Local contact homology and applications

TL;DR

This paper introduces a local contact homology framework for isolated Reeb flow orbits, establishing bounds and applications such as periodic orbit existence and resonance relations.

Contribution

It develops a local contact homology theory for isolated periodic orbits and applies it to prove new results in periodic orbit dynamics.

Findings

Rank of local contact homology is uniformly bounded for isolated iterations.

Generalization of Gromoll-Meyer's theorem on infinitely many simple periodic orbits.

Conditions for the existence of non-hyperbolic periodic orbits.

Abstract

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll-Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits.