# Nonuniform hyperbolicity

Post-publication activity

Curator: Boris Hasselblatt

## Introduction

The study of hyperbolic dynamics began with the study of uniformly hyperbolic dynamical systems, which is detailed in the entry on hyperbolic dynamics. This study was tremendously useful in the development of technical tools and insights, and in the shaping of a body of concepts suitable for the description and study of complicated dynamics.

However, it was clear early on that the assumption of uniform hyperbolicity is too stringent in several ways. For instance, not every manifold admits an Anosov diffeomorphism, and in applications it is not uncommon to encounter some hyperbolic behavior, but without uniformity of contraction and expansion. Nonuniform hyperbolicity allows the asymptotic expansion and contraction rates to depend on the point in a way that does not admit uniform bounds, which provides a generalization that is broad enough to include a wide range of applications but restrictive enough that to a remarkable degree the elements of the hyperbolic paradigm, such as positive entropy and strong ergodic properties, can be established here. In particular, nonuniformly hyperbolic systems can be found on any compact phase space. It is indeed the nonuniform version of hyperbolicity that allows for great applications outside of mathematics, such as to physics, biology, engineering, and so on, and that provides a mathematical basis for the theory of chaos.

This entry outlines essential elements of the theory of nonuniformly hyperbolic dynamical systems, which is also known as Pesin Theory, building on the notions and paradigms from the theory of uniformly hyperbolic dynamical systems. A different generalization of uniform hyperbolicity, partial hyperbolicity, is the subject of a separate entry.

## Lyapunov Exponents

Instead of prescribing bounds for the expansion and contraction of vectors, the theory of nonuniformly hyperbolic dynamical systems measures the infinitesimal asymptotic exponential relative behavior of points by the Lyapunov exponent. For a map $$f$$ the forward Lyapunov exponent of a vector $$v$$ at a point $$x$$ is defined by $\chi^+(x,v):=\varlimsup_{n\to\infty}\frac1n\log\|D_xf^n(v)\|.$

At any $$x$$ this takes only finitely many values $$\chi_1^+(x)<\dots<\chi_{p^+(x)}^+(x)$$ that determine subspaces $$V_i^+(x):=\{v\in T_xM\mid\chi^+(x,v)\leq\chi_i^+(x)\}$$ which are nested: $\{0\}=V_0^+(x)\subsetneq\dots\subsetneq V_{p^+(x)}^+(x)=T_xM.$

The multiplicity of $$\chi_i^+(x)$$ is defined as $$k_i^+(x):=\dim V_i^+(x)-\dim V_{i-1}^+(x)\ ;$$ both of the functions $$\chi_i^+$$ and $$k_i^+$$ are $$f$$-invariant. The sum of the positive Lyapunov exponents is $\tag{1} \Sigma(x):=\sum_{\chi^+_i(x)>0}k_i^+(x)\chi_i^+(x)\ .$

Taking the corresponding limits as $$n\to-\infty$$ yields the backwards Lyapunov exponent $$\chi^-\ ,$$ for which the corresponding results hold.

The Oseledec Multiplicative Ergodic Theorem. For a smooth diffeomorphism of a compact manifold $$M$$ and any invariant Borel probability measure almost every point $$x$$ is Lyapunov-Perron regular, i.e.,

• $$p^+(x)=p^-(x)=:p(x)\ ,$$
• $$T_xM=\oplus_{i=1}^{p(x)}E_i(x)\ ,$$ where $$E_i(x):= V_i^+(x)\cap V_i^-(x)\ ,$$
• $$\lim_{n\to\pm\infty}\frac1n\log\|D_xf^n(v)\|=\chi_i^+(x)=-\chi_i^-(x)=:\chi_i(x)$$ uniformly in $$\{v\in E_i(x)\mid\|v\|=1\}\ ,$$
• $$\lim_{n\to\pm\infty}\frac1n\log|\det D_xf^n|=\sum_{i=1}^{p(x)}\chi_i(x)\dim E_i(x)\ .$$

With analogous definitions one obtains the corresponding results for flows.

