Pesin theory

An important branch of the theory of dynamical systems (cf. Dynamical system) and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Ya.B. Pesin in the mid-1970{}s [a20], [a21], [a22]. Sometimes Pesin theory is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behaviour, or simply the theory of non-uniformly hyperbolic dynamical systems.

Introduction.

One of the paradigms of dynamical systems is that the local instability of trajectories influences the global behaviour of the system, and paves the way to the existence of stochastic behaviour. Mathematically, instability of trajectories corresponds to some degree of hyperbolicity (cf. Hyperbolic set). The "strongest possible" kind of hyperbolicity occurs in the important class of Anosov systems (also called -systems, cf. -system) [a1]. These are only known to occur in certain manifolds. Moreover, there are several results of topological nature showing that certain manifolds cannot carry Anosov systems.

Pesin theory deals with a "weaker" kind of hyperbolicity, a much more common property that is believed to be "typical" : non-uniform hyperbolicity. Among the most important features due to hyperbolicity is the existence of invariant families of stable and unstable manifolds and their "absolute continuity" . The combination of hyperbolicity with non-trivial recurrence produces a rich and complicated orbit structure. The theory also describes the ergodic properties of smooth dynamical systems possessing an absolutely continuous invariant measure in terms of the Lyapunov exponents. One of the most striking consequences is the Pesin entropy formula, which expresses the metric entropy of the dynamical system in terms of its Lyapunov exponents.

Non-uniform hyperbolicity.

Let be a diffeomorphism of a compact manifold. It induces the discrete dynamical system (or cascade) composed of the powers . Fix a Riemannian metric on . The trajectory of a point is called non-uniformly hyperbolic if there are positive numbers and splittings for each , and if for all sufficiently small there is a positive function on the trajectory such that for every :

1) ;

2) , ;

3) if and , then

4) if and , then

5) .

(The indices "s" and "u" refer, respectively, to "stable" and "unstable" .) The definition of non-uniformly partially hyperbolic trajectory is obtained by replacing the inequality by the weaker requirement that and .

If (respectively, and ) and the conditions 1)–5) hold for (i.e., if one can choose ), the trajectory is called uniformly hyperbolic (respectively, uniformly partially hyperbolic).

The term "non-uniformly" means that the estimates in 3) and 4) may differ from the "uniform" estimates and by at most slowly increasing terms along the trajectory, as in 1) (in the sense that the exponential rate in 1) is small in comparison to the number ); the term "partially" means that the hyperbolicity may hold only for a part of the tangent space.

One can similarly define the corresponding notions for a flow (continuous-time dynamical system) with replaced by , and the splitting of the tangent spaces replaced by , where is the one-dimensional subspace generated by the flow direction.

Stable and unstable manifolds.

Let be a non-uniformly partially hyperbolic trajectory of a -diffeomorphism (). Assume that . Then there is a local stable manifold such that , , and for every , , and ,

where is the distance induced by the Riemannian metric and is a positive constant. The size of can be chosen in such a way that for every , where is a positive constant. If (), then is of class .

The global stable manifold of at is defined by ; it is an immersed manifold with the same smoothness class as . One has if , if , and for every . The manifold is independent of the particular size of the local stable manifolds .

Similarly, when one can define a local (respectively, global) unstable manifold as a local (respectively, global) stable manifold of .

Non-uniformly hyperbolic dynamical systems and dynamical systems with non-zero Lyapunov exponents.

Let be a diffeomorphism and let be a (finite) Borel -invariant measure (cf. also Invariant measure). One calls non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) with respect to the measure if the set of points whose trajectories are non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) is such that . In this case , , , and are replaced by measurable functions , , , and , respectively.

The set is -invariant, i.e., it satisfies . Therefore, one can always assume that when ; this means that if , then the measure on defined by is -invariant and .

For , one defines the forward upper Lyapunov exponent of (with respect to ) by

(a1)

for each , and . For every , there exist a positive integer (the dimension of ) and collections of numbers and linear subspaces such that for every ,

and if , then .

The numbers are called the values of the forward upper Lyapunov exponent at , and the collection of linear subspaces is called the forward filtration at associated to . The number is the forward multiplicity of the exponent . One defines the forward spectrum of at as the collection of pairs for . Let be the values of the forward upper Lyapunov exponent at counted with multiplicities, i.e., in such a way that the exponent appears exactly a number of times. The functions and , for , are measurable and -invariant with respect to any -invariant measure.

One defines the backward upper Lyapunov exponent of (with respect to ) by an expression similar to (a1), with replaced by , and considers the corresponding backward spectrum.

A Lyapunov-regular trajectory (see, for example, [a3], Sect. 2) is non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) if and only if for all (respectively, for some ). For flows, a Lyapunov-regular trajectory is non-uniformly hyperbolic if and only if for all .

