View source for Pesin theory

An important branch of the theory of dynamical systems (cf. [[Dynamical system|Dynamical system]]) and of smooth [[Ergodic theory|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 [[#References|[a20]]], [[#References|[a21]]], [[#References|[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|Hyperbolic set]]). The  "strongest possible"  kind of hyperbolicity occurs in the important class of Anosov systems (also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101201.png" />-systems, cf. [[Y-system|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101202.png" />-system]]) [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101203.png" /> be a [[Diffeomorphism|diffeomorphism]] of a compact [[Manifold|manifold]]. It induces the discrete dynamical system (or [[Cascade|cascade]]) composed of the powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101204.png" />. Fix a [[Riemannian metric|Riemannian metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101205.png" />. The trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101206.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101207.png" /> is called non-uniformly hyperbolic if there are positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101208.png" /> and splittings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p1101209.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012010.png" />, and if for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012011.png" /> there is a positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012012.png" /> on the trajectory such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012013.png" />:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012014.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012016.png" />;

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012018.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012019.png" /></td> </tr></table>

4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012021.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012022.png" /></td> </tr></table>

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012023.png" />.

(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012024.png" /> by the weaker requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012026.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012027.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012029.png" />) and the conditions 1)–5) hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012030.png" /> (i.e., if one can choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012031.png" />), 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012033.png" /> by at most slowly increasing terms along the trajectory, as in 1) (in the sense that the exponential rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012034.png" /> in 1) is small in comparison to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012035.png" />); 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)|flow (continuous-time dynamical system)]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012036.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012037.png" />, and the splitting of the tangent spaces replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012039.png" /> is the one-dimensional subspace generated by the flow direction.

==Stable and unstable manifolds.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012040.png" /> be a non-uniformly partially hyperbolic trajectory of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012041.png" />-diffeomorphism (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012042.png" />). Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012043.png" />. Then there is a local stable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012046.png" />, and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012049.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012050.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012051.png" /> is the distance induced by the Riemannian metric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012052.png" /> is a positive constant. The size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012054.png" /> can be chosen in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012055.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012057.png" /> is a positive constant. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012058.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012059.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012060.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012061.png" />.

The global stable manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012062.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012063.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012064.png" />; it is an immersed manifold with the same smoothness class as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012065.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012066.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012068.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012070.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012071.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012072.png" /> is independent of the particular size of the local stable manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012073.png" />.

Similarly, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012074.png" /> one can define a local (respectively, global) unstable manifold as a local (respectively, global) stable manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012075.png" />.

==Non-uniformly hyperbolic dynamical systems and dynamical systems with non-zero Lyapunov exponents.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012076.png" /> be a [[Diffeomorphism|diffeomorphism]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012077.png" /> be a (finite) Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012078.png" />-invariant measure (cf. also [[Invariant measure|Invariant measure]]). One calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012079.png" /> non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012080.png" /> if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012081.png" /> of points whose trajectories are non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012082.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012085.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012086.png" /> are replaced by measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012089.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012090.png" />, respectively.

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012091.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012092.png" />-invariant, i.e., it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012093.png" />. Therefore, one can always assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012094.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012095.png" />; this means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012096.png" />, then the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012097.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012098.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p11012099.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120100.png" />-invariant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120101.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120102.png" />, one defines the forward upper Lyapunov exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120103.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120104.png" />) by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120107.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120108.png" />, there exist a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120109.png" /> (the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120110.png" />) and collections of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120111.png" /> and linear subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120112.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120113.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120114.png" /></td> </tr></table>

and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120115.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120116.png" />.

The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120117.png" /> are called the values of the forward upper Lyapunov exponent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120118.png" />, and the collection of linear subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120119.png" /> is called the forward filtration at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120120.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120121.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120122.png" /> is the forward multiplicity of the exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120123.png" />. One defines the forward spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120124.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120125.png" /> as the collection of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120126.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120127.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120128.png" /> be the values of the forward upper Lyapunov exponent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120129.png" /> counted with multiplicities, i.e., in such a way that the exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120130.png" /> appears exactly a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120131.png" /> of times. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120133.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120134.png" />, are measurable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120135.png" />-invariant with respect to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120136.png" />-invariant measure.

