Positive topological entropy of Reeb flows on spherizations

TL;DR

This paper proves that Reeb flows on spherizations of certain complex manifolds exhibit positive topological entropy, indicating chaotic dynamics, with exponential growth in Reeb orbits for most points and surfaces of higher genus.

Contribution

It establishes the positivity of topological entropy for Reeb flows on spherizations of manifolds with complicated loop spaces, extending known results to broader classes of manifolds.

Findings

Reeb flows on spherizations have positive topological entropy.

Number of Reeb orbits grows exponentially for almost all point pairs.

Higher genus surfaces exhibit exponential growth of closed Reeb orbits.

Abstract

Let M be a closed manifold whose based loop space is ``complicated''. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. We prove that the topological entropy of any Reeb flow on the spherization of T*M is positive. Moreover, given q in M, for almost every q' in M the number of Reeb orbits from the fiber over q to the fiber over q' grows exponentially in time. If M is a surface of higher genus, we also obtain exponential growth of the number of closed Reeb orbits.