Definition. A diffeomorphism $$f$$ is said to have nonzero exponents on an invariant set $$\Lambda$$ if for each $$x\in\Lambda$$ there is an $$s=s(x)$$ such that $\tag{2} \chi_1(x)<\dots<\chi_s(x)<0<\chi_{s+1}(x)<\dots<\chi_{p(x)}(x).$

An $$f$$-invariant Borel probability measure is said to be hyperbolic if equation (2) holds for almost every $$x\in M\ .$$

We say that a Borel set $$\Lambda\subset M$$ is nonuniformly partially hyperbolic (in the broad sense) if there exist

• a $$Df$$-invariant decomposition $$T_xM=E^1(x)\oplus E^2(x)$$ for $$x\in\Lambda$$ (i.e., $$Df E^i(x)=E^i(f(x))$$ for $$i=s,u$$) and
• $$f$$-invariant Borel functions $$\lambda_1,\lambda_2\colon \Lambda\to\mathbb{R}^+$$ (i.e., $$\lambda_i\circ f=\lambda_i$$ for $$i=1,2$$) such that either $$\lambda_1<\min\{1, \lambda_2\}$$ or $$\lambda_2>\max\{1,\lambda_1\}$$with the following properties:

For every $$\epsilon_0>0$$ there are an $$f$$-invariant Borel function $$\epsilon\colon \Lambda\to(0,\epsilon_0)$$ and positive Borel functions $$C,K$$ on $$\Lambda$$ such that for every $$x\in \Lambda$$ we have $\angle (E^2(x),E^1(x)) \ge K(x),$ $\|Df^n_x v\|\le C(x)(\lambda_1(x))^n\|v\|\hbox{ for }v\in E^1(x)\hbox{ and } n\in\mathbb{N},$ $\|Df^{-n}_xv\|\le C(x)(\lambda_2(x))^{-n}\|v\|\hbox{ for }v\in E^2(x)\hbox{ and }n\in\mathbb{N},$ and for every $$n\in\Z$$ and $$x\in \Lambda$$ we have $C(f^n(x))\le C(x)e^{\epsilon(x)|n|}\hbox{ and }K(f^n(x))\ge K(x)e^{\epsilon(x)|n|}.$

We say that $$\Lambda$$ is nonuniformly (completely) hyperbolic if $$\lambda_1<1<\lambda_2\ .$$

A set $$R$$ of Lyapunov-Perron regular points is nonuniformly partially hyperbolic (in the broad sense) if some Lyapunov exponents are nonzero at every point of $$R$$ and nonuniformly hyperbolic if all Lyapunov exponents are nonzero at every point of $$R\ .$$ Since there is no assumption that the limits in the definition of Lyaponov exponents converge uniformly, nonuniform hyperbolicity involves slowly changing functions $$C(x)$$ and $$K(x)\ ,$$ where uniformly hyperbolic systems have constants. That these parameters change more slowly than the expansion and contraction along the orbit makes the definition substantial nevertheless.

The set $$\Lambda$$ on which a diffeomorphism $$f$$ is nonuniformly hyperbolic is invariant but it is compact only if $$f$$ is uniformly hyperbolic on $$\Lambda\ .$$ This phenomenon is called Anosov rigidity.

## Stable Manifolds

Many results and techniques from the theory of uniformly hyperbolic dynamical systems extend to nonuniformly partially hyperbolic dynamical systems (in the broad sense), such as stable or unstable manifolds, cone techniques and $$u$$-measures. For nonuniformly completely hyperbolic dynamical systems, one obtains both stable and unstable manifolds. These are not known to be as uniform in size and alignment as in the uniform case. The problem is that although the hyperbolicity conditions may ensure contraction of stable separations in positive time and that of unstable directions in negative time, there is no mechanism to ensure that the reversal of time brings about any expansion. Accordingly, the stable and unstable manifolds, which in the uniformly hyperbolic case can be extended indefinitely, are not known to extend indefinitely. The local leaves are usually envisioned like a "fence" consisting of moderate-size posts between which there is a finer fence of somewhat crooked and smaller posts, between which in turn there is a yet finer fence of posts that are yet smaller and rather more crooked, and so on ad infinitum. The (noninvariant) sets of points for which the stable and unstable leaves have a certain minimal size and mutual angle, are called Pesin sets. Numerous arguments for the uniformly hyperbolic context that use invariant foliations can be extended to the present context, but they tend to be far more subtle and often have less uniform or global conclusions.

