Homoclinic classes for flows: ergodicity and SRB measures

TL;DR

This paper investigates homoclinic classes in flows, demonstrating conditions under which their intersections form ergodic components and exploring properties of SRB measures, extending prior results from the flow setting.

Contribution

It extends previous work by establishing ergodic properties of homoclinic class intersections and analyzing SRB measures in the context of flows.

Findings

Positive Lebesgue measure in both classes implies ergodic intersection

Results on regular SRB measures for flows

Extension of discrete flow results to continuous flows

Abstract

In this work we intend to study homoclinic classes for some classes of flows. To this end we obtain analogous results those obtained by Hertz-Hertz-Tahzibi-Ures in the flow setting. Namely we prove that if the Lesbegue measure gives positive measure to both stable and unstable homoclinic classes of a periodic hyperbolic orbit, then their intersection constitute an ergodic component. Futhermore, with similar techiniques we state several results concerning regular SRB measures.