Homological Resonances for Hamiltonian Diffeomorphisms and Reeb Flows

TL;DR

This paper demonstrates that Hamiltonian diffeomorphisms and Reeb flows with finitely many periodic orbits exhibit resonance relations among their mean indices, leading to invariants that distinguish contact structures.

Contribution

It establishes resonance relations for mean indices in finite periodic orbit cases and introduces mean Euler characteristics as contact invariants.

Findings

Resonance relations constrain mean indices of periodic orbits.

Mean Euler characteristics can distinguish contact structures.

Results apply under natural conditions on the ambient manifold.

Abstract

We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural requirements. In the case of Reeb flows, this leads to simple expressions (purely in terms of the mean indices) for the mean Euler characteristics. These are invariants of the underlying contact structure which are capable of distinguishing some contact structures that are homotopic but not diffeomorphic.