Symbolic dynamics for one dimensional maps with nonuniform expansion

Abstract

Given a piecewise $C^{1+\beta }$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, for each $\chi >0$ we construct a finite-to-one Hölder continuous map from a countable topological Markov shift to the natural extension of the interval map, that codes the lifts of all invariant probability measures as above with Lyapunov exponent greater than χ almost everywhere.

Introduction

The quadratic family ${\{f_a\}}_{0\le a\le 4}$ is the family of one dimensional interval maps $f_a:[0,1]\to [0,1]$, $f_a(x)=ax(1-x)$. Although simple to describe, it exhibits complicated dynamical behaviour: for a set of parameters of positive Lebesgue measure, $f_a$ has an absolutely continuous invariant measure with positive Lyapunov exponents [4], [5], [33], see also [58]. The idea to prove this is to construct a partition of the interval with good symbolic properties that allows to understand the orbit of the critical point $x=0.5$, so that for many parameters the critical value $f_a(0.5)$ has positive Lyapunov exponent (this latter property is known as the Collet-Eckmann condition). This idea has far reaching applications, see e.g. [25], [41], [42].

The present works goes in the reverse direction of the above idea: it considers piecewise $C^{1+\beta }$ maps $f:[0,1]\to [0,1]$ of the interval with positive Lyapunov exponent and constructs finite-to-one Hölder continuous symbolic extensions of the maps. We require f to satisfy the regularity conditions (A1)–(A3), that will be shortly described. These conditions allow f to have both critical points (where the first derivate vanishes) and discontinuities (where the first derivative can explode), and include the quadratic family, multimodal maps with non-flat critical points, piecewise continuous maps with discontinuities of polynomial type, and combinations of these classes, see Fig. 1 for examples.

Our main result is the construction of a symbolic model for the natural extension $\hat{f}:\widehat{[0,1]}\to \widehat{[0,1]}$ of f. See Section 1.5 for the definition of the natural extension and its main properties. Let us describe which maps we consider and which measures we are able to code. For the sake of simplicity, we consider maps defined on the interval $M=[0,0.5]$, since this interval has diameter less than one (and so we do not need to introduce multiplicative constants in the assumptions (A1)–(A3) below). Let $f:M\to M$ be a map with discontinuity set $\mathcal{D}$. We assume that f is $C^{1+\beta }$ in the set $M\backslash \mathcal{D}$, for some $\beta \in (0,1)$. Let $\mathcal{C}:=\{x\in M\backslash \mathcal{D}:d{f}_x=0\}$ denote the critical set of f.

Singular set: The singular set of f is $\mathcal{S}:=\mathcal{C}\cup \mathcal{D}$.

We allow $\mathcal{S}$ to be infinite, e.g. when f is the Gauss map. Let $B(x,r)\subset M$ denote the ball with centre x and radius r. We assume that f satisfies the following properties.

Regularity of f: There exist constants $a,\mathfrak{K}>1$ s.t. for all $x\in M$ with $x,f(x)\notin \mathcal{S}$ there is $\min \{d{(x,\mathcal{S})}^a,d{(f(x),\mathcal{S})}^a\}<\mathfrak{r}(x)<1$ s.t. for $D_x:=B(x,2\mathfrak{r}(x))$ and $E_x:=B(f(x),2\mathfrak{r}(x))$ the following holds:

• The restriction of f to $D_x$ is a diffeomorphism onto its image; the inverse branch of f taking $f(x)$ to x is a well-defined diffeomorphism from $E_x$ onto its image.

• For all $y\in D_x$ it holds $d{(x,\mathcal{S})}^a\le |d{f}_y|\le d{(x,\mathcal{S})}^{-a}$; for all $z\in E_x$ it holds $d{(x,\mathcal{S})}^a\le |d{g}_z|\le d{(x,\mathcal{S})}^{-a}$, where g is the inverse branch of f taking $f(x)$ to x.

• For all $y,z\in D_x$ it holds $|d{f}_y-d{f}_z|\le \mathfrak{K}|y-z{|}^{\beta }$; for all $y,z\in E_x$ it holds $|d{g}_y-d{g}_z|\le \mathfrak{K}|y-z{|}^{\beta }$.

Now we describe the measures that we code. We borrow the notation from [38]. Let μ be an f–invariant probability measure.

f–adapted measure: The measure μ is called f–adapted if $\log d(x,\mathcal{S})\in L^1(\mu )$. A fortiori, $\mu (\mathcal{S})=0$.

