Pesin Theory

Pesin Theory-An important branch of dynamical systems and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Yakov B. Pesin in the mid-seventies 20, 21, 22]. Sometimes it is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behavior, or simply theory of non-uniformly hyperbolic dynamical systems. A. Introduction. One of the paradigms of dynam-ical systems is that the local instability of trajecto-ries innuences the global behavior of the system, and paves the way to the existence of stochastic behavior. Mathematically, the instability of trajectories corresponds to some degree of hyperbolicity (cf. Hyper-bolic set). The \strongest possible" hyperbolicity occurs in the important class of Anosov systems (also called Y-systems) 1]. These are only known to occur in some manifolds. Moreover there are several results of topological nature showing that some manifolds cannot carry Anosov systems. Pesin Theory deals with a \weaker" 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 that expresses the metric entropy of the dy-namical system through its Lyapunov exponents. B. The Concept of Non-Uniform Hyperbolicity. Let f : M ! M be a diieomorphism of a compact mani-fold. It induces the discrete dynamical system (or cascade) composed of the powers ff n : n 2 Zg. We x a Riemannian metric on M. The trajectory ff n x : n 2 Zg of a point x 2 M is called non-uniformly hyperbolic if there are positive numbers < 1 < and splittings