One defines the backward upper Lyapunov exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120137.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120138.png" />) by an expression similar to (a1), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120139.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120140.png" />, and considers the corresponding backward spectrum.

A Lyapunov-regular trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120141.png" /> (see, for example, [[#References|[a3]]], Sect. 2) is non-uniformly hyperbolic (respectively, non-uniformly partially hyperbolic) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120142.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120143.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120144.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120145.png" />). For flows, a Lyapunov-regular trajectory is non-uniformly hyperbolic if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120146.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120147.png" />.

The [[Multiplicative ergodic theorem|multiplicative ergodic theorem]] of V. Oseledets [[#References|[a19]]] implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120148.png" />-almost all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120149.png" /> belong to a Lyapunov-regular trajectory. Therefore, for a given diffeomorphism, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120150.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120151.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120152.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120153.png" />) on a set of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120154.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120155.png" />) are precisely the diffeomorphisms with non-zero Lyapunov exponents (on a set of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120156.png" />-measure).

Furthermore, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120157.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120158.png" /> there exist subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120159.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120160.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120161.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120162.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120163.png" /></td> </tr></table>

for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120164.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120165.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120166.png" /></td> </tr></table>

==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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120167.png" /> be a non-uniformly hyperbolic diffeomorphism and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120168.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120169.png" />, one defines the measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120170.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120171.png" /></td> </tr></table>

One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120172.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120173.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120174.png" />. Each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120175.png" /> is closed but need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120176.png" />-invariant; for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120178.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120179.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120180.png" />. The distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120181.png" /> is, in general, only measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120182.png" /> but it is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120183.png" />. The local stable manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120184.png" /> depend continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120185.png" /> and their sizes are uniformly bounded below on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120186.png" />. Each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120187.png" /> is called a Pesin set.

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

==Lyapunov metrics and regular neighbourhoods.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120188.png" /> be the [[Riemannian metric|Riemannian metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120189.png" />. For each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120190.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120191.png" />, one defines a Lyapunov metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120192.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120193.png" /></td> </tr></table>

for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120194.png" />. One extends this metric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120195.png" /> by declaring orthogonal the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120196.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120197.png" />. The metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120198.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120199.png" />. The sequence of weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120200.png" /> is called a Pesin tempering kernel. Any linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120201.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120202.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120203.png" /></td> </tr></table>

is called a Lyapunov change of coordinates.

There exist a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120204.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120205.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120206.png" /> a collection of imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120207.png" />, defined on the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120208.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120209.png" />, such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120210.png" />, then:

1) the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120211.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120212.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120213.png" /> has the Lyapunov block form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120214.png" /></td> </tr></table>

where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120215.png" /> is an invertible linear operator between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120216.png" />-dimensional spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120217.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120218.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120219.png" />;

2) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120220.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120221.png" /></td> </tr></table>

3) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120222.png" />-distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120223.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120224.png" /> on the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120225.png" /> is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120226.png" />;

4) there exist a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120227.png" /> and a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120228.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120229.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120230.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120231.png" /></td> </tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120232.png" /> is bounded on each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120233.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120234.png" /> is called a regular neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120235.png" />.

==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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120236.png" /> be an absolutely continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120237.png" />-invariant measure, i.e., an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120238.png" />-invariant measure that is absolutely continuous with respect to [[Lebesgue measure|Lebesgue measure]] (cf. [[Absolute continuity|Absolute continuity]]). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120239.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120240.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120241.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120242.png" /> with size depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120243.png" /> and with the following properties (see [[#References|[a21]]]). Choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120244.png" />. Given two smooth manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120245.png" /> transversal to the local stable manifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120246.png" />, one defines

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120247.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120248.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120249.png" /> be the correspondence that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120250.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120251.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120252.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120253.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120254.png" /> is the measure induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120255.png" /> by the Riemannian metric, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120256.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120257.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120258.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120259.png" /> is sufficiently large, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120260.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120261.png" />).