χ–expanding measure: Given $\chi >0$, the measure μ is called χ–expanding if ${\lim }_{n\to \infty }\frac{1}{n}\log |d{f}_x^n|>\chi$ for μ–a.e. $x\in M$.

The next theorem is the main result of this paper. Below, $\hat{f}:\hat{M}\to \hat{M}$ is the natural extension of f, and $\hat{\mu }$ is the lift of μ, see Subsection 1.5 for the definition.

Theorem 1.1

Assume that f satisfies (A1)–(A3). For all $\chi >0$, there exists a countable topological Markov shift $(\mathit{\Sigma },\sigma )$ and $\pi :\mathit{\Sigma }\to \hat{M}$ Hölder continuous s.t.:

• $\pi \circ \sigma =\hat{f}\circ \pi$.

• $\pi [{\mathit{\Sigma }}^{\mathrm{\#}}]$ has full $\hat{\mu }$–measure for every f–adapted χ–expanding measure μ.

• For all $\hat{x}\in \pi [{\mathit{\Sigma }}^{\mathrm{\#}}]$, the set $\{\underset{\_}{v}\in {\mathit{\Sigma }}^{\mathrm{\#}}:\pi (\underset{\_}{v})=\hat{x}\}$ is finite.

Above, ${\mathrm{\Sigma }}^{\mathrm{\#}}$ is the recurrent set of Σ; it carries all σ–invariant probability measures, see Section 1.4. Therefore we are able to code simultaneously all the measures with nonuniform expansion greater than χ almost everywhere that do not approach the singular set exponentially fast.

It is important to make some comments on the assumption of f–adaptedness. By the Birkhoff ergodic theorem, if μ is f–adapted then ${\lim }_{n\to \infty }\frac{1}{n}\log d(f^n(x),\mathcal{S})=0$ μ–a.e. Ledrappier considered this latter condition for interval maps with critical points [37], where he used the terminology non-degenerate measure. Katok and Strelcyn implicitly used that the Lebesgue measure is adapted to billiard maps and then used that ${\lim }_{n\to \infty }\frac{1}{n}\log d(f^n(x),\mathcal{S})=0$ a.e. [35]. For an invariant measure of a three dimensional flow with positive speed, Lima and Sarig constructed a Poincaré section for which the respective Poincaré return map f satisfies ${\lim }_{n\to \infty }\frac{1}{n}\log d(f^n(x),\mathcal{S})=0$ almost surely with respect to the induced measure [39], and Lima and Matheus used the assumption of adaptability in their coding of billiard maps [38]. For one dimensional maps satisfying (A1)–(A3), if $\mathcal{D}=\varnothing$ and $\mathcal{C}$ is finite then every ergodic invariant probability measure that is not supported in an attracting periodic orbit satisfies ${\lim }_{n\to \infty }\frac{1}{n}\log d(f^n(x),\mathcal{S})=0$ a.e. [44], see also [49, Appendix A] for a proof that works under weaker assumptions. It would be interesting to obtain the same conclusion when $\mathcal{S}$ is finite.

Now we make a comparison of our method with the one developed by Hofbauer [30], [31], known as Hofbauer towers. These towers were first constructed to analyze measures of maximal entropy, and they provide a precise combinatorial description of one dimensional maps. In comparison to Hofbauer's method, our method explores the nonuniform expansion of χ–expanding measures. It constitutes the first implementation, for non-invertible systems, of the recent constructions of Markovian symbolic dynamics for nonuniformly hyperbolic systems [7], [38], [39], [53]. The novelty of the present paper is that, contrary to Hofbauer's method, our construction has the following advantages:

• The extension map π constructed in Theorem 1.1 is Hölder continuous.

• While Hofbauer's method only works in very specific higher dimensional cases (see Section 1.2 below), our method is more robust and will be extended, in a forthcoming work, to higher dimensional maps such as complex-valued functions, Viana maps, and general nonuniformly hyperbolic maps.

In addition to these, we emphasize that $\mathcal{S}$ can be infinite.

Assume that f satisfies (A1)–(A3) and has finite and positive topological entropy $h\in (0,\infty )$. In this section, we discuss two applications of Theorem 1.1:

• Estimates on the number of periodic points.

