SRB-like Measures for $C^0$ Dynamics

Volume 59 / 2011

Eleonora Catsigeras, Heber Enrich Bulletin Polish Acad. Sci. Math. 59 (2011), 151-164 MSC: Primary 37A05; Secondary 28D05. DOI: 10.4064/ba59-2-5

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.

Authors

  • Eleonora CatsigerasInstituto de Matemática y Estadística Prof. Ing. Rafael Laguardia
    Facultad de Ingeniería
    Universidad de la República
    Av. Herrera y Reissig 565
    C.P.11300, Montevideo, Uruguay
    e-mail
  • Heber EnrichInstituto de Matemática y Estadística Prof. Ing. Rafael Laguardia
    Facultad de Ingeniería
    Universidad de la República
    Av. Herrera y Reissig 565
    C.P.11300, Montevideo, Uruguay
    e-mail

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