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 $ Y $- systems, cf. $ Y $- 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 $ f : M \rightarrow M $ be a diffeomorphism of a compact manifold. It induces the discrete dynamical system (or cascade) composed of the powers $ \{ {f ^ {n} } : {n \in \mathbf Z } \} $. Fix a Riemannian metric on $ M $. The trajectory $ \{ {f ^ {n} x } : {n \in \mathbf Z } \} $ of a point $ x \in M $ is called non-uniformly hyperbolic if there are positive numbers $ \lambda < 1 < \mu $ and splittings $ T _ {f ^ {n} x } M = E ^ {u} ( f ^ {n} x ) \oplus E ^ {s} ( f ^ {n} x ) $ for each $ n \in \mathbf Z $, and if for all sufficiently small $ \epsilon > 0 $ there is a positive function $ C _ \epsilon $ on the trajectory such that for every $ k \in \mathbf Z $:

1) $ C _ \epsilon ( f ^ {k} x ) \leq e ^ {\epsilon | k | } C _ \epsilon ( x ) $;

2) $ Df ^ {k} E ^ {u} ( x ) = E ^ {u} ( f ^ {k} x ) $, $ Df ^ {k} E ^ {s} ( x ) = E ^ {s} ( f ^ {k} x ) $;

3) if $ v \in E ^ {u} ( f ^ {k} x ) $ and $ m < 0 $, then

$$ \left \| {Df ^ {m} v } \right \| \leq C _ \epsilon ( f ^ {m + k } x ) \mu ^ {m} \left \| v \right \| ; $$

4) if $ v \in E ^ {s} ( f ^ {k} x ) $ and $ m > 0 $, then

$$ \left \| {Df ^ {m} v } \right \| \leq C _ \epsilon ( f ^ {m + k } x ) \lambda ^ {m} \left \| v \right \| ; $$

5) $ { \mathop{\rm angle} } ( E ^ {u} ( f ^ {k} x ) ,E ^ {s} ( f ^ {k} x ) ) \geq C _ \epsilon ( f ^ {k} x ) ^ {- 1 } $.

(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 $ \lambda < 1 < \mu $ by the weaker requirement that $ \lambda < \mu $ and $ \min \{ \lambda, \mu ^ {- 1 } \} < 1 $.

If $ \lambda < 1 < \mu $( respectively, $ \lambda < \mu $ and $ \min \{ \lambda, \mu ^ {- 1 } \} < 1 $) and the conditions 1)–5) hold for $ \epsilon = 0 $( i.e., if one can choose $ C _ \epsilon = \textrm{ const } $), 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 $ \mu ^ {m} $ and $ \lambda ^ {m} $ by at most slowly increasing terms along the trajectory, as in 1) (in the sense that the exponential rate $ \epsilon $ in 1) is small in comparison to the number $ { \mathop{\rm log} } \mu, - { \mathop{\rm log} } \lambda $); 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 $ k \in \mathbf Z $ replaced by $ k \in \mathbf R $, and the splitting of the tangent spaces replaced by $ T _ {x} M = E ^ {u} ( x ) \oplus E ^ {s} ( x ) \oplus X ( x ) $, where $ X ( x ) $ is the one-dimensional subspace generated by the flow direction.

Stable and unstable manifolds.

Let $ \{ {f ^ {n} x } : {n \in \mathbf Z } \} $ be a non-uniformly partially hyperbolic trajectory of a $ C ^ {1 + \alpha } $- diffeomorphism ( $ \alpha > 0 $). Assume that $ \lambda < 1 $. Then there is a local stable manifold $ V ^ {s} ( x ) $ such that $ x \in V ^ {s} ( x ) $, $ T _ {x} V ^ {s} ( x ) = E ^ {s} ( x ) $, and for every $ y \in V ^ {s} ( x ) $, $ k \in \mathbf Z $, and $ m > 0 $,

