Continuity of topological entropy for perturbations of time-one maps of hyperbolic flows

TL;DR

This paper proves that the topological entropy of diffeomorphisms near the time-one map of a hyperbolic flow varies continuously, extending to all known partially hyperbolic systems with one-dimensional center.

Contribution

It establishes the continuity of topological entropy in a $C^1$ neighborhood of the time-one map of hyperbolic flows, including all known partially hyperbolic examples with one-dimensional center.

Findings

Topological entropy varies continuously in the specified neighborhood.

Continuity holds for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center.

The result applies to perturbations of time-one maps of hyperbolic flows.

Abstract

We consider a $C^1$ neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle.