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

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.