Invariant probability measures from pseudoholomorphic curves I

TL;DR

This paper presents a novel method using pseudoholomorphic curves to construct invariant probability measures for a broad class of volume-preserving flows on odd-dimensional manifolds, with applications to ergodic theory.

Contribution

It introduces a new technique from symplectic geometry to analyze invariant measures of volume-preserving flows, extending previous results to higher dimensions.

Findings

Constructed invariant measures for various volume-preserving flows

Identified obstructions to unique ergodicity in these flows

Generalized earlier results by Taubes and Ginzburg-Niche

Abstract

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.