Dimension and ergodic decompositions for hyperbolic flows

Luis Barreira,Christian Wolf

doi:10.3934/dcds.2007.17.201

Abstract

For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.

Highlights

In the theory of dynamical systems, in the study of chaotic behavior, each global quantity can often be “constructed” with the help of a certain local quantity
In the case of dimension there exists an appropriate version of this statement: namely, the Hausdorff dimension of a measure is given by the essential supremum of the pointwise dimension
Our main objective in this paper is to extend this study to the case of hyperbolic flows

Summary

Introduction

In the theory of dynamical systems, in the study of chaotic behavior, each global quantity can often be “constructed” with the help of a certain local quantity. We want to obtain explicit formulas for the pointwise dimension and the Hausdorff dimension of invariant measures of conformal hyperbolic flows that are not necessarily ergodic. Let μ be a compactly supported finite measure invariant under a C1+α diffeomorphism f It follows from work of Ledrappier and Young in [10] and work of Barreira, Pesin and Schmeling in [2] that if the measure μ is hyperbolic the pointwise dimension exists almost everywhere. In the present paper we establish an analogous formula to that in (4) for the pointwise dimension in the case of conformal hyperbolic flows. The identity in (5) can be used to describe how the Hausdorff dimension dimH μ of an invariant measure μ on a hyperbolic set Λ behaves under an ergodic decomposition.

We assume that there exists ε

Let now