SRB-like Measures for C⁰ Dynamics

Eleonora Catsigeras, and Heber Enrich. "SRB-like Measures for C⁰ Dynamics." Bulletin of the Polish Academy of Sciences. Mathematics 59.2 (2011): 151-164. <http://eudml.org/doc/286615>.

@article{EleonoraCatsigeras2011,
abstract = {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.},
author = {Eleonora Catsigeras, Heber Enrich},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {dynamical systems and ergodic theory; invariant measures},
language = {eng},
number = {2},
pages = {151-164},
title = {SRB-like Measures for C⁰ Dynamics},
url = {http://eudml.org/doc/286615},
volume = {59},
year = {2011},
}

TY - JOUR
AU - Eleonora Catsigeras
AU - Heber Enrich
TI - SRB-like Measures for C⁰ Dynamics
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2011
VL - 59
IS - 2
SP - 151
EP - 164
AB - 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.
LA - eng
KW - dynamical systems and ergodic theory; invariant measures
UR - http://eudml.org/doc/286615
ER -