$$ d ( f ^ {m + k } x,f ^ {m + k } y ) \leq KC _ \epsilon ( f ^ {k} x ) ^ {2} \lambda ^ {m} e ^ {\epsilon m } d ( f ^ {k} x,f ^ {k} y ) , $$

where $ d $ is the distance induced by the Riemannian metric and $ K $ is a positive constant. The size $ r ( x ) $ of $ V ^ {s} ( x ) $ can be chosen in such a way that $ r ( f ^ {k} x ) \geq K ^ \prime e ^ {- \epsilon | k | } r ( x ) $ for every $ k \in \mathbf Z $, where $ K ^ \prime $ is a positive constant. If $ f \in C ^ {r + \alpha } $( $ \alpha > 0 $), then $ V ^ {s} ( x ) $ is of class $ C ^ {r} $.

The global stable manifold of $ f $ at $ x $ is defined by $ W ^ {s} ( x ) = \cup _ {k \in \mathbf Z } f ^ {- k } ( V ^ {s} ( f ^ {k} x ) ) $; it is an immersed manifold with the same smoothness class as $ V ^ {s} ( x ) $. One has $ W ^ {s} ( x ) \cap W ^ {s} ( y ) = \emptyset $ if $ y \notin W ^ {s} ( x ) $, $ W ^ {s} ( x ) = W ^ {s} ( y ) $ if $ y \in W ^ {s} ( x ) $, and $ f ^ {n} W ^ {s} ( x ) = W ^ {s} ( f ^ {n} x ) $ for every $ n \in \mathbf Z $. The manifold $ W ^ {s} ( x ) $ is independent of the particular size of the local stable manifolds $ V ^ {s} ( y ) $.

Similarly, when $ \mu > 1 $ one can define a local (respectively, global) unstable manifold as a local (respectively, global) stable manifold of $ f ^ {- 1 } $.

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

Let $ f : M \rightarrow M $ be a diffeomorphism and let $ \nu $ be a (finite) Borel $ f $- invariant measure (cf. also Invariant measure). One calls $ f $ non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) with respect to the measure $ \nu $ if the set $ \Lambda \subset M $ of points whose trajectories are non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) is such that $ \nu ( \Lambda ) > 0 $. In this case $ \lambda $, $ \mu $, $ \epsilon $, and $ C _ \epsilon $ are replaced by measurable functions $ \lambda ( x ) $, $ \mu ( x ) $, $ \epsilon ( x ) $, and $ C _ \epsilon ( x ) $, respectively.

The set $ \Lambda $ is $ f $- invariant, i.e., it satisfies $ f \Lambda = \Lambda $. Therefore, one can always assume that $ \nu ( \Lambda ) = 1 $ when $ \nu ( \Lambda ) > 0 $; this means that if $ \nu ( \Lambda ) > 0 $, then the measure $ {\widehat \nu } $ on $ \Lambda $ defined by $ {\widehat \nu } ( B ) = { {\nu ( B ) } / {\nu ( \Lambda ) } } $ is $ f $- invariant and $ {\widehat \nu } ( \Lambda ) = 1 $.

For $ ( x, v ) \in M \times T _ {x} M $, one defines the forward upper Lyapunov exponent of $ ( x, v ) $( with respect to $ f $) by

$$ \tag{a1 } \chi ( x, v ) = {\lim\limits \sup } _ {m \rightarrow + \infty } { \frac{1}{m} } { \mathop{\rm log} } \left \| {Df ^ {m} v } \right \| $$

for each $ v \neq 0 $, and $ \chi ( x,0 ) = - \infty $. For every $ x \in M $, there exist a positive integer $ s ( x ) \leq { \mathop{\rm dim} } M $( the dimension of $ M $) and collections of numbers $ \chi _ {1} ( x ) < \dots < \chi _ {s ( x ) } ( x ) $ and linear subspaces $ E _ {1} ( x ) \subset \dots \subset E _ {s ( x ) } ( x ) = T _ {x} M $ such that for every $ i = 1 \dots s ( x ) $,

$$ E _ {i} ( x ) = \left \{ {v \in T _ {x} M } : {\chi ( x,v ) \leq \chi _ {i} ( x ) } \right \} , $$

and if $ v \in E _ {i} ( x ) \setminus E _ {i - 1 } ( x ) $, then $ \chi ( x,v ) = \chi _ {i} ( x ) $.

The numbers $ \chi _ {i} ( x ) $ are called the values of the forward upper Lyapunov exponent at $ x $, and the collection of linear subspaces $ E _ {i} ( x ) $ is called the forward filtration at $ x $ associated to $ f $. The number $ k _ {i} ( x ) = { \mathop{\rm dim} } E _ {i} ( x ) - { \mathop{\rm dim} } E _ {i - 1 } ( x ) $ is the forward multiplicity of the exponent $ \chi _ {i} ( x ) $. One defines the forward spectrum of $ f $ at $ x $ as the collection of pairs $ ( \chi _ {i} ( x ) ,k _ {i} ( x ) ) $ for $ i = 1 \dots s ( x ) $. Let $ \chi _ {1} ^ \prime ( x ) \leq \dots \leq \chi _ { { \mathop{\rm dim} } M } ^ \prime ( x ) $ be the values of the forward upper Lyapunov exponent at $ x $ counted with multiplicities, i.e., in such a way that the exponent $ \chi _ {i} ( x ) $ appears exactly a number $ k _ {i} ( x ) $ of times. The functions $ s ( x ) $ and $ \chi _ {i} ^ \prime ( x ) $, for $ i = 1 \dots { \mathop{\rm dim} } M $, are measurable and $ f $- invariant with respect to any $ f $- invariant measure.

One defines the backward upper Lyapunov exponent of $ ( x,v ) $( with respect to $ f $) by an expression similar to (a1), with $ m \rightarrow + \infty $ replaced by $ m \rightarrow - \infty $, and considers the corresponding backward spectrum.

A Lyapunov-regular trajectory $ \{ {f ^ {n} x } : {n \in \mathbf Z } \} $( see, for example, [a3], Sect. 2) is non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) if and only if $ \chi ( x,v ) \neq 0 $ for all $ v \in T _ {x} M $( respectively, $ \chi ( x,v ) \neq 0 $ for some $ v \in T _ {x} M $). For flows, a Lyapunov-regular trajectory is non-uniformly hyperbolic if and only if $ \chi ( x,v ) \neq 0 $ for all $ v \notin X ( x ) $.

The multiplicative ergodic theorem of V. Oseledets [a19] implies that $ \nu $- almost all points of $ M $ belong to a Lyapunov-regular trajectory. Therefore, for a given diffeomorphism, one has $ \chi ( x,v ) \neq 0 $ for all $ v \in T _ {x} M $( respectively $ \chi ( x,v ) \neq 0 $ for some $ v \in T _ {x} M $) on a set of positive $ \nu $- 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 $ \nu $) are precisely the diffeomorphisms with non-zero Lyapunov exponents (on a set of positive $ \nu $- measure).

Furthermore, for $ \nu $- almost every $ x \in \Lambda $ there exist subspaces $ H _ {j} ( x ) $, for $ j = 1 \dots s ( x ) $, such that for every $ i = 1 \dots s ( x ) $ one has $ E _ {i} ( x ) = \oplus _ {j = 1 } ^ {i} H _ {j} ( x ) $,

$$ {\lim\limits } _ {m \rightarrow \pm \infty } { \frac{1}{m} } { \mathop{\rm log} } \left \| {Df ^ {m} v } \right \| = \chi _ {i} ( x ) $$

for every $ v \in H _ {i} ( x ) \setminus \{ 0 \} $, and if $ i \neq j $, then

$$ {\lim\limits } _ {m \rightarrow \pm \infty } { \frac{1}{m} } { \mathop{\rm log} } \left | { { \mathop{\rm angle} } ( H _ {i} ( f ^ {m} x ) ,H _ {j} ( f ^ {m} x ) ) } \right | = 0. $$

