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 Λ is entropy expansive, and the topological entropy varies continuously with the flow. Furthermore, if Λ is Lyapunov stable, then it has positive entropy; in addition, if Λ 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.