Lyapunov introduced Lyapunov exponents, and the notion of regularity appears in his work and that of Perron in connection with the study of stability of solutions of linear ordinary differential equation with nonconstant coefficients. Oseledec published the Multiplicative Ergodic Theorem in 1968, and the relation between Lyapunov exponents and nonuniform hyperbolicity was established by Pesin in 1977.

## Examples

There are nonuniformly hyperbolic systems on any manifold whose dimension is not too small:

Theorem. On any compact smooth Riemannian manifold other than $$S^1$$ there is a $$C^\infty$$ volume-preserving Bernoulli diffeomorphism with nonzero Lyapunov exponents almost everywhere. Every compact smooth Riemannian manifold of dimension at least 3 admits a $$C^\infty$$ volume-preserving Bernoulli flow with nonzero Lyapunov exponents almost everywhere (except for the zero Lyapunov exponent in the flow direction).

In fact, it is conjectured that among $$C^2$$ volume-preserving maps and flows those that are nonuniformly hyperbolic are in some sense generic, and there are substantial partial results in this direction.

In constructing particular examples, two main techniques are used: slow-down and cone-twisting.

### Slow-down

A basic example of a nonuniformly hyperbolic dynamical systems that is not uniformly hyperbolic is given by a suitably constructed deformation of the map induced by Arnold's cat map of the 2-torus, that is the map generated by the matrix $$\begin{pmatrix}2&1\\1&1\end{pmatrix}\ .$$ The mechanism of the construction is to slow down the motion at the origin in order to produce zero Lyapunov exponent. Specifically, the map is carefully altered in a neighborhood of the fixed point 0 in such a way that afterwards 0 is an indifferent fixed point. At this point, and at all points on the stable and unstable separatrices through it, all Lyapunov exponents are therefore equal to 0, but for almost every other point one can ensure that the Lyapunov exponents are nonzero. Since there are orbits that spend large amounts of time near 0 they produce Lyapunov exponents close to 0.

If this construction is carried out with enough symmetry then it projects to the 2-sphere via the identification $$x\sim-x\ .$$ A variant of this construction (known as blow-up) produces analytic examples as well.

The scheme of slowing down the motion near a particular point can also be used to produce a corresponding flow example. Here a volume-preserving Anosov flow on a 3-manifold is altered in a small box by a slowdown in parts of a small solid torus. Deleting the core of this torus and gluing the boundary to that of an identical copy produces a volume-preserving nonuniformly hyperbolic flow on a 3-manifold.

Nonuniformly hyperbolic flows also arise in passing from negatively curved Riemannian manifolds, whose geodesic flows are Anosov flows, to nonpositively curved manifolds. The study of these was a motivating factor in the development of the theory of nonuniformly hyperbolic dynamical systems.

### Cone-twisting

A different mechanism for producing nonuniformly hyperbolic systems from uniformly hyperbolic ones is the twisting of cone fields. Analogously to the uniformly hyperbolic situation, nonuniform hyperbolicity can be described in terms of cone families, but in this new situation there is no uniform control over the opening angles or the mutual angles of the cones, and these families need not be continuous. Therefore constructions that alter a uniformly hyperbolic diffeomorphism in such a way that in some places an expanding cone has large overlap with a contracting one can result in complete or partial cancellation of the expansion of vectors in the expanding cone by the subsequent contraction.

Cone-twisting occurs in the Hénon attractor. Near the apex of the arc of the Hénon attractor horizontal expanding cones become close to vertical, and are therefore subject to the extreme vertical contraction of the Hénon map. This very phenomenon makes this system difficult to study. (A related noninvertible family of examples is given by the map $$x\mapsto ax(1-x)$$ for $$a\le4$$ of the unit interval; the Jakobson Theorem gives information about its hyperbolicity for a set of $$a$$ of positive Lebesgue measure.)

### Eventually strictly invariant cones

Invariant cone fields provide a versatile method for establishing nonuniform hyperbolicity. To deal with the weak expansion and contraction, Wojtkowski introduced the notion of eventually strictly invariant cone fields. He showed that nonuniform hyperbolicity can be established using cone fields even if the requirement that the image of a cone be strictly in the next is weakened to the requirement that the image of each cone lie in the next, and that for every orbit there is some time at which this inclusion is strict. This is the most powerful method for establishing nonuniform hyperbolicity, and it generalizes the initial approach by Sinai and Alekseev.

