Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SRB measures for partially hyperbolic systems with one-dimensional center subbundles

David Burguet

Abstract

For a partially hyperbolic attractor with a center bundle splitting in a dominatedway into one-dimensional subbundles we show that for Lebesgue almost every point there is anempirical measure from $x$ with a SRB component. Moreover if the center exponents are nonzero, then $x$ lies in the basin of an ergodic hyperbolic SRB measure and there are only finitely many such measures. This gives another proof of the existence of SRB measures in this context, which was established firstly in [11] by using random perturbations. Moreover this generalizes results of [15,18] which deal with a single one-dimensional center subbundle.

SRB measures for partially hyperbolic systems with one-dimensional center subbundles

Abstract

For a partially hyperbolic attractor with a center bundle splitting in a dominatedway into one-dimensional subbundles we show that for Lebesgue almost every point there is anempirical measure from with a SRB component. Moreover if the center exponents are nonzero, then lies in the basin of an ergodic hyperbolic SRB measure and there are only finitely many such measures. This gives another proof of the existence of SRB measures in this context, which was established firstly in [11] by using random perturbations. Moreover this generalizes results of [15,18] which deal with a single one-dimensional center subbundle.
Paper Structure (17 sections, 23 theorems, 78 equations)

This paper contains 17 sections, 23 theorems, 78 equations.

Table of Contents

  1. Empirical measures
  2. General setting
  3. Bounding from below the entropy of empirical measures
  4. $\phi$-hyperbolic empirical measures
  5. Hyperbolic times
  6. Hyperbolic empirical measures
  7. Weakly hyperbolic empirical measures
  8. Midly hyperbolic empirical measures
  9. Empirical measures for partially hyperbolic systems
  10. Gibbs property for empirical measures at hyperbolic times
  11. Dynamical density on hyperbolic times
  12. Proof of Proposition \ref{['fond']}
  13. The hyperbolic case
  14. The non-hyperbolic case
  15. Proof of Proposition \ref{['prop:second']}
  16. ...and 2 more sections

Key Result

Theorem 1

With the above notations, for Lebesgue almost every $x\in U$, we have the following dichotomy :

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 3
  • Lemma 4
  • proof : Sketch of Proof
  • ...and 31 more