The multiplicative ergodic theorem of V. Oseledets [a19] implies that -almost all points of belong to a Lyapunov-regular trajectory. Therefore, for a given diffeomorphism, one has for all (respectively for some ) on a set of positive -measure if and only if the diffeomorphism is non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic). Hence, the non-uniformly hyperbolic diffeomorphisms (with respect to the measure ) are precisely the diffeomorphisms with non-zero Lyapunov exponents (on a set of positive -measure).

Furthermore, for -almost every there exist subspaces , for , such that for every one has ,

for every , and if , then

Pesin sets.

To a non-uniformly partially hyperbolic diffeomorphism one associates a filtration of measurable sets (not necessarily invariant) on which the estimates 3)–5) are uniform.

Let be a non-uniformly hyperbolic diffeomorphism and let . Given , one defines the measurable set by

One has when , and . Each set is closed but need not be -invariant; for every and there exists an such that . The distribution is, in general, only measurable on but it is continuous on . The local stable manifolds depend continuously on and their sizes are uniformly bounded below on . Each set is called a Pesin set.

One similarly defines Pesin sets for arbitrary non-uniformly partially hyperbolic diffeomorphisms.

Lyapunov metrics and regular neighbourhoods.

Let be the Riemannian metric on . For each fixed and every , one defines a Lyapunov metric on by

for each . One extends this metric to by declaring orthogonal the subspaces for . The metric is continuous on . The sequence of weights is called a Pesin tempering kernel. Any linear operator on such that

is called a Lyapunov change of coordinates.

There exist a measurable function satisfying , and for each a collection of imbeddings , defined on the ball by , such that if , then:

1) the derivative of at the point has the Lyapunov block form

where each is an invertible linear operator between the -dimensional spaces and , for ;

2) for each ,

3) the -distance between and on the ball is at most ;

4) there exist a constant and a measurable function satisfying such that for every ,

The function is bounded on each . The set is called a regular neighbourhood of the point .

Absolute continuity.

A property playing a crucial role in the study of the ergodic properties of (uniformly and non-uniformly) hyperbolic dynamical systems is the absolute continuity of the families of stable and unstable manifolds. It allows one to pass from the local properties of the system to the study of its global behaviour.

Let be an absolutely continuous -invariant measure, i.e., an -invariant measure that is absolutely continuous with respect to Lebesgue measure (cf. Absolute continuity). For each and there exists a neighbourhood of with size depending only on and with the following properties (see [a21]). Choose . Given two smooth manifolds transversal to the local stable manifolds in , one defines

for . Let be the correspondence that takes to the point such that for some . If is the measure induced on by the Riemannian metric, for , then is absolutely continuous with respect to (if is sufficiently large, then for ).

This result has the following consequences (see [a21]). For each measurable set , let be the union of all the sets such that and . The partition of into the submanifolds is a measurable partition (also called measurable decomposition), and the corresponding conditional measure of on is absolutely continuous with respect to the measure induced on by the Riemannian metric, for each such that . In addition, for -almost all , and the measure on defined for each measurable set by , is absolutely continuous with respect to .

Smooth ergodic theory.

Let be a non-uniformly hyperbolic -diffeomorphism () with respect to a Sinai–Ruelle–Bowen measure , i.e., an -invariant measure that has a non-zero Lyapunov exponent -almost everywhere and has absolutely continuous conditional measures on stable (or unstable) manifolds with respect to Lebesgue measure (in particular, this holds if is absolutely continuous with respect to Lebesgue measure and has no zero Lyapunov exponents [a21]; see also above: "Absolute continuity" ). Then there is at most a countable number of disjoint -invariant sets (the ergodic components) such that [a21], [a11]:

1) , , and and is ergodic (see Ergodicity) with respect to for every ;

2) each set is a disjoint union of sets such that for each , and ;

3) for every and , there is a metric isomorphism between and a Bernoulli automorphism (in particular, the mapping is a -system).

If is an absolutely continuous -invariant measure and the foliation (or ) of is -continuous (i.e., for each there is a neighbourhood of in that is the image of an injective -mapping , defined on the ball with centre at and of radius , and the mapping from into the family of -mappings is continuous), then any ergodic component of positive -measure is an open set (mod ); if, in addition, is topologically transitive (cf. Topological transitivity; Chaos), then is ergodic [a21].

If is ergodic, then for Lebesgue-almost-every point and every continuous function , one has

There is a measurable partition of with the following properties:

1) for -almost every , the element containing is an open subset (mod ) of ;

2) is a refinement of , and is the partition of into points;

3) coincides with the measurable hull of , as well as with the maximal partition with zero entropy (the -partition for ; see Entropy of a measurable decomposition);

4) (cf. Entropy theory of a dynamical system).

