SRB-like Measures for $C^0$ Dynamics
Volume 59 / 2011
Abstract
For any continuous map $f\colon M \to M$ 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 ${\mathcal O}$ 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) \sum_{j= 0}^{n-1}\delta_{f^j(x)}\}_{n \geq 1}$, for Lebesgue almost all $x \in M$. We prove that any isolated measure in ${\mathcal O}$ is SRB. Finally we conclude that if ${\mathcal O}$ is finite or countably infinite, then there exist (countably many) SRB measures such that the union of their basins covers $M$ Lebesgue a.e.