## Ergodic Properties

Hyperbolicity usually implies an abundance of invariant Borel probability measures. Some of these are not of great interest in themselves, such as point masses concentrated on fixed points or periodic orbits. On the other hand, a measure of maximal entropy is clearly of dynamical interest; in the uniformly hyperbolic context this is the Bowen-Margulis Measure. If a hyperbolic dynamical system preserves volume or another smooth measure, this is of evident interest, and when present, so is a Sinai-Ruelle-Bowen measure. These three constitute the principal measures of interest for nonuniformly hyperbolic dynamical systems.

The ergodic decomposition of a smooth volume-preserving dynamical system may consist of an uncountable family of null sets. However, for nonuniformly hyperbolic dynamical systems there is an effective decomposition:

Theorem. A smooth hyperbolic invariant measure for a $$C^{1+\alpha}$$ diffeomorphism of a compact manifold $$M$$ decomposes into countably many ergodic components of positive measure.

A central ingredient in this proof is the Hopf argument, which rests on the fact that even for nonuniformly hyperbolic systems the stable and unstable foliations have the absolute continuity property discussed above. Examples show that there may indeed be countably many ergodic components.

If the unstable foliation extends to a continuous foliation of $$M$$ with smooth leaves and if every local unstable leaf (regardless of its size) eventually expands to a certain uniform size when pushed forward then each ergodic component can be taken to be an open set. This is referred to as local ergodicity.

Another approach to proving local ergodicity (due to Burns-Gerber and Katok-Burns) is based on Lyapunov function techniques and is closely related to the approach of Wojtkowski on computing Lyapunov exponents via an infinitesimally eventually strict cone family. One can also establish local ergodicity for partially hyperbolic systems with negative central exponents (either volume-preserving or with partially hyperbolic attractors).

As in the uniformly hyperbolic case, hyperbolic invariant measures tend to have much stronger ergodic properties than ergodicity. The ergodic components in the preceding theorem each have a spectral decomposition into finitely many pieces that are permuted by $$f$$ in such a way that the return map to each piece has the Bernoulli property. In particular, if this decomposition is trivial, that is, for a weakly mixing smooth invariant measure, the diffeomorphism is a Bernoulli automorphism.

There are intimate connections between Lyapunov exponents and entropy. The first to among these was the Margulis-Ruelle inequality $h_\nu(f)\leq\int_M\Sigma(x)d\nu(x),$

for any invariant Borel probability measure. Here $$\Sigma$$ is the sum of the positive Lyapunov exponents from equation (1) above. The central pertinent result is the Pesin Entropy Formula, which has been developed rather extensively and is therefore the subject of a separate entry. The content of this formula and related ones is that the entropy of a measure is given exactly by the total expansion in the system, and these also establish close relations to fractal dimensions of measures and their stable or unstable conditionals.

A hyperbolic measure $$\nu$$ is said to be a Sinai-Ruelle-Bowen measure for a diffeomorphism $$f$$ if it is absolutely continuous on unstable manifolds. This means, roughly, that restricting $$\nu$$ to small neighborhoods of a piece of an unstable manifold and keeping this restriction normalized as the neighborhood shrinks down to the small piece of stable manifold one obtains in the limit the normalized Riemannian measure on the piece of unstable manifold. As a direct consequence of the definition one obtains

Theorem. A $$C^{1+\alpha}$$ diffeomorphism of a compact smooth manifold has at most countably many Sinai-Ruelle-Bowen measures, and the basin of attraction of every Sinai-Ruelle-Bowen measure has positive Lebesgue measure.

This says that Sinai-Ruelle-Bowen measures are "physical measures" in that there is a positive probability that a point chosen at random (with respect to Lebesgue measure) will display asymptotics that reflect the Sinai-Ruelle-Bowen measure.

Accordingly, establishing the existence of a Sinai-Ruelle-Bowen measure on an attractor is a major step that immediately provides a detailed probabilistic picture of the dynamics on the attractor.

