Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces

TL;DR

This paper proves that almost every smooth closed 1-form with saddle loops on a compact surface induces mixing flows on minimal components, providing quantitative decay estimates for correlations, extending previous results on suspension flows over interval exchange transformations.

Contribution

It establishes a quantitative mixing result for locally Hamiltonian flows with saddle loops, generalizing Ulcigrai's theorem to multiple singularities and applying to asymmetric logarithmic suspension flows.

Findings

Almost every such flow is mixing on minimal components.

Provides decay rate estimates for correlations.

Extends Ulcigrai's theorem to multiple singularities.

Abstract

Given a compact surface $\mathcal{M}$ with a smooth area form $\omega$, we consider an open and dense subset of the set of smooth closed 1-forms on $\mathcal{M}$ with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate of the speed of the decay of correlations for smooth functions with compact support on the complement of the set of singularities. This result is achieved by proving a quantitative version for the case of finitely many singularities of a theorem by Ulcigrai (ETDS, 2007), stating that any suspension flow with one asymmetric logarithmic singularity over almost every interval exchange transformation is mixing. In particular, the quantitative mixing estimate we prove applies to asymmetric logarithmic suspension flows over rotations, which were shown to be mixing by Sinai and Khanin.