Symbolic dynamics for three dimensional flows with positive topological entropy

TL;DR

This paper develops symbolic dynamics for three-dimensional flows with positive entropy, providing new insights into the structure of geodesic flows on surfaces and their closed orbits.

Contribution

It constructs symbolic dynamics for 3D flows with positive entropy on compact manifolds, extending previous results to a broader class of flows.

Findings

Existence of symbolic dynamics on sets of full measure for these flows

Lower bounds on the number of closed orbits of the geodesic flow

Applicability to flows with positive topological entropy without curvature assumptions

Abstract

We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+\epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.