• Understanding of equilibrium measures.

The first can be obtained in great generality, but the second is subtler due to the presence of discontinuities and eventual non-finiteness of $\mathcal{S}$. As a matter of fact, the measure of maximal entropy may not exist, even for $C^r$ maps [19], [51]. When $\mathcal{D}=\varnothing$ and $\mathcal{C}$ is finite, f is called a multimodal map. There are many results on ergodic and statistical properties of equilibrium measures for one dimensional maps in various contexts, such as uniformly expanding maps [11], [40], [56], [57], piecewise continuous maps [30], [31], piecewise monotone maps [28], and multimodal maps [6], [15], [16], [22], [32], [37], [45].

As mentioned above, Theorem 1.1 allows $\mathcal{S}$ to be infinite. When there is an f–adapted measure of maximal entropy, we obtain exponential estimates on the number of periodic points.

Theorem 1.2

Assume f satisfies (A1)–(A3) with topological entropy $h\in (0,\infty )$. If there is an f–adapted measure of maximal entropy, then $\exists C>0,p\ge 1$ s.t. $\mathrm{\#}\{x\in [0,1]:f^{pn}(x)=x\}\ge C{e}^{hpn}$ for all $n\ge 1$.

Proof

The set of periodic points of f is in bijection with the set of periodic points of its natural extension $\hat{f}$. Since $(\mathrm{\Sigma },\sigma )$ is a finite extension of $\hat{f}$, the growth rate of periodic points of $\hat{f}$ is at least that of σ. Let μ be an ergodic f–adapted measure of maximal entropy for f. Its lift $\hat{\mu }$ is an ergodic measure of maximal entropy for $\hat{f}$ satisfying the assumptions of Theorem 1.1. Proceeding as in [53, §13], $\hat{\mu }$ lifts to an ergodic measure of maximal entropy ν for $(\mathrm{\Sigma },\sigma )$. By [26], [27], ν is supported on a topologically transitive countable topological Markov shift, and there is $p\ge 1$ s.t. for every vertex v it holds $\mathrm{\#}\{\underset{\_}{v}\in \mathrm{\Sigma }:{\sigma }^{pn}(\underset{\_}{v})=\underset{\_}{v},v_0=v\}\asymp e^{hpn}$. Hence, there exists $C>0$ s.t. $\mathrm{\#}\{x\in [0,1]:f^{pn}(x)=x\}\ge C{e}^{hpn}$. □

The number p equals the period of the transitive component supporting the measure of maximal entropy. For surface maps, Buzzi gave conditions for which $p=1$ [17], and we expect his method also applies here.

Now we turn attention to equilibrium measures. Let $(X,\mathcal{A},T)$ be a probability measure-preserving system, and $\phi :X\to \mathbb{R}$ a continuous potential. The following definitions are standard.

Topological pressure: The topological pressure of φ is $P_{\mathrm{top}}(\phi ):=\sup \{h_{\mu }(T)+\int \phi d\mu \}$, where the supremum ranges over all T–invariant probability measures.

Equilibrium measure: An equilibrium measure for φ is a T–invariant probability measure μ s.t. $P_{\mathrm{top}}(\phi )=h_{\mu }(T)+\int \phi d\mu$.

Due to the presence of discontinuities, the topological pressure might be infinite (e.g. the Gauss map has infinite topological entropy), and equilibrium measures may not exist. Theorem 1.1 provides countability results for a class of potentials and equilibrium measures: we require φ to be a bounded Hölder continuous potential with finite topological pressure, and consider equilibrium measures that are f–adapted and χ–expanding for some $\chi >0$. Call a measure expanding if it is χ–expanding for some $\chi >0$. Observe that, when the Ruelle inequality applies, every ergodic measure with positive metric entropy is expanding.

Theorem 1.3

Assume that f satisfies (A1)–(A3). Every equilibrium measure of a bounded Hölder continuous potential with finite topological pressure has at most countably many f–adapted expanding ergodic components. Furthermore, the lift to $\hat{M}$ of each such ergodic component is Bernoulli up to a period.

Proof

Let $\phi :M\to \mathbb{R}$ be bounded, Hölder continuous, with $P_{\mathrm{top}}(\phi )<\infty$. We prove that, for each $\chi >0$, φ possesses at most countably many f–adapted χ–expanding equilibrium measures. The first part of the theorem follows by taking the union of these measures for ${\chi }_n=\frac{1}{n}$, $n>0$.