For transitive Anosov diffeomorphisms or hyperbolic attractors there is a Sinai-Ruelle-Bowen measure, and it is obtained as the limit of push-forwards of Lebesgue measure, that is, it is given by $$\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f_*^km\ ,$$ where $$m$$ is Lebesgue measure and $$f_*^km(A):=m(f^{-1}(A))\ .$$ Even for nonuniformly hyperbolic systems obtained by "slowing down" the map $$\begin{pmatrix}2&1\\1&1\end{pmatrix}$$ on $$\mathbb{T}^2$$ the existence of a Sinai-Ruelle-Bowen measure is subtle.

For the Hénon attractor the existence of a Sinai-Ruelle-Bowen measure was established for a substantial set of parameters:

Benedicks-Young Theorem. There exist $$\epsilon,\delta>0$$ such that for every $$b\in(0,\delta)$$ there is set $$\Delta_b\subset(2-\epsilon,2)$$ of positive Lebesgue measure such that for $$a\in\Delta_b$$ the Hénon map $$(x,y)\mapsto(1-ax^2-y,x)$$ admits a unique Sinai-Ruelle-Bowen measure.

Geodesic flows of manifolds of nonpositive curvature are amenable to the techniques of Pesin theory as well and provide important examples.

Theorem. Let $$M$$ be a compact surface of nonpositive curvature and genus $$> 1\ .$$ Then the geodesic flow $$g_t$$ on $$M$$ possesses an open $$\pmod 0$$ dense invariant set $$\Lambda$$ such that $$g_t\upharpoonright\Lambda$$ is nonuniformly hyperbolic, ergodic and indeed a Bernoulli flow.

This result extends to multidimensional manifolds with no focal points.

A highly important recent application is to the Teichmüller flow: this flow is also nonuniformly hyperbolic.

## Topological Properties

The Anosov Closing Lemma generalizes to orbit segments of nonuniformly hyperbolic diffeomorphisms that almost close up. The required closeness of the beginning and end points depend on the Pesin set in which they lie, as does the degree of approximation by the resulting hyperbolic periodic orbit.

As a result, hyperbolic periodic points are dense in the support of every hyperbolic measure. Moreover, if there is any hyperbolic measure then the measures supported on hyperbolic periodic orbits are weakly dense in the set of hyperbolic measures.

Similarly, the Shadowing Lemma extends to this context, and the allowed $$\epsilon$$ depends on the desired closeness of the shadowing orbit as well as the Pesin set in which the pseudo-orbit lies.

Moreover, the Livshitz Theorem generalizes to this setting as well, but with lower regularity of the solution of the cohomological equation:

Livshitz Theorem. If $$\nu$$ is a hyperbolic measure for a diffeomorphism $$f$$ and $$\varphi$$ is a Hölder continuous function such that $$\sum_{i-0}^{n-1}\varphi(f^i(x))=0$$ whenever $$f^n(x)=x$$ then there is a Borel-measurable function $$h$$ such that $$\varphi(x)=h(f(x))-h(x)$$ for $$\nu$$-almost every $$x\ .$$

There is also a spectral decomposition for continuous nonatomic hyperbolic invariant Borel probability measures: Each Pesin set is contained in a finite union of orbit closures.

Indeed, whenever there is a continuous nonatomic hyperbolic invariant Borel probability measure then there is a horseshoe, that is, a compact locally maximal uniformly hyperbolic set (on which the diffeomorphism corresponds to a topological Markov chain via a Markov partition). In particular, the diffeomorphism has positive topological entropy.

More strongly, and without the asumption of hyperbolicity, given any ergodic invariant Borel probability measure $$\nu$$ with positive entropy $$h_\nu(f)$$ and any $$\epsilon>0$$ there is a hyperbolic horseshoe $$\Lambda$$ in an $$\epsilon$$-neighborhood of the support of $$\mu$$ such that the topological entropy of $$f$$ on $$\Lambda$$ exceeds $$h_\nu(f)-\epsilon\ .$$ Indeed, there is a sequence $$\nu_n$$ of $$f$$-invariant Borel probability measures supported on hyperbolic horseshoes such that $$\nu_n\to\nu$$ weakly and $$h_{\nu_n}(f)\to h_\nu(f)\ .$$