ( ǫ ) be large enough to ensure that if k ≥ K then O ( o k ) ≤ ǫ |{ ρ : | ρ |≤ P }| . Then, sincethis is true for every ρ with | ρ | ≤ P , we see that for k ≥ max { K , K } , putting this into (40) weget |F q ( ν ) | ≤ X | ρ |≤ P Z ξ ∈ ˜ A η Z kχ + D ′ kχ (cid:12)(cid:12) F q (cid:0) M e − x ◦ f ¯ η ′ ◦ f ρ ν (cid:1)(cid:12)(cid:12) d Γ A k ′ k,ℓ ( ξ ) ( x ) d P ( ξ ) + 2 ǫ Recall that by Lemma 2.7, the probability measure Γ A k ′ k,ℓ ( ξ ) is absolutely continuous with respectto the Lebesgue measure on [ kχ, kχ + D ′ ], such that the norm of its density function is uniformlybounded by D > k ≥ max { K , K } then |F q ( ν ) | ≤ X | ρ |≤ P Z ξ ∈ ˜ A η Z kχ + D ′ kχ (cid:12)(cid:12) F q (cid:0) M e − z ◦ f ¯ η ′ ◦ f ρ ν (cid:1)(cid:12)(cid:12) · D dz ! d P ( ξ ) + 2 ǫ At this point we are finally in position to invoke Lemma 3.6. Taking the measure(s) to be f ¯ η ′ ◦ f ρ ν , for any r >
0, we get the inequality (as long as k is large enough) |F q ( ν ) | ≤ X | ρ |≤ P Z ξ ∈ ˜ A η D ′ D · (cid:18) e r · | q | + Z f ¯ η ′ ◦ f ρ ν ( B e χk · r ( y )) d (cid:0) f ¯ η ′ ◦ f ρ ν ( y ) (cid:1)(cid:19) d P ( ξ ) + 2 ǫ
25y Corollary 4.4, all the maps (cid:0) f ¯ η ′ ◦ f ρ (cid:1) − as above are O ( e k ′ χ ) Lipschitz (with the implied constantin the O ( · ) being uniform). Therefore, there is some T > η ′ and y ∈ [0 , (cid:0) f ¯ η ′ ◦ f ρ (cid:1) − (cid:0) B e χk · r ( f ¯ η ′ ◦ f ρ ( y ) (cid:1) ⊆ B T · e χ ( k + k ′ ) · r ( y )so, for a fixed r > ρ and ξ , and get |F q ( ν ) | ≤ |{ ρ : | ρ | ≤ P }| · (cid:18) e r · | q | + Z ν ( B T · e χ ( k + k ′ ) · r ( y )) dν ( y ) (cid:19) · D ′ D + 2 ǫ By Lemma 3.2 there exists some δ = δ ( ǫ ) > ν ( B δ ( y )) < ǫ · D · D ′ · |{ ρ : | ρ | ≤ P }| , ∀ y ∈ R Now, we choose r so that T · e χ ( k + k ′ ) · r = δ . This implies that ν ( B T · e χ ( k + k ′ ) · r ( y )) ≤ ǫ · D · D ′ ·|{ ρ : | ρ |≤ P }| for every y . Therefore, r = T · e χ ( k + k ′ ) δ . So, as | q | = Θ C (cid:18) o − k · e ( k + k ′ ) χ (cid:19) , e r · | q | = e · T · e χ ( k + k ′ ) δ · | q | ≤ C · e · T · o k · e χ ( k + k ′ ) δ · e χ ( k + k ′ ) = o k · e · T · Cδ So, as long as k ≥ K = K ( ǫ ), e r · | q | ≤ ǫ · D · D ′ · |{ ρ : | ρ | ≤ P }| Finally, if k ≥ max { K , K , K } we see that |F q ( ν ) | ≤ ǫ which implies the Theorem. Let Φ be a C γ ([0 , c ( · , · ) be the derivative cocycle. In the following, let X = { , ..., n } N and let H κ denote the space of κ -Holder continuous maps X → C . Recall that we defineΛ c = { θ : ∃ φ θ ∈ H κ with | φ θ | = 1 and u θ ∈ S such that ∀ ( a, ω ) ∈ { , ...n } × { , ..., n } N ,φ θ ( ι a ( ω )) = u θ exp( − iθ · c ( a, ω )) · φ θ ( ω ) } Following Benoist and Quint [2], we say that c is an aperiodic cocycle ifΛ c = { } Next, writing Φ = { f , ..., f n } , we define F Φ = (cid:8) − log (cid:12)(cid:12) f ′ i ( y i ) (cid:12)(cid:12) : where f i ( y i ) = y i , i ∈ { , ..., n } (cid:9) Notice that F Φ is precisely the set that appears in condition (3) of Theorem 1.1. In fact, aperiodic cocycles are defined in [2, equation (15.8)] in a different way, by a certain spectral gap property.However, It is a consequence of [2, Lemma 15.3] that the two definitions are equivalent. emma 5.1. If c is not aperiodic (i.e. it is periodic) then F Φ belongs to a translation of a lattice.Proof. The assumption that c is not aperiodic means that there exists 0 = θ ∈ Λ c . So, there exists φ ∈ H κ ( X ) with | φ | = 1 and u ∈ S such that ∀ ( a, ω ) ∈ { , ...n } × X , φ ( ι a ( ω )) = u · exp( − iθ · c ( a, ω )) · φ ( ω )Now, fix 1 ≤ a ≤ n and let ω = ( a, a, a.... ) ∈ X . Plugging these into the equation above, φ ( ω ) = u · exp( − iθ · c ( a, ω )) · φ ( ω ) ⇒ u exp( − iθ · ( − log f ′ a ( x ω )))Noting that that f a ( x ω ) = x ω , the equation above implies that F Φ belongs to a translation (deter-mined by u ) of the lattice πθ Z . Proof of Theorem 1.1
This is immediate from Lemma 5.1 and Theorem 1.4.
In this Section we prove Corollary 1.2. We begin with some general observations regarding whenare self similar IFS’s aperiodic:
Lemma 5.2.
Suppose that Φ is a self similar IFS. If F Φ belongs to a translation of a lattice then c is not aperiodic.Proof. By definition, it suffices to find 0 = θ , φ ∈ H κ ( X ) with | φ | = 1 and u ∈ S such that for all( a, ω ) ∈ { , ...n } × X , φ ( ι a ( ω )) = u · exp( − iθ · c ( a, ω )) · φ ( ω )Now, by our assumption, F Φ ⊆ πθ Z + t for some θ = 0 and t ∈ R . Since all the maps in Φ areaffine, F Φ = {− log | r a | : r a = f ′ a , f a ∈ Φ } Choose u = exp( iθt ). Then for all ( a, ω ) ∈ { , ...n } × X , u · exp( − iθ · c ( a, ω )) = exp( iθt ) · exp( − i · θ ( − log f ′ a ))= exp( iθt ) · exp( − i · θ ( 2 πk a θ + t ))= 1Let φ be the constant function 1. Then the calculation above shows that for all ( a, ω ) ∈ { , ...n }× X , φ ( ι a ( ω )) = u · exp( − iθ · c ( a, ω )) · φ ( ω )It follows that 0 = θ ∈ Λ c , as claimed.The following Proposition now relates our definition of periodic self similar IFS’s with theperiodicity of their derivative cocycle: Proposition 5.3.
Let Φ be a self similar IFS with contractions { r , ..., r n } . Then c is periodic ifand only if there are some r, t ∈ R and ℓ i ∈ Z such that for every i , r i = t · r ℓ i roof. This is a combination of Lemma 5.2 and of Lemma 5.1. The only additional observationneeded is that F Φ belongs to a translation of a lattice if and only if there are t, r as in the statementof the Proposition, and this is immediate.Before we prove Theorem 1.2, we need one more Lemma: Lemma 5.4.