This result has the following consequences (see [[#References|[a21]]]). For each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120262.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120263.png" /> be the union of all the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120264.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120265.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120266.png" />. The partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120267.png" /> into the submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120268.png" /> is a measurable partition (also called [[Measurable decomposition|measurable decomposition]]), and the corresponding conditional measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120269.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120270.png" /> is absolutely continuous with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120271.png" /> induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120272.png" /> by the Riemannian metric, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120273.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120274.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120275.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120276.png" />-almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120277.png" />, and the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120278.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120279.png" /> defined for each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120280.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120281.png" />, is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120282.png" />.

==Smooth ergodic theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120283.png" /> be a non-uniformly hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120284.png" />-diffeomorphism (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120285.png" />) with respect to a Sinai–Ruelle–Bowen measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120286.png" />, i.e., an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120287.png" />-invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120288.png" /> that has a non-zero Lyapunov exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120289.png" />-almost everywhere and has absolutely continuous conditional measures on stable (or unstable) manifolds with respect to Lebesgue measure (in particular, this holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120290.png" /> is absolutely continuous with respect to Lebesgue measure and has no zero Lyapunov exponents [[#References|[a21]]]; see also above:  "Absolute continuity" ). Then there is at most a countable number of disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120291.png" />-invariant sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120292.png" /> (the ergodic components) such that [[#References|[a21]]], [[#References|[a11]]]:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120293.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120294.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120295.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120296.png" /> is ergodic (see [[Ergodicity|Ergodicity]]) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120297.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120298.png" />;

2) each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120299.png" /> is a disjoint union of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120300.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120301.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120302.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120303.png" />;

3) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120305.png" />, there is a [[Metric isomorphism|metric isomorphism]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120306.png" /> and a [[Bernoulli automorphism|Bernoulli automorphism]] (in particular, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120307.png" /> is a [[K-system(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120308.png" />-system]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120309.png" /> is an absolutely continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120310.png" />-invariant measure and the [[Foliation|foliation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120311.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120312.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120313.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120316.png" />-continuous (i.e., for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120317.png" /> there is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120318.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120319.png" /> that is the image of an injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120320.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120321.png" />, defined on the ball with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120322.png" /> and of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120323.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120324.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120325.png" /> into the family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120326.png" />-mappings is continuous), then any ergodic component of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120327.png" />-measure is an open set (mod <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120328.png" />); if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120329.png" /> is topologically transitive (cf. [[Topological transitivity|Topological transitivity]]; [[Chaos|Chaos]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120330.png" /> is ergodic [[#References|[a21]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120331.png" /> is ergodic, then for Lebesgue-almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120332.png" /> and every continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120333.png" />, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120334.png" /></td> </tr></table>

There is a measurable partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120335.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120336.png" /> with the following properties:

1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120337.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120338.png" />, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120339.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120340.png" /> is an open subset (mod <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120341.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120342.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120343.png" /> is a refinement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120344.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120345.png" /> is the partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120346.png" /> into points;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120347.png" /> coincides with the measurable hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120348.png" />, as well as with the maximal partition with zero entropy (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120349.png" />-partition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120350.png" />; see [[Entropy of a measurable decomposition|Entropy of a measurable decomposition]]);

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120351.png" /> (cf. [[Entropy theory of a dynamical system|Entropy theory of a dynamical system]]).

==Pesin entropy formula.==
For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120352.png" />-diffeomorphism (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120353.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120354.png" /> of a compact manifold and an absolutely continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120355.png" />-invariant probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120356.png" />, the [[Metric entropy|metric entropy]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120357.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120358.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120359.png" /> is given by the Pesin entropy formula [[#References|[a21]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120360.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120361.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120362.png" /> form the forward spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120363.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120364.png" />.

