Slow volume growth for Reeb flows on spherizations and contact Bott--Samelson theorems

TL;DR

This paper establishes a lower bound on the polynomial complexity of Reeb flows on spherizations, linking it to the homology growth of loop spaces, and extends classical theorems to Reeb flows.

Contribution

It introduces a uniform lower bound for Reeb flow complexity based on homology growth and generalizes Bott--Samelson theorems from geodesic flows to Reeb flows.

Findings

Reeb flows have polynomial complexity bounded below by homology growth.

If a Reeb flow is periodic, the manifold is a sphere or has finite fundamental group.

Extension of Bott--Samelson theorem to Reeb flows on spherizations.

Abstract

We give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S*M,\xi) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our uniform bound is in terms of the polynomial growth of the homology of the based loops space of M. As an application, we extend the Bott--Samelson theorem from geodesic flows to Reeb flows: If (S*M,\xi) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fibre S*_q M, then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M is the one of a compact rank one symmetric space.