Smooth Livsic regularity for piecewise expanding maps

We consider the regularity of measurable solutions $\chi$ to the cohomological equation \[ \phi = \chi \circ T -\chi, \] where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $\beta$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $\chi$ possess $C^k$ versions. In particular we show that if $(T,X,\mu)$ is a $\beta$-transformation then $\chi$ has a $C^k$ version, thus improving a result of Pollicott et al.~\cite{Pollicott-Yuri}.


Introduction
In this note we consider the regularity of solutions χ to the cohomological equation where (T, X, µ) is a dynamical system and φ : X → R is a C k valued cocycle.
In particular we are interested in the setting in which T : X → X is a piecewise C k Gibbs-Markov map, an affine β-transformation of the unit interval or more generally a piecewise C k uniformly expanding map of an interval. Rigidity in this context means that a solution χ with a certain degree of regularity is forced by the dynamics to have a higher degree of regularity. Cohomological equations arise frequently in ergodic theory and dynamics and, for example, determine whether observations φ have positive variance in the central limit theorem and and have implication for other distributional limits (for examples see [20,2]). Related cohomological equations to Equation (1) decide on stable ergodicity and weak-mixing of compact group extensions of hyperbolic systems [11,20,19] and also play a role in determining whether two dynamical systems are (Hölder, smoothly) conjugate to each other. Livšic [13,14] gave seminal results on the regularity of measurable solutions to cohomological equations for Abelian group extensions of Anosov systems with an absolutely continuous invariant measure. Theorems which establish that a priori measurable solutions to cohomological equations must have a higher degree of regularity are often called measurable Livšic theorems in honor of his work.
We say that χ : X → R has a C k version (with respect to µ) if there exists a C k function h : Pollicott and Yuri [23] prove Livšic theorems for Hölder R-extensions of β-transformations (T : [0, 1) → [0, 1), T (x) = βx (mod 1) where β > 1) via transfer operator techniques. They show that any essentially bounded measurable solution χ to Equation (1) is of bounded variation on [0, 1 − ǫ) for any ǫ > 0. In this paper we improve this result to show that measurable coboundaries χ for C k R-valued cocycles φ over β-transformations have C k versions (see Theorem 2).
Jenkinson [10] proves that integrable measurable coboundaries χ for Rvalued smooth cocycles φ (i.e. again solutions to φ = χ • T − χ) over smooth expanding Markov maps T of S 1 have versions which are smooth on each partition element.
Nicol and Scott [15] have obtained measurable Livšic theorems for certain discontinuous hyperbolic systems, including β-transformations, Markov maps, mixing Lasota-Yorke maps, a simple class of toral-linked twist map and Sinai dispersing billiards. They show that a measurable solution χ to Equation (1) has a Lipschitz version for β-transformations and a simple class of toral-linked twist map. For mixing Lasota-Yorke maps and Sinai dispersing billiards they show that such a χ is Lipschitz on an open set. There is an error in [15,Theorem 1] in the setting of C 2 Markov maps -they only prove measurable solutions χ to Equation (1) are Lipschitz on each element T α, α ∈ P, where P is the defining partition for the Markov map, and not that the solutions are Lipschitz on X, as Theorem 1 erroneously states. The error arose in the following way: if χ is Lipschitz on α ∈ P it is possible to extend χ as a Lipschitz function to T α by defining χ(T x) = φ(x) + χ(x), however extending χ as a Lipschitz function from α to T 2 α via the relation χ(T 2 x) = φ(T x) + χ(T x) may not be possible, as φ • T may have discontinuities on T α. In this paper we give an example, (see Section 3), which shows that for Markov maps this result cannot be improved on. Gouëzel [7] has obtained similar results to Nicol and Scott [15] for cocycles into Abelian groups over one-dimensional Gibbs-Markov systems. In the setting of Gibbs-Markov system with countable partition he proves any measurable solution χ to Equation (1) is Lipschitz on each element T α, α ∈ P, where P is the defining partition for the Gibbs-Markov map.
In related work, Aaronson and Denker [1,Corollary 2.3] have shown that if (T, X, µ, P) is a mixing Gibbs-Markov map with countable Markov partition P preserving a probability measure µ and φ : X → R d is Lipschitz (with respect to a metric ρ on X derived from the symbolic dynamics) then any measurable solution χ : X → R d to φ = χ • T − χ has a versionχ which is Lipschitz continuous, i.e. there exists C > 0 such that d(χ(x),χ(y)) ≤ Cρ(x, y) for all x, y ∈ T (α) and each α ∈ P.
Bruin et al. [4] prove measurable Livšic theorems for dynamical systems modelled by Young towers and Hofbauer towers. Their regularity results apply to solutions of cohomological equations posed on Hénon-like mappings and a wide variety of non-uniformly hyperbolic systems. We note that Corollary 1 of [4, Theorem 1] is not correct -the solution is Hölder only on M k and T M k rather than T j M k for j > 1 as stated for reasons similar to those given above for the result in Nicol et al. [15].

