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 (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.
Highlights
Let us consider a smooth compact connected orientable surface M, together with a smooth area form ω
Orbits of locally Hamiltonian flows can be seen as hyperplane sections of periodic manifolds, as pointed out by Arnold [1], who studied the case when M is the 2-dimensional torus T2
He proved that T2 can be decomposed into finitely many regions filled with periodic trajectories and one minimal ergodic component; in the same paper he asked whether the restriction of the flow to this ergodic component is mixing
Summary
Let us consider a smooth compact connected orientable surface M, together with a smooth area form ω. 2 for definitions); it is called locally Hamiltonian flow or multi-valued Hamiltonian flow. Orbits of locally Hamiltonian flows can be seen as hyperplane sections of periodic manifolds, as pointed out by Arnold [1], who studied the case when M is the 2-dimensional torus T2. He proved that T2 can be decomposed into finitely many regions filled with periodic trajectories and one minimal ergodic component; in the same paper he asked whether the restriction of the flow to this ergodic component is mixing.
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