SRB measures for partially hyperbolic flows with mostly expanding center

TL;DR

This paper establishes the existence and finiteness of SRB measures for certain partially hyperbolic flows with expanding center behavior, extending results known for diffeomorphisms to flows.

Contribution

It proves the existence of SRB measures for partially hyperbolic attractors in $C^1$ flows and finiteness under $C^2$ conditions with sectional expansion, addressing flow-specific challenges.

Findings

Partially hyperbolic attractors support SRB measures.

Finitely many SRB measures cover almost all points in the basin.

Results extend known diffeomorphism results to flows.

Abstract

We prove that a partially hyperbolic attractor for a $C^1$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $C^2$, and the center bundle admits the sectional expanding condition w.r.t. any Gibbs $u$-state, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.