Main results
We first describe one-dimensional Gibbs-Markov maps. Let I ⊂ R be a bounded interval, and P a countable partition of I into intervals. We let m denote Lebesgue measure. Let T : I → I be a piecewise C k , k ≥ 2, expanding map such that T is C k on the interior of each element of P with |T ′ | > λ > 1, and for each α ∈ P, T α is a union of elements in P. Let P n := n j=0 T −j P and J T := d(m•T ) dm . We assume: (i) (Big images property) There exists C 1 > 0 such that m(T α) > C 1 for all α ∈ P.
(iii) (Bounded distortion) There exists 0 < γ 2 < 1 and C 2 > 0 such that Under these assumptions T has an invariant absolutely continuous probability measure µ and the density of µ, h = dµ dm is bounded above and below by a constant 0 Note that a Markov map satisfies (i), (ii) and (iii) for finite partition P.
It is proved in [15] for the Markov case (finite P), and in [7] for the Gibbs-Markov case (countable P) that if φ : I → R is Hölder continuous or Lipschitz continuous, and φ = χ • T − χ for some measurable function χ : I → R, then there exists a function χ 0 : I → R that is Hölder or Lipschitz on each of the elements of P respectively, and χ 0 = χ holds µ (or m) a.e. A related result to [7] is given in [4,Theorem 7] where T is the base map of a Young Tower, which has a Gibbs-Markov structure.
Fried [6] has shown that the transfer operator of a graph directed Markov system with C k,α -contractions, acting on a space of C k,α -functions, has a spectral gap. If we apply his result to our setting, letting the contractions be the inverse branches of a Gibbs-Markov map we can conclude that the transfer operator of a Gibbs-Markov map acting on C k -functions has a spectral gap. As in Jenkinson's paper [10] and with the same proof, this gives us immediately the following proposition, which is implied by the results of Fried and Jenkinson: Proposition 1. Let T : T → I be a mixing Gibbs-Markov map such that T is C k on each partition element and T −1 : T (α) → α is C k on each partition element α ∈ P. Let φ : I → R be uniformly C k on each of the partition elements α ∈ P. Suppose χ : I → R is a measurable function such that φ = χ • T − χ. Then there exists a function χ 0 : I → R such that χ 0 is uniformly C k on T α for each partition element of α ∈ P, and χ 0 = χ almost everywhere.

A counterexample
We remark that in general, if φ = χ • T − χ, one cannot expect χ to be continuous on I if φ is C k on I. We give an example of a Markov map T with Markov partition P, a function φ that is C k on I, and a function χ that is C k on each element α of P such that φ = χ • T − χ, yet χ has no version that is continuous on I.
If c = 1 8 , then the partition is a Markov partition for T . Define χ such that χ is 0 on [ 1 2 − 1 4d , 1 2 ] and 1 on χ(x) = 1. For any natural number k, this can be done so that χ is C k except at the point 1 2 where it has a jump. One easily check that φ defined by φ = χ • T − χ is C k . This is illustrated in Figures 1-4.