Fix $\chi >0$. Let $\vartheta :\hat{M}\to M$ be the projection into the zeroth coordinate, see Subsection 1.5, and define $\hat{\phi }:\hat{M}\to \mathbb{R}$ by $\hat{\phi }=\phi \circ \vartheta$. Then $\hat{\phi }$ is bounded, Hölder continuous and has finite topological pressure $P_{\mathrm{top}}(\hat{\phi })=P_{\mathrm{top}}(\phi )$. Furthermore, μ is an ergodic equilibrium measure for φ iff $\hat{\mu }$ is an ergodic equilibrium measure for $\hat{\phi }$. When μ is additionally f–adapted and χ–expanding, we can apply the procedure in [53, §13] and Theorem 1.1 to lift $\hat{\mu }$ to an ergodic equilibrium measure ν in $(\mathrm{\Sigma },\sigma )$. By ergodicity, ν is carried by a topologically transitive countable topological Markov shift. The potential associated to ν is $\mathrm{\Phi }=\hat{\phi }\circ \pi$, which is bounded, Hölder continuous and has finite topological pressure $P_{\mathrm{top}}(\mathrm{\Phi })=P_{\mathrm{top}}(\hat{\phi })=P_{\mathrm{top}}(\phi )$. By [13, Thm. 1.1], each topologically transitive countable topological Markov shift carries at most one equilibrium measure for Φ, hence there are at most countably many such ν. This proves the first part of the theorem. By [52], each such ν is Bernoulli up to a period. Since this latter property is preserved by finite-to-one extensions, the same occurs to $\hat{\mu }$. This concludes the proof of the theorem. □

Symbolic models in dynamics have a longstanding history that can be traced back to the work of Hadamard on closed geodesics of hyperbolic surfaces, see e.g. [36]. The late sixties and early seventies saw a great deal of development of symbolic dynamics for uniformly hyperbolic diffeomorphisms and flows, through the works of Adler & Weiss [2], [3], Sinaĭ [54], [55], Bowen [8], [9], Ratner [47], [48]. Below we discuss other relevant contexts.

Hofbauer towers: Takahashi developed a combinatorial method to construct an isomorphism between a large subset X of the natural extension of β–shifts and countable topological Markov shifts [57]. Hofbauer proved that X carries all measures of positive entropy and hence β–shifts have a unique measure of maximal entropy [29]. Hofbauer later extended his construction to piecewise continuous interval maps [30], [31]. The symbolic models obtained by his methods are called Hofbauer towers, and they have been extensively used to establish ergodic properties of one dimensional maps.

Higher dimensional Hofbauer towers: Buzzi constructed Hofbauer towers for piecewise expanding affine maps in any dimension [18], for perturbations of fibred products of one dimensional maps [20], and for arbitrary piecewise invertible maps whose entropy generated by the boundary of some dynamically relevant partition is less than the topological entropy of the map [21]. These Hofbauer towers carry all invariant measures with entropy close enough to the topological entropy of the system. We remark that, contrary to us, Buzzi's conditions make no reference to the nonuniform hyperbolicity of the system.

Inducing schemes: Many systems, although not hyperbolic, do have sets on which it is possible to define a (not necessarily first) return map on which the map becomes uniformly hyperbolic. This process is known as inducing. Indeed, Hofbauer towers can be seen as inducing schemes for which the map becomes uniformly expanding, see [12] for this relation. It is possible to understand ergodic theoretical properties of invariant measures that lift to an inducing scheme, as done for one-dimensional maps [45], higher dimensional ones that do not have full “boundary entropy” [46], and expanding measures [43].

Yoccoz puzzles: Yoccoz constructed Markov structures for quadratic maps of the complex plane, nowadays called Yoccoz puzzles, and used them to establish the MLC conjecture for finitely renormalizable parameters as well as a proof of Jakobson's theorem, see [58].

Nonuniformly hyperbolic diffeomorphisms: Katok constructed horseshoes of large topological entropy for $C^{1+\beta }$ diffeomorphisms [34]. These horseshoes usually have zero measure for measures of maximal entropy. Sarig constructed a “horseshoe” of full entropy for $C^{1+\beta }$ surface diffeomorphisms [53]: for each $\chi >0$ there is a countable topological Markov shift that is an extension of the diffeomorphism and codes all χ–hyperbolic measures simultaneously. Ben Ovadia extended the work of Sarig to higher dimension [7].