For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120365.png" />-diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120366.png" /> of a compact manifold and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120367.png" />-invariant probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120368.png" />, the Ruelle inequality holds [[#References|[a25]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120369.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

An important consequence of (a3) is that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120370.png" />-diffeomorphism with positive topological entropy has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120371.png" />-invariant measure with at least one positive and one negative Lyapunov exponent; in particular, for surface diffeomorphisms there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120372.png" />-invariant measure with every exponent non-zero. For arbitrary invariant measures the inequality (a3) may be strict [[#References|[a7]]].

The formula (a2) was first established by Pesin in [[#References|[a21]]]. A proof which does not use the theory of invariant manifolds and absolute continuity was given by R. Mañé [[#References|[a17]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120373.png" />-diffeomorphisms, (a2) holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120374.png" /> has absolutely continuous conditional measures on unstable manifolds [[#References|[a13]]], [[#References|[a12]]].

The formula (a2) has been extended to mappings with singularities [[#References|[a12]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120375.png" />-diffeomorphisms and arbitrary invariant measures, results of F. Ledrappier and L.-S. Young [[#References|[a14]]] show that the possible defect between the left- and right-hand sides of (a3) is due to the defects between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120376.png" /> and the [[Hausdorff dimension|Hausdorff dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120377.png" />  "in the direction of Eix"  for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120378.png" />.

==Hyperbolic measures.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120379.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120380.png" />-diffeomorphism (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120381.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120382.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120383.png" />-invariant measure. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120384.png" /> is hyperbolic (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120385.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120386.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120387.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120388.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120389.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120390.png" /> is hyperbolic (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120391.png" />) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120392.png" /> is non-uniformly hyperbolic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120393.png" /> (and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120394.png" /> has full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120395.png" />-measure). The fundamental work of A. Katok has revealed a rich and complicated orbit structure for diffeomorphisms possessing a hyperbolic measure.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120396.png" /> be a hyperbolic measure. The support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120397.png" /> is contained in the closure of the set of periodic points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120398.png" /> is ergodic and not concentrated on a periodic orbit, then [[#References|[a7]]], [[#References|[a9]]]:

1) the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120399.png" /> is contained in the closure of the set of hyperbolic periodic points possessing a transversal homoclinic point;

2) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120400.png" /> there exists a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120401.png" />-invariant hyperbolic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120402.png" /> such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120403.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120404.png" /> is topologically conjugate to a topological [[Markov chain|Markov chain]] with topological entropy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120405.png" />, i.e., the entropy of a hyperbolic measure can be approximated by the topological entropies of invariant hyperbolic sets.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120406.png" /> possesses a hyperbolic measure, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120407.png" /> satisfies a closing lemma: given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120408.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120409.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120410.png" /> and each integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120411.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120412.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120413.png" />, there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120414.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120415.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120416.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120417.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120418.png" /> is a hyperbolic periodic point [[#References|[a7]]]. The diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120419.png" /> also satisfies a shadowing lemma (see [[#References|[a9]]]) and a Lifschitz-type theorem [[#References|[a9]]]: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120420.png" /> is a Hölder-continuous function (cf. [[Hölder condition|Hölder condition]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120421.png" /> for each periodic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120422.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120423.png" />, then there is a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120424.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120425.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120426.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120427.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120428.png" /> be the number of periodic points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120429.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120430.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120431.png" /> possesses a hyperbolic measure or is a surface diffeomorphism, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120432.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120433.png" /> is the topological entropy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120434.png" /> [[#References|[a7]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120435.png" /> be a hyperbolic ergodic measure. L.M. Barreira, Pesin and J. Schmeling [[#References|[a2]]] have shown that there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120436.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120437.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120438.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120439.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120440.png" /> is the ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120441.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120442.png" /> and of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120443.png" /> (this claim was known as the Eckmann–Ruelle conjecture); this implies that the [[Hausdorff dimension|Hausdorff dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120444.png" /> and the lower and upper box dimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120445.png" /> coincide and are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120446.png" /> (see [[#References|[a2]]]). Ledrappier and Young [[#References|[a14]]] have shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120447.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120448.png" />) are the conditional measures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120449.png" /> with respect to the stable (respectively, unstable) manifolds, then there are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120450.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120451.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120452.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120453.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120454.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120455.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120456.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120457.png" />) is the ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120458.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120459.png" />) with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120460.png" /> and of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120461.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120462.png" /> [[#References|[a2]]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120463.png" /> has an  "almost product structure"  (see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120464.png" /> on the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120465.png" /> is called an eventually strict Lyapunov function if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120466.png" />-almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120467.png" />:

1) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120468.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120469.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120470.png" /> are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120471.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120472.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120473.png" /> is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120474.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120475.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120476.png" />;

4) there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120477.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120478.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120479.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120480.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120481.png" /> possesses an eventually strict Lyapunov function, then there exist exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120482.png" /> positive Lyapunov exponents and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120483.png" /> negative ones [[#References|[a8]]] (see also [[#References|[a28]]]).

Another method to estimate the Lyapunov exponents was presented in [[#References|[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 [[#References|[a10]]], including billiard systems and other physical models; infinite-dimensional versions of results on stable and unstable manifolds in Hilbert spaces [[#References|[a27]]] and Banach spaces [[#References|[a18]]], given certain compactness assumptions; some results have been extended to random mappings [[#References|[a15]]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Anosov,   "Geodesic flows on closed Riemann manifolds with negative curvature"  ''Proc. Steklov Inst. Math.'' , '''90'''  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Barreira,   Ya. Pesin,   J. Schmeling,   "On the pointwise dimension of hyperbolic measures: A proof of the Eckmann–Ruelle conjecture"  ''Electronic Research Announc. Amer. Math. Soc.'' , '''2'''  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Cornfeld,   Ya. Sinai,   "Basic notions of ergodic theory and examples of dynamical systems"  Ya. Sinai (ed.) , ''Dynamical Systems II'' , ''Encycl. Math. Sci.'' , '''2''' , Springer  (1989)  pp. 2–27  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Fathi,   M. Herman,   J. Yoccoz,   "A proof of Pesin's stable manifold theorem"  J. Palis (ed.) , ''Geometric Dynamics'' , ''Lecture Notes in Mathematics'' , '''1007''' , Springer  (1983)  pp. 177–215</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Goldsheid,   G. Margulis,   "Lyapunov exponents of a product of random matrices"  ''Russian Math. Surveys'' , '''44'''  (1989)  pp. 11–71  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Herman,   "Une méthode pour minorer les exposants de Lyapunov et quelques examples montrant le caractére local d'un théorèm d'Arnold et de Moser sur le tore de dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110120/p110120484.png" />"  ''Comment. Math. Helv.'' , '''58'''  (1983)  pp. 453–502</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Katok,   "Lyapunov exponents, entropy and periodic orbits for diffeomorphisms"  ''IHES Publ. Math.'' , '''51'''  (1980)  pp. 137–173</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Katok,   K. Burns,   "Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems"  ''Ergodic Th. Dynamical Systems'' , '''14'''  (1994)  pp. 757–785</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Katok,   L. Mendoza,   "Dynamical systems with nonuniformly hyperbolic behavior"  A. Katok (ed.)  B. Hasselblatt (ed.) , ''Introduction to the Modern Theory of Dynamical Systems'' , Cambridge Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Katok,   J.-M. Strelcyn,   "Invariant manifolds, entropy and billiards; smooth maps with singularities" , ''Lecture Notes in Mathematics'' , '''1222''' , Springer  (1986)  (with the collaboration of F. Ledrappier and F. Przytycki)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Ledrappier,   "Propriétés ergodiques des mesures de Sinaï"  ''IHES Publ. Math.'' , '''59'''  (1984)  pp. 163–188</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  F. Ledrappier,   J.-M. Strelcyn,   "A proof of the estimate from below in Pesin's entropy formula"  ''Ergodic Th. Dynamical Systems'' , '''2'''  (1982)  pp. 203–219</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  F. Ledrappier,   L.-S. Young,   "The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin's entropy formula"  ''Ann. of Math. (2)'' , '''122'''  (1985)  pp. 509–539</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  F. Ledrappier,   L.-S. Young,   "The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension"  ''Ann. of Math. (2)'' , '''122'''  (1985)  pp. 540–574</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  P.-D. Liu,   M. 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