SRB-like measures for C0 dynamics

Abstract

For any continuous map f on a compact manifold M, we define the SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f 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 every initial state. Precisely, the never empty set O of all the observable measures, is the minimal weak*-compact set of Borel probabilities in M that contains the limits (in the weak* topology) of all the convergent subsequences of the empiric probabilities for Lebesgue almost all x in M. We prove that any isolated measure in O is SRB. Finally we conclude that if O is finite or countable infinite, then there exist (up to countable many) SRB measures such that the union of their basins cover M Lebesgue a.e.