C^0-stability of topological entropy for Reeb flows in dimension 3

TL;DR

This paper investigates the stability of topological entropy for Reeb flows on contact 3-manifolds and geodesic flows on surfaces, showing generic lower semi-continuity under C^0 perturbations.

Contribution

It establishes that generic contact forms and non-degenerate Riemannian metrics are lower semi-continuity points for topological entropy in the C^0 topology.

Findings

C^ 0-generic contact forms are lower semi-continuity points for entropy.

Non-degenerate Riemannian metrics are lower semi-continuity points for entropy.

Stability results apply to Reeb flows and geodesic flows on surfaces.

Abstract

We study stability properties of the topological entropy of Reeb flows on contact 3-manifolds with respect to the C^0-distance on the space of contact forms. Our main results show that a C^\infty-generic contact form on a closed co-oriented contact 3-manifold (Y,\xi) is a lower semi-continuity point for the topological entropy, seen as a functional on the space of contact forms of (Y,\xi) endowed with the C^0-distance. We also study the stability of the topological entropy of geodesic flows of Riemannian metrics on closed surfaces. In this setting, we show that a non-degenerate Riemannian metric on a closed surface S is a lower semi-continuity point of the topological entropy, seen as a functional on the space of Riemannian metrics on S endowed with the C^0-distance.