SRB-like Measures for C⁰ Dynamics

Eleonora Catsigeras ; Heber Enrich

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011), p. 151-164 / Harvested from The Polish Digital Mathematics Library

Résumé

For any continuous map f: M → M on a compact manifold M, we define SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f always has observable measures, even if SRB measures do not exist. We prove that the definition of observability is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost all initial states. Precisely, the never empty set of all observable measures is the minimal weak* compact set of Borel probabilities in M that contains the limits (in the weak* topology) of all convergent subsequences of the empirical probabilities ${(1/n) ∑_{j=0}^{n-1} δ_{f^{j}(x)}}_{n≥1}$, for Lebesgue almost all x ∈ M. We prove that any isolated measure in is SRB. Finally we conclude that if is finite or countably infinite, then there exist (countably many) SRB measures such that the union of their basins covers M Lebesgue a.e.

Détail

Publié le : 2011-01-01

Zbl 1248.37007

EUDML-ID : urn:eudml:doc:286615