SRB-like measures for C0 dynamics

TL;DR

This paper introduces SRB-like measures for continuous maps on compact manifolds, establishing their existence even without traditional SRB measures and characterizing their properties and significance in describing asymptotic statistical behavior.

Contribution

It generalizes the concept of SRB measures to include observable measures, proving their existence and optimality for describing asymptotic statistics in C0 dynamics.

Findings

Existence of observable measures even when SRB measures do not exist

Observable measures form a minimal weak*-compact set containing limits of empirical measures

Isolated measures in the set are SRB measures

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.