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

Abstract

We give a uniform lower bound for the polynomial complexity of Reeb flows on the spherization (S*M, ξ) 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 lower bound is in terms of the polynomial growth of the homology of the based loop space of M. As an application, we extend the Bott–Samelson theorem from geodesic flows to Reeb flows: If (S*M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.