SRB-like Measures for C 0 Dynamics

For any continuous map f : M ! M on a compact manifold M, we dene the SRB-like probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f has SRB-like measures, even if SRB measures do not exist. We prove that the denition of SRB-like measures is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost every initial state. We prove that any isolated measure in the setO of SRBlike measures is SRB. Finally we conclude that ifO is nite or countable innite, then there exist (up to countable many) SRB measures such that the union of their basins cover M Lebesgue a.e. MSC2010: Primary 37A05; Secondary 28D05

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