Livšic theorems for piecewise expanding maps of an interval
Let I = [0, 1) and let m denote Lebesgue measure on I. We consider piecewise expanding maps T : I → I, satisfying the following assumptions: (i) There is a number λ > 1, and a finite partition P of I into intervals, such that the restriction of T to any interval in P can be extended to a C 2 -function on the closure, and |T ′ | > λ on this interval.
(ii) T has an absolutely continuous invariant measure µ with respect to which T is mixing.
(iii) T has the property of being weakly covering, as defined by Liverani in [12], namely that there exists an n 0 such that for any element α ∈ P n 0 j=0 T j (α) = I.
For any n ≥ 0 we define the partition P n = P ∨ · · · ∨ T −n+1 P. The partition elements of P n are called n-cylinders, and P n is called the partition of I into n-cylinders.
We prove the following two theorems.    Theorem 1. Let (T, I, µ) be a piecewise expanding map satisfying assumptions (i), (ii) and (iii). Let φ : I → R be a Hölder continuous function, such that φ = χ • T − χ for some measurable function χ, with e −χ ∈ L 1 (m). Then there exists a function χ 0 such that χ 0 has bounded variation and χ 0 = χ almost everywhere.
For the next theorem we need some more definitions. Let A be a set, and denote by int A the interior of the set A. We assume that the open sets T (int α), where α is an element in P, cover int I.
We will now define a new partition Q. For a point x in the interior of some element of P, we let Q(x) be the largest open set such that for any x 2 ∈ Q(x), and any m-cylinder C m , there are points (y 1,k ) n k=1 and (y 2,k ) n k=1 , such that y 1,k and y 2,k are in the same element of P, T (y i,k+1 ) = y i,k , T (y 1,1 ) = x, T (y 2,1 ) = x 2 , and y 1,n , y 2,n ∈ C m . (This forces n ≥ m.) Note that if Q(x) ∩ Q(y) = ∅, then for z ∈ Q(x) ∩ Q(y) we have Q(z) = Q(x) ∪ Q(y). We let Q be the coarsest collection of connected sets, such that any element of Q can be represented as a union of sets Q(x).
Theorem 2. Let (T, I, µ) be a piecewise expanding map satisfying assumptions (i), (ii) and (iii). If φ : I → R is a continuously differentiable function, such that φ = χ • T − χ for some function χ with e −χ ∈ L 1 (m), then there exists a function χ 0 such that χ 0 is continuously differentiable on each element of Q and χ 0 = χ almost everywhere. If T ′ is constant on the elements of P, then χ 0 is piecewise C k on Q if φ is in C k . If for each r, 1 (T r ) ′ is in C k with derivatives up to order k uniformly bounded, then It is not always clear how big the elements in the partition Q are. The following lemma gives a lower bound on the diameter of the elements in Q. Lemma 1. Assume that the sets { T (int α) : α ∈ P } cover (0, 1). Let δ be the Lebesgue number of the cover. Then the diameter of Q(x) is at least δ/2 for all x.
Proof. Let C m be a cylinder of generation m. We need to show that for some n ≥ m there are sequences (y 1,k ) n k=1 and (y 2,k ) n k=1 as in the definition of Q above.
Take n 0 such that µ(T n 0 (C m )) = 1. Write C m as a finite union of cylinders of generation n 0 , C m = i D i . Then R := [0, 1] \ T n 0 (∪ i int D i ) consists of finitely many points. Let ε be the smallest distance between two of these points.
Let I δ be an open interval of diameter δ. Let n 1 be such that δλ −n 1 < ε. Consider the full pre-images of I δ under T n 1 . By the definition of δ, there is at least one such pre-image, and any such pre-image is of diameter less than ε. Hence any pre-image contains at most one point from R.
If the pre-image does not contain any point of R, then I δ is contained in some element of Q and we are done. Assume that there is a point z in I δ corresponding to the point of R in the pre-image of I δ . Assume that z is in the right half of I δ . The case when z is in the left part is treated in a similar way. Take a new open interval J δ of length δ, such that the left half of J δ coincides with the right half of I δ .
Arguing in the same way as for I δ , we find that a pre-image of J δ contains at most one point of R. If there is no such point, or the corresponding point z J ∈ J δ is not equal to z, then I δ ∪ J δ is contained in an element in Q and we are done.
It remains to consider the case z = z J . Let I δ = (a, b) and J δ = (c, d). Then the intervals (a, z) and (z, d) are both of length at least δ/2, and both are contained in some element of Q. This finishes the proof. Corollary 1. If β > 1 and T : x → βx (mod 1) is a β-transformation then clearly T is weakly covering and Q = {(0, 1)}, so in this case Theorem 2 and Theorem 1 of [15] imply that is an affine β-transformation, then Q = {(0, 1)}, and hence if e −χ is in L 1 (m) then χ has a C k version.