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 $ f $ be a non-uniformly hyperbolic diffeomorphism and let $ C ( x ) = C _ {\epsilon ( x ) } ( x ) $. Given $ l > 0 $, one defines the measurable set $ \Lambda _ {l} $ by

$$ \left \{ {x \in \Lambda } : {C ( x ) \leq l, \lambda ( x ) \leq { \frac{l - 1 }{l} } < { \frac{l + 1 }{l} } \leq \mu ( x ) } \right \} . $$

One has $ \Lambda _ {l} \subset \Lambda _ {L} $ when $ l \leq L $, and $ \cup _ {l > 0 } \Lambda _ {l} = \Lambda ( { \mathop{\rm mod} } 0 ) $. Each set $ \Lambda _ {l} $ is closed but need not be $ f $- invariant; for every $ m \in \mathbf Z $ and $ l > 0 $ there exists an $ L = L ( m,l ) $ such that $ f ^ {m} \Lambda _ {l} \subset \Lambda _ {L} $. The distribution $ E ^ {s} ( x ) $ is, in general, only measurable on $ \Lambda $ but it is continuous on $ \Lambda _ {l} $. The local stable manifolds $ V ^ {s} ( x ) $ depend continuously on $ x \in \Lambda _ {l} $ and their sizes are uniformly bounded below on $ \Lambda _ {l} $. Each set $ \Lambda _ {l} $ is called a Pesin set.

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

Lyapunov metrics and regular neighbourhoods.

Let $ \langle {\cdot, \cdot } \rangle $ be the Riemannian metric on $ TM $. For each fixed $ \epsilon > 0 $ and every $ x \in \Lambda $, one defines a Lyapunov metric on $ H _ {i} ( x ) $ by

$$ \left \langle {u,v } \right \rangle _ {x} ^ \prime = \sum _ {m \in \mathbf Z } \left \langle {Df ^ {m} u,Df ^ {m} v } \right \rangle _ {x} e ^ {- 2m \chi _ {i} ( x ) - 2 \epsilon \left | m \right | } , $$

for each $ u, v \in H _ {i} ( x ) $. One extends this metric to $ T _ {x} M $ by declaring orthogonal the subspaces $ H _ {i} ( x ) $ for $ i = 1 \dots s ( x ) $. The metric $ \langle {\cdot, \cdot } \rangle ^ \prime $ is continuous on $ \Lambda _ {l} $. The sequence of weights $ \{ e ^ {- 2m \chi _ {i} ( x ) - 2 \epsilon | m | } \} _ {m \in \mathbf Z } $ is called a Pesin tempering kernel. Any linear operator $ L _ \epsilon ( x ) $ on $ T _ {x} M $ such that

$$ \left \langle {u,v } \right \rangle _ {x} ^ \prime = \left \langle {L _ \epsilon ( x ) u,L _ \epsilon ( x ) v } \right \rangle _ {x} $$

is called a Lyapunov change of coordinates.

There exist a measurable function $ q : \Lambda \rightarrow {( 0,1 ] } $ satisfying $ e ^ {- \epsilon } \leq { {q ( fx ) } / {q ( x ) } } \leq e ^ \epsilon $, and for each $ x \in \Lambda $ a collection of imbeddings $ {\Psi _ {x} } : {B ( 0,q ( x ) ) } \rightarrow M $, defined on the ball $ B ( 0,q ( x ) ) \subset T _ {x} M $ by $ \Psi _ {x} = { \mathop{\rm exp} } _ {x} \circ L _ \epsilon ( x ) ^ {- 1 } $, such that if $ f _ {x} = \Psi _ {fx } ^ {- 1 } \circ f \circ \Psi _ {x} $, then:

