SRB and equilibrium measures via dimension theory

TL;DR

This paper develops a new method to construct SRB and equilibrium measures for hyperbolic flows using a dimension theory approach, extending previous discrete-time results to continuous-time systems.

Contribution

It introduces a novel construction of leaf measures for hyperbolic flows based on Hausdorff measure concepts, providing a more complete description of scaling properties and directly producing equilibrium measures.

Findings

Constructed leaf measures analogous to Hausdorff measure for hyperbolic flows.

Extended the dimension theory approach from discrete to continuous-time systems.

Provided a direct method to produce equilibrium measures using Bowen balls.

Abstract

It is well-known that SRB and equilibrium measures for uniformly hyperbolic flows admit a product structure in terms of measures on stable and unstable leaves with scaling properties given by the potential function. We describe a construction of these leaf measures analogous to the definition of Hausdorff measure, relying on the Pesin-Pitskel' description of topological pressure as a dimensional characteristic using Bowen balls. These leaf measures were constructed for discrete-time systems by the author, Ya. Pesin, and A. Zelerowicz. In the continuous-time setting here, the description of the scaling properties is more complete, and we use a similar procedure with two-sided Bowen balls to directly produce the equilibrium measure itself.