A proof of Pesin’s stable manifold theorem. (English) Zbl 0532.58012

Fathi, A.; Herman, M. R.; Yoccoz, Jean-Christophe

Geometric dynamics, Proc. int. Symp., Rio de Janeiro/Brasil 1981, Lect. Notes Math. 1007, 177-215 (1983).

This paper is a set of lecture notes given in the ”Séminaire de théorie ergodique” of Paris VI in 1978-79 which was devoted that year to Pesin’s work. The part of Pesin’s work for which we provide a proof may be formulated in the following way: Given a \(C^ 2\) diffeomorphism f of the manifold M, and an f-invariant probability measure \(\mu\), for \(\mu\)-almost every x the stable set of x \(W^ s_ x=\{y\in M| \lim \sup \frac{1}{n} \log d(f^ n(x),f^ n(y))<0\}\) is in fact an immersed Euclidean space.

MSC:

37A99	Ergodic theory
37D99	Dynamical systems with hyperbolic behavior
28D10	One-parameter continuous families of measure-preserving transformations

Keywords:

f-invariant probability measure; Oseledec’s ergodic multiplicative theorem; tangent cocycle; stable manifold