Proof of Theorem 1
We continue to assume that (T, I, µ) is a piecewise expanding map satisfying assumptions (i), (ii) and (iii). For a function ψ : I → R we define the weighted transfer operator L ψ by The proof is based on the following two facts, that can be found in Hofbauer and Keller's papers [8,9]. The first fact is There is a function h ≥ 0 of bounded variation such that if f ∈ L 1 with f ≥ 0 and f = 0, then L n 0 f converges to h f dm in L 1 . (2) The second fact is Let f ∈ L 1 with f ≥ 0 and f = 0 be fixed. There is a function w ≥ 0 with bounded variation, a measure ν, and a number a > 0, depending on φ, such that a n L n φ f → w f dν, in L 1 . (3) For f of bounded variation, these facts are proved as follows. Theorem 1 of [8] gives us the desired spectral decomposition for the transfer operator acting of functions of bounded variation. Proposition 3.6 of Baladi's book [3] gives us that there is a unique maximal eigenvalue. This proves the two facts for f of bounded variation. The case of a general f in L 1 follows since such an f can be approximated by functions of bounded variation.
Using that T is weakly covering, we can conclude by Lemma 4.2 in [12], that h > γ > 0. The proof of this fact in [12] goes through also for w, and so we may also conclude that w > γ > 0.
Let us now see how Theorem 1 follows from these facts. The following argument is analogous to the argument used by Pollicott and Yuri in [23] for β-expansions. We first observe that φ = χ • f − χ implies that Since a n L n φ 1 → w and e −χ L n φ 1 = L n 0 e −χ → h e −χ dm we have that a n L n φ 1 converges to w in L 1 and L n φ 1 converges to he χ e −χ dm in L 1 . By taking a subsequence, we can achieve that the convergences are a.e. Therefore, we must have a = 1 and It follows that almost everywhere. Since h and w are bounded away from zero, their logarithms are of bounded variation. This proves the theorem.

Proof of Theorem 2
We first note that it is sufficient to prove that χ 0 is continuously differentiable on elements of the form Q(x).
Let x 1 be a point in an element of Q, and take x 2 ∈ Q(x 1 ). We choose preimages y 1,j and y 2,j of x 1 and x 2 such that T (y i,1 ) = x i and T (y i,j ) = y i,j−1 . We then have We would like to let n → ∞ and conclude that χ(y 1,n ) − χ(y 2,n ) → 0. By Theorem 1 we know that χ has bounded variation. Assume for contradiction that no matter how we choose y 1,j and y 2,j we cannot make |χ(y 1,n ) − χ(y 2,n )| smaller than some ε > 0. Let m be large and consider the cylinders of generation m. For any such cylinder C m , we can choose y 1,j and y 2,j such that y 1,n and y 2,n both are in C m . Since |χ(y 1,n ) − χ(y 2,n )| ≥ ε, the variation of χ on C m is at least ε. Summing over all cylinders of generation m, we conclude that the variation of χ on I is at least N(m)ε. Since m is arbitrary and N(m) → ∞ as m → ∞, we get a contradiction to the fact that χ is of bounded variation.
The series converges since |(T j ) ′ | > λ j . This shows that χ ′ (x 1 ) exists and satisfies If T ′ is constant on the elements of P, then (4) implies that χ is in C k provided that φ is in C k .
Let us now assume that 1 (T r ) ′ is in C k with derivatives up to order k uniformly bounded in r. We proceed by induction. Let g n = 1 (T n ) ′ . Assume that ψ n,m (y n )g n (y n ), where (ψ n,m ) ∞ n=1 is in C n−m with derivatives up to order n − m uniformly bounded. Then n,m (y n )g n (y n )+ψ n,m (y n )g ′ n (y n ) g n (y n ) = ∞ n=1 ψ n,m+1 g n (y n ).
This proves that there are uniformly bounded functions ψ n,m such that (5) holds for 1 ≤ m ≤ k. The series in (5) converges uniformly since g n decays with exponential speed. This proves that χ is in C k .