A Remark on Unique Ergodicity and the Contact Type Condition

TL;DR

This paper demonstrates that Hamiltonian flows on certain symplectic manifolds, including Euclidean space, cannot be uniquely ergodic due to contact type conditions and the Weinstein conjecture, with implications for twisted geodesic flows.

Contribution

It establishes a link between contact type conditions, the Weinstein conjecture, and non-unique ergodicity of Hamiltonian flows on broad classes of symplectic manifolds.

Findings

Hamiltonian flows on regular energy levels are not uniquely ergodic.

Exact twisted geodesic flows cannot be uniquely ergodic under certain conditions.

Hamiltonian structures with non-vanishing self-linking number must have contact type.

Abstract

We prove that for a broad class of exact symplectic manifolds including ${\mathbb R}^{2m}$ the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and an observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.