On growth rate and contact homology

TL;DR

This paper investigates the growth rate of Reeb periodic orbits and contact homology in different geometric settings, proving exponential growth in hyperbolic cases and polynomial growth in certain non-hyperbolic cases.

Contribution

It proves exponential growth of Reeb orbits and contact homology for hyperbolic fibered manifolds, and computes contact homology in specific non-hyperbolic cases.

Findings

Exponential growth of Reeb orbits in hyperbolic fibered manifolds.

Polynomial growth of contact homology in non-hyperbolic circle bundle cases.

Infinite non-isomorphic contact structures with exponential Reeb orbit growth.

Abstract

It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of Reeb periodic orbits of any non-degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non-hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non-transverse to the fibers on a circle bundle.