1) the derivative $ D _ {0} f _ {x} $ of $ f _ {x} $ at the point $ 0 $ has the Lyapunov block form

$$ D _ {0} f _ {x} = \left ( \begin{array}{ccc} A _ {1} ( x ) &{} &{} \\ {} &\dvd &{} \\ {} &{} &A _ {s ( x ) } ( x ) \\ \end{array} \right ) , $$

where each $ A _ {i} ( x ) $ is an invertible linear operator between the $ k _ {i} ( x ) $- dimensional spaces $ L _ \epsilon ( x ) H _ {i} ( x ) $ and $ L _ \epsilon ( fx ) H _ {i} ( fx ) $, for $ i = 1 \dots s ( x ) $;

2) for each $ i = 1 \dots s ( x ) $,

$$ e ^ {\chi _ {i} ( x ) - \epsilon } \leq \left \| {A _ {i} ( x ) ^ {- 1 } } \right \| ^ {- 1 } \leq \left \| {A _ {i} ( x ) } \right \| \leq e ^ {\chi _ {i} ( x ) + \epsilon } ; $$

3) the $ C ^ {1} $- distance between $ f _ {x} $ and $ d _ {0} f _ {x} $ on the ball $ B ( 0,q ( x ) ) $ is at most $ \epsilon $;

4) there exist a constant $ K $ and a measurable function $ A : \Lambda \rightarrow \mathbf R $ satisfying $ e ^ {- \epsilon } \leq { {A ( fx ) } / {A ( x ) } } \leq e ^ \epsilon $ such that for every $ y,z \in B ( 0,q ( x ) ) $,

$$ Kd ( \Psi _ {x} y, \Psi _ {x} z ) \leq \left \| {y - z } \right \| \leq A ( x ) d ( \Psi _ {x} y, \Psi _ {x} z ) . $$

The function $ A ( x ) $ is bounded on each $ \Lambda _ {l} $. The set $ \Psi _ {x} ( B ( 0,q ( x ) ) ) \subset M $ is called a regular neighbourhood of the point $ x $.

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 $ \nu $ be an absolutely continuous $ f $- invariant measure, i.e., an $ f $- invariant measure that is absolutely continuous with respect to Lebesgue measure (cf. Absolute continuity). For each $ x \in \Lambda $ and $ l > 0 $ there exists a neighbourhood $ U ( x ) $ of $ x $ with size depending only on $ l $ and with the following properties (see [a21]). Choose $ y \in \Lambda _ {l} \cap U ( x ) $. Given two smooth manifolds $ W _ {1} , W _ {2} \subset U ( x ) $ transversal to the local stable manifolds in $ U ( x ) $, one defines

$$ A _ {i} = \left \{ {w \in W _ {i} \cap V ^ {s} ( z ) } : {z \in \Lambda _ {l} \cap U ( x ) } \right \} $$

for $ i = 1,2 $. Let $ p : {A _ {1} } \rightarrow {A _ {2} } $ be the correspondence that takes $ w \in W _ {1} $ to the point $ p ( w ) \in W _ {2} $ such that $ w,p ( w ) \in V ^ {s} ( z ) $ for some $ z $. If $ \nu _ {i} $ is the measure induced on $ W _ {i} $ by the Riemannian metric, for $ i = 1,2 $, then $ p ^ {*} \nu _ {1} $ is absolutely continuous with respect to $ \nu _ {2} $( if $ l $ is sufficiently large, then $ \nu _ {i} ( A _ {i} ) > 0 $ for $ i = 1,2 $).

This result has the following consequences (see [a21]). For each measurable set $ B \subset W _ {1} \cap \Lambda _ {l} $, let $ {\widehat{B} } $ be the union of all the sets $ V ^ {s} ( z ) \cap U ( x ) $ such that $ z \in \Lambda _ {l} $ and $ V ^ {s} ( z ) \cap B \neq \emptyset $. The partition of $ {\widehat{B} } $ into the submanifolds $ V ^ {s} ( z ) $ is a measurable partition (also called measurable decomposition), and the corresponding conditional measure of $ \nu $ on $ V ^ {s} ( z ) $ is absolutely continuous with respect to the measure $ \nu _ {z} $ induced on $ V ^ {s} ( z ) $ by the Riemannian metric, for each $ z \in \Lambda _ {l} $ such that $ V ^ {s} ( z ) \cap B \neq \emptyset $. In addition, $ \nu _ {z} ( V ^ {s} ( z ) ) > 0 $ for $ \nu $- almost all $ z \in {\widehat{B} } \cap \Lambda _ {l} $, and the measure $ {\widehat \nu } $ on $ W _ {1} $ defined for each measurable set $ B $ by $ {\widehat \nu } ( B ) = \nu ( {\widehat{B} } ) $, is absolutely continuous with respect to $ \nu _ {1} $.

Smooth ergodic theory.

Let $ f : M \rightarrow M $ be a non-uniformly hyperbolic $ C ^ {1 + \alpha } $- diffeomorphism ( $ \alpha > 0 $) with respect to a Sinai–Ruelle–Bowen measure $ \nu $, i.e., an $ f $- invariant measure $ \nu $ that has a non-zero Lyapunov exponent $ \nu $- almost everywhere and has absolutely continuous conditional measures on stable (or unstable) manifolds with respect to Lebesgue measure (in particular, this holds if $ \nu $ 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 $ f $- invariant sets $ \Lambda _ {0} , \Lambda _ {1} , \dots $( the ergodic components) such that [a21], [a11]:

1) $ \cup _ {i \geq 0 } \Lambda _ {i} = \Lambda $, $ \nu ( \Lambda _ {0} ) = 0 $, and $ \nu ( \Lambda _ {i} ) > 0 $ and $ f \mid _ {\Lambda _ {i} } $ is ergodic (see Ergodicity) with respect to $ \nu \mid _ {\Lambda _ {i} } $ for every $ i > 0 $;

2) each set $ \Lambda _ {i} $ is a disjoint union of sets $ \Lambda _ {i1 } \dots \Lambda _ {in _ {i} } $ such that $ f ( \Lambda _ {ij } ) = \Lambda _ {i,j + 1 } $ for each $ j < n _ {i} $, and $ f ( \Lambda _ {in _ {i} } ) = \Lambda _ {i1 } $;

3) for every $ i $ and $ j $, there is a metric isomorphism between $ f ^ {n _ {i} } \mid _ {\Lambda _ {ij } } $ and a Bernoulli automorphism (in particular, the mapping $ f ^ {n _ {i} } \mid _ {\Lambda _ {ij } } $ is a $ K $- system).

If $ \nu $ is an absolutely continuous $ f $- invariant measure and the foliation $ W ^ {s} $( or $ W ^ {u} $) of $ \Lambda $ is $ C ^ {1} $- continuous (i.e., for each $ x \in \Lambda $ there is a neighbourhood of $ x $ in $ W ^ {s} ( x ) $ that is the image of an injective $ C ^ {1} $- mapping $ \varphi _ {x} $, defined on the ball with centre at $ 0 $ and of radius $ 1 $, and the mapping $ x \mapsto \varphi _ {x} $ from $ \Lambda $ into the family of $ C ^ {1} $- mappings is continuous), then any ergodic component of positive $ \nu $- measure is an open set (mod $ 0 $); if, in addition, $ f \mid _ \Lambda $ is topologically transitive (cf. Topological transitivity; Chaos), then $ f \mid _ \Lambda $ is ergodic [a21].

If $ f \mid _ \Lambda $ is ergodic, then for Lebesgue-almost-every point $ x \in M $ and every continuous function $ g $, one has

$$ { \frac{1}{n} } \sum _ {k = 0 } ^ { {n } - 1 } g ( f ^ {k} x ) \rightarrow \int\limits _ { M } g {d \nu } \textrm{ as } n \rightarrow + \infty. $$

There is a measurable partition $ \eta $ of $ M $ with the following properties:

1) for $ \nu $- almost every $ x \in M $, the element $ \eta ( x ) \in \eta $ containing $ x $ is an open subset (mod $ 0 $) of $ W ^ {s} ( x ) $;

2) $ f \eta $ is a refinement of $ \eta $, and $ \lor _ {k = 0 } ^ \infty f ^ {k} \eta $ is the partition of $ M $ into points;

3) $ \wedge _ {k = 0 } ^ \infty f ^ {- k } \eta $ coincides with the measurable hull of $ W ^ {s} $, as well as with the maximal partition with zero entropy (the $ \pi $- partition for $ f $; see Entropy of a measurable decomposition);

4) $ h _ \nu ( f ) = h _ \nu ( f, \eta ) $( cf. Entropy theory of a dynamical system).

