Quantitative Mixing for Locally Hamiltonian Flows with Saddle Loops on Compact Surfaces

Davide Ravotti*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

9 Citations (Scopus)
247 Downloads (Pure)

Abstract

Given a compact surface M with a smooth area form ω, we consider an open and dense subset of the set of smooth closed 1-forms on 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 (Ergod Theory Dyn Syst 27(3):991–1035, 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.

Original languageEnglish
Pages (from-to)3815-3861
Number of pages47
JournalAnnales Henri Poincaré
Volume18
Issue number12
Early online date25 Oct 2017
DOIs
Publication statusPublished - 1 Dec 2017

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