Nonuniformly hyperbolic three-dimensional flows: Lima and Sarig constructed symbolic models for nonuniformly hyperbolic three dimensional flows with positive speed [39]. The idea is to build a “good” Poincaré section and construct a Markov partition for the Poincaré return map f.

Billiards: Dynamical billiards are maps with discontinuities. Katok and Strelcyn constructed invariant manifolds for nonuniformly hyperbolic billiard maps [35]. Bunimovich, Chernov, and Sinaĭ constructed countable Markov partitions for two dimensional dispersing billiard maps [14]. Young constructed an inducing scheme for certain two dimensional dispersing billiard maps and used it to prove exponential decay of correlations [59]. All these results are for Liouville measures. Lima and Matheus constructed countable Markov partitions for two dimensional billiard maps and nonuniformly hyperbolic (not necessarily Liouville) measures that are adapted to the billiard map [38].

The proof of Theorem 1.1 is based on [53], [39] and [38], and follows the steps below:

• The derivative cocycle df induces an invertible cocycle $\widehat{df}$, defined on a fibre bundle over the natural extension space $\hat{M}$, with the same spectrum as df.

• If μ is f–adapted and χ–expanding, then $\hat{\mu }$–a.e. $\hat{x}\in \hat{M}$ has a Pesin chart ${\mathrm{\Psi }}_{\hat{x}}:[-Q_{\epsilon }(\hat{x}),Q_{\epsilon }(\hat{x})]\to M$ s.t. ${\lim }_{n\to \infty }\frac{1}{n}\log Q_{\epsilon }({\hat{f}}^n(\hat{x}))=0$. In Pesin charts, the inverse branches of f are uniform contractions.

• Introduce the ε–chart ${\mathrm{\Psi }}_{\hat{x}}^p$ as the restriction of ${\mathrm{\Psi }}_{\hat{x}}$ to $[-p,p]$. The parameter p gives a definite size for the unstable manifold at $\hat{x}$, hence different ε–charts ${\mathrm{\Psi }}_{\hat{x}}^p,{\mathrm{\Psi }}_{\hat{x}}^q$ define unstable manifolds of different sizes.

• Construct a countable collection of ε–charts that are dense in the space of all ε–charts, where denseness is defined in terms of finitely many parameters of $\hat{x}$.

• Draw an edge ${\mathrm{\Psi }}_{\hat{x}}^p\leftarrow {\mathrm{\Psi }}_{\hat{y}}^q$ between ε–charts when an inverse branch of f can be represented in these charts by a uniform contraction and the parameter q is as large as possible. Each path of ε–charts defines an element of $\hat{M}$, and this coding induces a countable cover on a subset of $\hat{M}$. The requirement that q is as large as possible guarantees that this cover is locally finite.

• Apply a refinement procedure to this cover. The resulting partition defines a countable topological Markov shift $(\mathrm{\Sigma },\sigma )$ and a coding $\pi :\mathrm{\Sigma }\to \hat{M}$ that satisfy Theorem 1.1.

It is important to stress the importance of having a countable locally finite cover in step (5). If not, its refinement could be an uncountable partition (imagine e.g. the cover of $\mathbb{R}$ by intervals with rational endpoints). When the countable cover is locally finite, i.e. each element of the cover intersects at most finitely many others, then its refinement is again countable. Local finiteness is crucial, and it is the reason to choose q as large as possible.

Contrary to [38], [39], [53], we find no difficulty on the control of the geometry of M (all exponential maps are identities) neither with the geometry of stable and unstable directions (the stable direction is trivial). Hence the methods we use in steps (2)–(5) are more clear and more easily implemented than those in [38], [39], [53]. For example, we do not make use of graph transforms. On the other hand, a difficulty for the implementation of steps (2)–(5) is that neither $\hat{M}$ nor $\hat{f}$ are smooth objects. This is not a big issue, since what we want is to control the action of f and its inverse branches, and this can be made by controlling the action of ${\hat{f}}^{\pm 1}$ in the zeroth coordinate. Working with natural extensions makes step (1) heavier in notation, and step (6) more complicated to implement.