Pesin entropy formula.

For a $ C ^ {1 + \alpha } $- diffeomorphism ( $ \alpha > 0 $) $ f : M \rightarrow M $ of a compact manifold and an absolutely continuous $ f $- invariant probability measure $ \nu $, the metric entropy $ h _ \nu ( f ) $ of $ f $ with respect to $ \nu $ is given by the Pesin entropy formula [a21]

$$ \tag{a2 } h _ \nu ( f ) = \int\limits _ { M } {\sum _ {i = 1 } ^ { {s } ( x ) } \chi _ {i} ^ {+} ( x ) k _ {i} ( x ) } {d \nu ( x ) } , $$

where $ \chi _ {i} ^ {+} ( x ) = \max \{ \chi _ {i} ( x ) ,0 \} $ and $ ( \chi _ {i} ( x ) ,k _ {i} ( x ) ) $ form the forward spectrum of $ f $ at $ x $.

For a $ C ^ {1} $- diffeomorphism $ f : M \rightarrow M $ of a compact manifold and an $ f $- invariant probability measure $ \nu $, the Ruelle inequality holds [a25]:

$$ \tag{a3 } h _ \nu ( f ) \leq \int\limits _ { M } {\sum _ {i = 1 } ^ { {s } ( x ) } \chi _ {i} ^ {+} ( x ) k _ {i} ( x ) } {d \nu ( x ) } . $$

An important consequence of (a3) is that any $ C ^ {1} $- diffeomorphism with positive topological entropy has an $ f $- invariant measure with at least one positive and one negative Lyapunov exponent; in particular, for surface diffeomorphisms there is an $ f $- 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 $ C ^ {2} $- diffeomorphisms, (a2) holds if and only if $ \nu $ has absolutely continuous conditional measures on unstable manifolds [a13], [a12].

The formula (a2) has been extended to mappings with singularities [a12]. For $ C ^ {2} $- 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 $ { \mathop{\rm dim} } E _ {i} ( x ) $ and the Hausdorff dimension of $ \nu $" in the direction of Eix" for each $ i $.

Hyperbolic measures.

Let $ f $ be a $ C ^ {1 + \alpha } $- diffeomorphism ( $ \alpha > 0 $) and let $ \nu $ be an $ f $- invariant measure. One says that $ \nu $ is hyperbolic (with respect to $ f $) if $ \chi _ {i} ( x ) \neq 0 $ for $ \nu $- almost every $ x \in M $ and all $ i = 1 \dots s ( x ) $. The measure $ \nu $ is hyperbolic (with respect to $ f $) if and only if $ f $ is non-uniformly hyperbolic with respect to $ \nu $( and the set $ \Lambda $ has full $ \nu $- measure). The fundamental work of A. Katok has revealed a rich and complicated orbit structure for diffeomorphisms possessing a hyperbolic measure.

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

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

2) for every $ \epsilon > 0 $ there exists a closed $ f $- invariant hyperbolic set $ \Gamma $ such that the restriction of $ f $ to $ \Gamma $ is topologically conjugate to a topological Markov chain with topological entropy $ h ( f \mid _ \Gamma ) \geq h _ \nu ( f ) - \epsilon $, i.e., the entropy of a hyperbolic measure can be approximated by the topological entropies of invariant hyperbolic sets.

