Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread

TL;DR

This paper investigates Hamiltonian systems on contact manifolds, demonstrating a decomposition into Reeb and Liouville dynamics, and establishing conditions for invariant measures using symplectic sandwiches with contact structures.

Contribution

It introduces a novel splitting of contact Hamiltonian systems into Reeb and Liouville parts and characterizes invariant measures via symplectic sandwiches, advancing understanding of contact dynamics.

Findings

Invariant measure exists for Reeb dynamics on contact manifolds.

Decomposition of Hamiltonian systems into Reeb and Liouville parts.

Invariant measure for Liouville dynamics characterized by symplectic sandwiches.

Abstract

We prove that, under some natural conditions, Hamiltonian systems on a contact manifold $C$ can be split into a Reeb dynamics on an open subset of $C$ and a Liouville dynamics on a submanifold of $C$ of codimension 1. For the Reeb dynamics we find an invariant measure. Moreover, we show that, under certain completeness conditions, the existence of an invariant measure for the Liouville dynamics can be characterized using the notion of a symplectic sandwich with contact bread.