Natural extensions have been previously used to investigate nonuniformly expanding systems. Up to the author's knowledge, the first one to use this approach was Ledrappier, in the context of absolutely continuous invariant measures of interval maps [37]. Other employments of this approach are [23], [24].

The methods employed in this article require some familiarity with the articles [38], [39], [53], and a first reading might be difficulty for those not familiar with the referred literature. Unfortunately, a self-contained exposition would lead to a lengthy manuscript, thus preventing to focus on the novelty of the work.

Let $\mathcal{G}=(V,E)$ be an oriented graph, where V= vertex set and E= edge set. We denote edges by $v\to w$, and we assume that V is countable.

Topological Markov shift (TMS): A topological Markov shift (TMS) is a pair $(\mathrm{\Sigma },\sigma )$ where

$$\mathrm{\Sigma }:=\{\mathbb{Z}\text{–indexed paths on }\mathcal{G}\}=\{\underset{\_}{v}={\{v_n\}}_{n\in \mathbb{Z}}\in V^{\mathbb{Z}}:v_n\to v_{n+1},\forall n\in \mathbb{Z}\}$$

and $\sigma :\mathrm{\Sigma }\to \mathrm{\Sigma }$ is the left shift, ${[\sigma (\underset{\_}{v})]}_n=v_{n+1}$. The recurrent set of Σ is

$${\mathrm{\Sigma }}^{\mathrm{\#}}:=\{\underset{\_}{v}\in \mathrm{\Sigma }:\exists v,w\in V\text{ s.t. }\begin{array}{c}{v}_n=v\text{ for infinitely many }n>0\hfill \\ v_n=w\text{ for infinitely many }n<0\hfill \end{array}\}.$$

We endow Σ with the distance $d(\underset{\_}{v},\underset{\_}{w}):=\exp [-\min \{|n|:n\in \mathbb{Z}\text{ s.t. }{v}_n\ne w_n\}]$. This choice is not canonical, and affects the Hölder regularity of π in Theorem 1.1.

Write $a=e^{\pm \epsilon }b$ when $e^{-\epsilon }\le \frac{a}{b}\le e^{\epsilon }$, and $a=\pm b$ when $-|b|\le a\le |b|$. Given an open set $U\subset \mathbb{R}$ and $h:U\to \mathbb{R}$, let ${\Vert h\Vert }_0:={\sup }_{x\in U}\Vert h(x)\Vert$ denote the $C^0$ norm of h. For $0<\beta <1$, let ${\mathrm{Hol}}_{\beta }(h):=\sup \frac{\Vert h(x)-h(y)\Vert }{{\Vert x-y\Vert }^{\beta }}$ where the supremum ranges over distinct elements $x,y\in U$. If h is differentiable, let ${\Vert h\Vert }_1:={\Vert h\Vert }_0+{\Vert dh\Vert }_0$ denote its $C^1$ norm, and ${\Vert h\Vert }_{1+\beta }:={\Vert h\Vert }_1+{\mathrm{Hol}}_{\beta }(dh)$ its $C^{1+\beta }$ norm. For $r>0$, define $R[r]:=[-r,r]\subset \mathbb{R}$, where $\mathbb{R}$ is endowed with the usual euclidean distance.

Most of the discussion below is classical, see e.g. [50] or [1, §3.1]. Given a (possibly non-invertible) map $f:M\to M$, let

$$\hat{M}:=\{\hat{x}={(x_n)}_{n\in \mathbb{Z}}:f(x_{n-1})=x_n,\forall n\in \mathbb{Z}\}.$$

Although $\hat{M}$ does depend on f, we do not write this dependence. Endow $\hat{M}$ with the distance $\hat{d}(\hat{x},\hat{y}):=\sup \{2^{n}d(x_n,y_n):n\le 0\}$; then $\hat{M}$ is a compact metric space. As for TMS, the definition of $\hat{d}$ is not canonical and reflects the Hölder regularity of π in Theorem 1.1. For each $n\in \mathbb{Z}$, let ${\vartheta }_n:\hat{M}\to M$ be the projection into the n–th coordinate, ${\vartheta }_n[\hat{x}]=x_n$. Let $\hat{\mathcal{B}}$ be the sigma-algebra in $\hat{M}$ generated by $\{{\vartheta }_n:n\le 0\}$, i.e. $\hat{\mathcal{B}}$ is the smallest sigma-algebra that makes all ${\vartheta }_n$, $n\le 0$, measurable.