If $ f $ possesses a hyperbolic measure, then $ f $ satisfies a closing lemma: given $ \epsilon > 0 $, there exists a $ \delta = \delta ( l, \epsilon ) > 0 $ such that for each $ x \in \Lambda _ {l} $ and each integer $ m $ satisfying $ f ^ {m} x \in \Lambda _ {l} $ and $ d ( x,f ^ {m} x ) < \delta $, there exists a point $ y $ such that $ f ^ {m} y = y $, $ d ( f ^ {k} x,f ^ {k} y ) < \epsilon $ for every $ k = 0 \dots m $, and $ y $ is a hyperbolic periodic point [a7]. The diffeomorphism $ f $ also satisfies a shadowing lemma (see [a9]) and a Lifschitz-type theorem [a9]: if $ \varphi $ is a Hölder-continuous function (cf. Hölder condition) such that $ \sum _ {k = 0 } ^ {m - 1 } \varphi ( f ^ {k} p ) = 0 $ for each periodic point $ p $ with $ f ^ {m} p = p $, then there is a measurable function $ h $ such that $ \varphi ( x ) = h ( fx ) - h ( x ) $ for $ \nu $- almost every $ x $.

Let $ P _ {n} ( f ) $ be the number of periodic points of $ f $ with period $ n $. If $ f $ possesses a hyperbolic measure or is a surface diffeomorphism, then

$$ {\lim\limits \sup } _ {n \rightarrow + \infty } { \frac{1}{n} } { \mathop{\rm log} } ^ {+} P _ {n} ( f ) \geq h ( f ) , $$

where $ h ( f ) $ is the topological entropy of $ f $[a7].

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

$$ {\lim\limits } _ {r \rightarrow 0 } { \frac{ { \mathop{\rm log} } \nu ( B ( x,r ) ) }{ { \mathop{\rm log} } r } } = d, $$

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

$$ {\lim\limits } _ {r \rightarrow 0 } { \frac{ { \mathop{\rm log} } \nu _ {x} ^ {s} ( B ^ {s} ( x,r ) ) }{ { \mathop{\rm log} } r } } = d ^ {s} , $$

$$ {\lim\limits } _ {r \rightarrow 0 } { \frac{ { \mathop{\rm log} } \nu _ {x} ^ {u} ( B ^ {u} ( x,r ) ) }{ { \mathop{\rm log} } r } } = d ^ {u} , $$

where $ B ^ {s} ( x,r ) $( respectively, $ B ^ {u} ( x,r ) $) is the ball in $ V ^ {s} ( x ) $( respectively, $ V ^ {u} ( x ) $) with centre at $ x $ and of radius $ r $. Moreover, $ d = d ^ {s} + d ^ {u} $[a2] and $ \nu $ 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 $ Q $ on the tangent bundle $ TM $ is called an eventually strict Lyapunov function if for $ \nu $- almost every $ x \in M $:

1) the function $ Q _ {x} ( v ) = Q ( x,v ) $ 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 $ \{ 0 \} \cup Q _ {x} ^ {- 1 } ( 0, + \infty ) $ and $ \{ 0 \} \cup Q _ {x} ^ {- 1 } ( - \infty,0 ) $ are constants $ r ^ {+} ( Q ) $ and $ r ^ {-} ( Q ) $, and $ r ^ {+} ( Q ) + r ^ {-} ( Q ) $ is the dimension of $ M $;

3) $ Q _ {fx } ( Dfv ) \geq Q _ {x} ( v ) $ for all $ v \in T _ {x} M $;

4) there exists a positive integer $ m = m ( x ) $ such that for all $ v \in T _ {x} M \setminus \{ 0 \} $,

$$ Q _ {f ^ {m} x } ( Df ^ {m} v ) > Q _ {x} ( v ) , $$

$$ Q _ {f ^ {- m } x } ( Df ^ {- m } v ) < Q _ {x} ( v ) . $$

If $ f $ possesses an eventually strict Lyapunov function, then there exist exactly $ r ^ {+} ( Q ) $ positive Lyapunov exponents and $ r ^ {-} ( Q ) $ 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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