Pesin entropy formula.

For a -diffeomorphism () of a compact manifold and an absolutely continuous -invariant probability measure , the metric entropy of with respect to is given by the Pesin entropy formula [a21]

(a2)

where and form the forward spectrum of at .

For a -diffeomorphism of a compact manifold and an -invariant probability measure , the Ruelle inequality holds [a25]:

(a3)

An important consequence of (a3) is that any -diffeomorphism with positive topological entropy has an -invariant measure with at least one positive and one negative Lyapunov exponent; in particular, for surface diffeomorphisms there is an -invariant measure with every exponent non-zero. For arbitrary invariant measures the inequality (a3) may be strict [a7].

The formula (a2) was first established by Pesin in [a21]. A proof which does not use the theory of invariant manifolds and absolute continuity was given by R. Mañé [a17]. For -diffeomorphisms, (a2) holds if and only if has absolutely continuous conditional measures on unstable manifolds [a13], [a12].

The formula (a2) has been extended to mappings with singularities [a12]. For -diffeomorphisms and arbitrary invariant measures, results of F. Ledrappier and L.-S. Young [a14] show that the possible defect between the left- and right-hand sides of (a3) is due to the defects between and the Hausdorff dimension of "in the direction of Eix" for each .

Hyperbolic measures.

Let be a -diffeomorphism () and let be an -invariant measure. One says that is hyperbolic (with respect to ) if for -almost every and all . The measure is hyperbolic (with respect to ) if and only if is non-uniformly hyperbolic with respect to (and the set has full -measure). The fundamental work of A. Katok has revealed a rich and complicated orbit structure for diffeomorphisms possessing a hyperbolic measure.

Let be a hyperbolic measure. The support of is contained in the closure of the set of periodic points. If is ergodic and not concentrated on a periodic orbit, then [a7], [a9]:

1) the support of is contained in the closure of the set of hyperbolic periodic points possessing a transversal homoclinic point;

2) for every there exists a closed -invariant hyperbolic set such that the restriction of to is topologically conjugate to a topological Markov chain with topological entropy , i.e., the entropy of a hyperbolic measure can be approximated by the topological entropies of invariant hyperbolic sets.

If possesses a hyperbolic measure, then satisfies a closing lemma: given , there exists a such that for each and each integer satisfying and , there exists a point such that , for every , and is a hyperbolic periodic point [a7]. The diffeomorphism also satisfies a shadowing lemma (see [a9]) and a Lifschitz-type theorem [a9]: if is a Hölder-continuous function (cf. Hölder condition) such that for each periodic point with , then there is a measurable function such that for -almost every .

Let be the number of periodic points of with period . If possesses a hyperbolic measure or is a surface diffeomorphism, then

where is the topological entropy of [a7].

Let be a hyperbolic ergodic measure. L.M. Barreira, Pesin and J. Schmeling [a2] have shown that there is a constant such that for -almost every ,

where is the ball in with centre at and of radius (this claim was known as the Eckmann–Ruelle conjecture); this implies that the Hausdorff dimension of and the lower and upper box dimensions of coincide and are equal to (see [a2]). Ledrappier and Young [a14] have shown that if (respectively, ) are the conditional measures of with respect to the stable (respectively, unstable) manifolds, then there are constants and such that for -almost every ,

where (respectively, ) is the ball in (respectively, ) with centre at and of radius . Moreover, [a2] and has an "almost product structure" (see [a2]).

Criteria for having non-zero Lyapunov exponents.

Above it has been shown that non-uniformly hyperbolic dynamical systems possess strong ergodic properties, as well as many other important properties. Therefore, it is of primary interest to have verifiable methods for checking the non-vanishing of Lyapunov exponents.

The following Katok–Burns criterion holds: A real-valued measurable function on the tangent bundle is called an eventually strict Lyapunov function if for -almost every :

1) the function is continuous, homogeneous of degree one and takes both positive and negative values;

2) the maximal dimensions of the linear subspaces contained, respectively, in the sets and are constants and , and is the dimension of ;

3) for all ;

4) there exists a positive integer such that for all ,

If possesses an eventually strict Lyapunov function, then there exist exactly positive Lyapunov exponents and negative ones [a8] (see also [a28]).

Another method to estimate the Lyapunov exponents was presented in [a6].

Generalizations.

There are several natural and important generalizations of Pesin theory. Examples of these are: generalizations to non-invertible mappings; extensions of the main results of Pesin's work to mappings with singularities [a10], including billiard systems and other physical models; infinite-dimensional versions of results on stable and unstable manifolds in Hilbert spaces [a27] and Banach spaces [a18], given certain compactness assumptions; some results have been extended to random mappings [a15].

Related results have been obtained for products of random matrices (see [a5] and the references therein).

References

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