Natural extension of f: The natural extension of f is the map $\hat{f}:\hat{M}\to \hat{M}$ defined by $\hat{f}(\dots ,x_{-1};x_0,\dots )=(\dots ,x_0;f(x_0),\dots )$. It is an invertible map, with inverse ${\hat{f}}^{-1}(\dots ,x_{-1};x_0,\dots )=(\dots ,x_{-2};x_{-1},\dots )$.

Note that $\hat{f}$ is indeed an extension of f, since ${\vartheta }_0\circ \hat{f}=f\circ {\vartheta }_0$. It is the smallest invertible extension of f: any other invertible extension of f is an extension of $\hat{f}$. The benefit of considering the natural extension is that, in addition to having an invertible map explicitly defined, its complexity is the same as that of f: there is a bijection between f–invariant and $\hat{f}$–invariant probability measures, as follows.

Projection of a measure: If $\hat{\mu }$ is an $\hat{f}$–invariant probability measure, then $\mu =\hat{\mu }\circ {\vartheta }_0^{-1}$ is an f–invariant probability measure.

Lift of a measure: If μ is an f–invariant probability measure, let $\hat{\mu }$ be the unique probability measure on $\hat{M}$ s.t. $\hat{\mu }[\{\hat{x}\in \hat{M}:x_n\in A\}]=\mu [A]$ for all $A\subset M$ Borel and all $n\le 0$.

It is clear that $\hat{\mu }$ is $\hat{f}$–invariant. What is less clear is that the projection and lift procedures above are inverse operations, and that they preserve the Kolmogorov-Sinaĭ entropy, see [50]. Here is one consequence of this fact: μ is an equilibrium measure for a potential $\phi :M\to \mathbb{R}$ iff $\hat{\mu }$ is an equilibrium measure for $\phi \circ {\vartheta }_0:\hat{M}\to \mathbb{R}$. In particular, the topological entropies of f and $\hat{f}$ coincide, and μ is a measure of maximal entropy for f iff $\hat{\mu }$ is a measure of maximal entropy for $\hat{f}$.

Now let $N={⨆}_{x\in M}{N}_x$ be a vector bundle over M, and let $A:N\to N$ measurable s.t. for every $x\in M$ the restriction $A{\mathrm{\upharpoonright }}_{N_x}$ is a linear isomorphism $A_x:N_x\to N_{f(x)}$. For example, if f is a differentiable endomorphism on a manifold M, then we can take $N=TM$ and $A=df$. The map A defines a (possibly non-invertible) cocycle ${(A^{(n)})}_{n\ge 0}$ over f by $A_x^{(n)}=A_{f^{n-1}(x)}\cdots A_{f(x)}{A}_x$ for $x\in M$, $n\ge 0$. There is a way of extending ${(A^{(n)})}_{n\ge 0}$ to an invertible cocycle over $\hat{f}$. For $\hat{x}\in \hat{M}$, let $N_{\hat{x}}:=N_{{\vartheta }_0[\hat{x}]}$ and let $\hat{N}:={⨆}_{\hat{x}\in \hat{M}}{N}_{\hat{x}}$, a vector bundle over $\hat{M}$. Define the map $\hat{A}:\hat{N}\to \hat{N}$, ${\hat{A}}_{\hat{x}}:=A_{{\vartheta }_0[\hat{x}]}$. For $\hat{x}={(x_n)}_{n\in \mathbb{Z}}$, define

$${\hat{A}}_{\hat{x}}^{(n)}:=\{\begin{array}{cc}{A}_{x_0}^{(n)}\hfill & ,\text{ if }n\ge 0\hfill \\ A_{x_{-n}}^{-1}\cdots A_{x_{-2}}^{-1}{A}_{x_{-1}}^{-1}\hfill & ,\text{ if }n\le 0.\hfill \end{array}$$

By definition, ${\hat{A}}_{\hat{x}}^{(m+n)}={\hat{A}}_{{\hat{f}}^n(\hat{x})}^{(m)}{\hat{A}}_{\hat{x}}^{(n)}$ for all $\hat{x}\in \hat{M}$ and all $m,n\in \mathbb{Z}$, hence ${({\hat{A}}^{(n)})}_{n\in \mathbb{Z}}$ is an invertible cocycle over $\hat{f}$.

Whenever it is convenient, we will write ϑ to represent ${\vartheta }_0$.