Entropy theory for sectional hyperbolic flows

Abstract

We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for $C^1$ flows, every sectional hyperbolic set $\Lambda$ is entropy expansive, and the topological entropy varies continuously with the flow. Furthermore, if $\Lambda$ is Lyapunov stable, then it has positive entropy; in addition, if $\Lambda$ is a chain recurrent class, then it contains a periodic orbit. As a corollary, we prove that for $C^1$ generic flows, every Lorenz-like class is an attractor.