SRB-like Measures for C⁰ Dynamics

• Autorzy: Eleonora Catsigeras , Heber Enrich
• Języki publikacji: EN

Abstrakty

EN

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 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 ${(1/n) ∑_{j=0}^{n-1} δ_{f^{j}(x)}}_{n≥1}$, for Lebesgue almost all x ∈ M. We prove that any isolated measure in 𝓞 is SRB. Finally we conclude that if 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.

Kategorie tematyczne

• 37A05: Measure-preserving transformations
• 28D05: Measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2011
• Tom: 59
• Numer: 2
• Strony: 151-164

Daty

wydano

2011

Twórcy

autor

Eleonora Catsigeras

• Instituto 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

autor

Heber Enrich

• Instituto 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

Identyfikatory

DOI

10.4064/ba59-2-5