What Are SRB Measures, and Which Dynamical Systems Have Them?

@article{Young2002WhatAS,
  title={What Are SRB Measures, and Which Dynamical Systems Have Them?},
  author={Lai-Sang Young},
  journal={Journal of Statistical Physics},
  year={2002},
  volume={108},
  pages={733-754}
}
  • L. Young
  • Published 1 December 2002
  • Physics
  • Journal of Statistical Physics
This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking, 
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References

SHOWING 1-10 OF 99 REFERENCES
The Dynamics of the Henon Map
The purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the Henon map with expansion combined with strong contraction. Our study
GIBBS MEASURES IN ERGODIC THEORY
In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of
Ergodic Theory of Differentiable Dynamical Systems
These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, µ), where f is a diffeomorphism of a Riemannian manifold and µ is an
Strange Attractors with One Direction of Instability
Abstract: We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only
Abundance of strange attractors
In 1976 [He] H~non performed a numerical study of the family of diffeomorphisms of the plane ha,b(X, y)=(1-ax2+y, bx) and detected for parameter values a=l.4, b=0.3, what seemed to be a non-trivial
SRB measures for partially hyperbolic systems whose central direction is mostly expanding
Abstract.We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms – the tangent bundle splits into two invariant subbundles, one of which is uniformly
Almost every real quadratic map is either regular or stochastic
In this paper we complete a program to study measurable dynamics in the real quadratic family. Our goal was to prove that almost any real quadratic map Pc : z t- x2 + c, c c [-2,1/4], has either an
Regular points for ergodic Sinai measures
Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure p if it is generic for p. and the Lyapunov exponents at it coincide with those of p..
Solution of the basin problem for Hénon-like attractors
Abstract.For a large class of non-uniformly hyperbolic attractors of dissipative diffeomorphisms, we prove that there are no “holes” in the basin of attraction: stable manifolds of points in the
A MEASURE ASSOCIATED WITH AXIOM-A ATTRACTORS.
The future orbits of a diffeomorphism near an Axiom-A attrac- tor are investigated. It is found that their asymptotic behavior is in general described by a fixed probability measure yt carried by the
...
1
2
3
4
5
...