Invariant probability measures from pseudoholomorphic curves Ⅰ

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.

Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

“…We also prove in [18] that, under some mild assumptions on the framed Hamiltonian structure, that these probability measures are "interesting". That and may satisfy additional properties depending on the situation.…”

“…Fix a framed Hamiltonian manifold (M , η = (λ, ω)) of dimension 2n +1 and let X denote its associated Hamiltonian vector field. The main theorem (Theorem 1.4) of [18] uses pseudoholomorphic curves inside the cylinder R × M , equipped with a special kind of almost-complex structure, to construct X -invariant probability measures.…”

“…That and may satisfy additional properties depending on the situation. These results, contained in Theorems 1.5, 1.6 and 1.8 of [18], make use of the geometric and topological properties of pseudoholomorphic curves.…”

“…Section 1.2 specializes the discussion to pseudoholomorphic curves in R×M , where (M , η = (λ, ω)) is a framed Hamiltonian manifold. It defines ω-finite pseudoholomorphic curves in R × M , which are the pseudoholomorphic curves required by [18] to construct interesting invariant measures of the Hamiltonian vector field X associated to (M , η). It also defines Fish-Hofer's feral pseudoholomorphic curves, which are special cases of ω-finite pseudoholomorphic curves that we construct in this paper.…”

“…This paper is a companion paper to the work [18] by the author. This work presents a method for constructing invariant probability measures of the autonomous flow associated to a framed Hamiltonian structure η = (λ, ω) on a closed, oriented smooth manifold M of dimension 2n + 1 for n ≥ 1.…”