Let Φ be an aperiodic self similar IFS on an interval J . Let g : J → g ( J ) be a C γ ( J ) map with non vanishing derivative. Then the conjugated IFS Ψ = g ◦ Φ ◦ g − is alsoaperiodic.Proof. It suffices to show that under these assumptions, F Φ = F Ψ . Indeed, if this is true, then F Ψ equals F Φ that does not lie on a translation of a lattice by Proposition 5.3, and so F Ψ does not lieon a translation of a lattice. By Lemma 5.1 this means that the derivative cocycle of Ψ is aperiodic.To prove that F Φ = F Ψ , let g ◦ f ◦ g − ∈ Ψ with f ∈ Φ, and let y be its fixed point. Then D g ◦ f ◦ g − ( y ) = D g (cid:0) f ◦ g − ( y ) (cid:1) ◦ D f (cid:0) g − ( y ) (cid:1) D g − ( y ) = f ′ where we have used that f ◦ g − ( y ) = g − ( y ), which follows since y is the fixed point of g ◦ f ◦ g − .This gives the equality F Φ = F ψ , and hence the Lemma. Proof of Corollary 1.2
We need to verify that if Φ is an aperiodic self similar IFS such that K ⊂ J and g : J → g ( J ) is a C γ ( J ) map with non vanishing derivative, then the conjugated IFSΨ = g ◦ Φ ◦ g − satisfies the conditions of Theorem 1.1. It is clear that Ψ is a C γ smooth IFS.The second condition might a-priori not hold, since a general upper bound on | g ′ | for g ∈ Ψ is onlygiven by sup ≤ a ≤ n | r a | · max x,y ∈ J (cid:12)(cid:12)(cid:12) g ′ ( x ) g ′ ( y ) (cid:12)(cid:12)(cid:12) . However, this is not substantial since still | g ′ | is uniformlybounded away from 0, and since Ψ will clearly be uniformly contracting. Finally, by Lemma 5.4, F Ψ does not lie on a translation of a lattice. This complete the proof. Let Φ = { f , ..., f n } be an orientation preserving self similar IFS with contractions r r . Let ν be a non atomic self conformal measure. In this Section we will show that ν is Rajchman, even ifΦ is periodic. Now, if Φ is aperiodic then Theorem 1.2 gives the result. Otherwise, it does not -and the main issue is that we may not apply Theorem 2.9. However, since the IFS is self similar,we may substitute this for the renewal Theorem proved by Li and Sahlsten [25, Proposition 2.1].Other than that, the proof is essentially the same - and now also resembles that of Li and Sahlsten.Here are some more details:Let X ( ω ) = − log | f ′ ω (0) | , let X n = X ◦ σ n − , and let S n = P n X i . Let ρ be the law of X .We also use the same notation as before for P , E , etc. Define a stopping time τ k ( ω ) = min { m : S m ( ω ) ≥ k } Let M = max { r : r ∈ supp( ρ ) } and define a local C norm on ( − , M + 1) via || g || C = sup {| g ( x ) | + | g ′ ( x ) | : x ∈ ( − , M + 1) } Theorem 5.5. [25, Proposition 2.1] For every k ≥ M + 1 and C function g on R , the followingholds as k → ∞ : E [ g ( S τ k − k )] = 1 σ Z R + g ( x ) p ( x ) dx + o k · || g || C where o k → as k → ∞ , p ( x ) = ρ ( x, ∞ ) , and σ is the mean of ρ . ν is Rajchman:1. Let o k be the rate in Theorem 5.5, fix q and choose k that satisfies | q | = Θ C (cid:18) o − k · e k (cid:19) that such a k can be found follows from a similar argument as in Lemma 3.4.2. Let g k be the C ∞ function g k ( t ) = |F q ( M e − ( t + k ) ◦ ν ) | Then by the choice of k and q || g || C ≤ O ( o − k )3. Now, as in Section 4, since this is a self similar IFS | F q ( ν ) | ≤ E (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) f ω | τk ( ω ) ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) = E (cid:20)(cid:12)(cid:12)(cid:12) F q (cid:16) M e − Sτk ( ω ) ◦ ν (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) = E (cid:2) g k ( S τ k ( ω ) − k ) (cid:3) = 1 σ Z R + g k ( x ) p ( x ) dx + o k · || g k || C ≤ C · Z |F q ( M e − ( t + k ) ◦ ν ) | dt + O ( o k )4. Finally, the latter integral can be shown to be arbitrarily small by Lemma 3.6 using the sameideas as in the last part of Section 4. In this Section we prove Theorem 1.3: Let { f , ..., f n } be a C r smooth IFS on an interval J , whereeither r = 1 + γ for γ > r = ω , such that:1. We have 0 < inf {| f ′ ( x ) | : f ∈ Φ , x ∈ J } ≤ sup {| f ′ ( x ) | : f ∈ Φ , x ∈ J } < r = ω or K is an interval.3. There exists a non-atomic self conformal measure ν that is not a Rajchman measure.Then Theorem 1.4 implies that Λ c = { } . Therefore, there exists 0 = θ ∈ Λ c . This means thatthere is a κ - H¨older continuous map φ : { , ..., n } N → S and some u ∈ S such that φ ( ι a ( ω )) = u · exp( iθ · c ( a, ω )) · φ ( ω )We first rephrase this in terms of functions on the interval J . Recall that K is the attractor of ourIFS. Lemma 5.6.
There exists a H¨older continuous function ϕ : J → R and some α ∈ R such that forevery ≤ i ≤ n and every x ∈ Kϕ ( f i ( x )) = α + θ · ( − log | f ′ i ( x ) | ) + ϕ ( x ) mod 129 roof. First, that such a function ϕ exists, that is defined on { , ..., n } N and takes values in T , isimmediate from the last displayed equation, taking α to be a value that corresponds to u . Now,notice that we can define a derivative cocycle directly on { , ..., n } × J by taking( i, x )
7→ − log | f ′ i ( x ) | which is completely analogues to the symbolic version we have been working with so far. Thus, θ ∈ Λ c means that θ belongs to the Λ set of this new cocycle on the support of ν which is K . So,we may assume that ϕ : J → T is such that for every 1 ≤ i ≤ n and x ∈ Kϕ ( f i ( x )) = α + θ · ( − log | f ′ i ( x ) | ) + ϕ ( x ) mod 1Finally, a standard argument, using that the the derivative is homotopic to 0 on the convex hull of f i ( K ) for all i , shows that we may assume ϕ is real valued. Lemma 5.7.
Let ϕ and α be as in Lemma 5.6. If Φ is C γ ( J ) and K is an interval, then forevery ≤ i ≤ n there is some n i ∈ Z such that for every x ∈ Jϕ ( f i ( x )) = α + θ · ( − log | f ′ i ( x ) | ) + ϕ ( x ) + n i If Φ is a C ω IFS then every y ∈ K admits a neighbourhood N y in J and n y,i ∈ Z such that forevery x ∈ N y and every i ϕ ( f i ( x )) = α + θ · ( − log | f ′ i ( x ) | ) + ϕ ( x ) + n y,i Proof.
Notice that the difference between this Lemma and Lemma 5.6 is the set on which thecohomological equation holds. This is where we apply assumption (2) of Theorem 1.3. First, byLemma 5.6, for every 1 ≤ i ≤ n and for all x ∈ Kϕ ( f i ( x )) − (cid:0) α + θ · ( − log | f ′ i ( x ) | ) + ϕ ( x ) (cid:1) ∈ Z Assuming K is an interval, the function on the left hand side is a continuous function taking valuesin Z , so it must be constantly n i ∈ Z on K .If the IFS is C ω then it is standard that ϕ ∈ C ω ( J ). So, the function on the left hand side inthe last displayed equation is a C ω function that takes values in Z on K . Since K is compact andinfinite, this implies the Lemma. Proof of Theorem 1.3
Assume first that Φ is C γ smooth and that K is an interval. Let h : J → R be a C γ ( J ) smooth function that is a primitive of exp( ϕ ( x ) θ ) on J . Now, for every i define g i ( x ) = h ◦ f i ◦ h − : h ( J ) → h ( J )and let Ψ be the IFS consisting of the maps g i . By Lemma 5.7, for every i and every y ∈ h ( J ) g ′ i ( y ) = (cid:0) h ◦ f i ◦ h − (cid:1) ′ ( y )= h ′ (cid:0) f i ◦ h − ( y ) (cid:1) · f ′ i ( h − ( y )) h ′ ( h − ( y ))= exp (cid:18) ϕ ◦ f i ( h − ( y )) θ + log | f ′ i ( h − ( y )) | ) − ϕ ( h − ( y )) θ (cid:19) · sign (cid:0) f ′ i ( h − ( y )) (cid:1) = exp (cid:16) n i θ + αθ (cid:17) · sign (cid:0) f ′ i ( h − ( y )) (cid:1) t = exp( αθ ) and β = exp( θ ). Noticing that assumption (1) of Theorem 1.3 meansthat sign (cid:0) f ′ i ( h − ( y )) (cid:1) is constant, this implies the Theorem.If the IFS is C ω smooth then the same proof shows that h ( K ) can be covered by finitely manyintervals on which every map in Ψ acts like an affine map, with contractions of the form t · β n i . Weleave the verification to the reader. References [1] Artur Avila, Jairo Bochi, and Jean-Christophe Yoccoz. Uniformly hyperbolic finite-valuedSL(2 , R )-cocycles. Comment. Math. Helv. , 85(4):813–884, 2010.[2] Yves Benoist and Jean-Fran¸cois Quint.
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Department of Mathematics, the Pennsylvania State University, University Park, PA 16802, USA
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