Absence of mixing in area-preserving flows on surfaces | Annals of Mathematics

Abstract We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus g≥2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. As a corollary, minimal flows given by multi-valued Hamiltonians on higher genus surfaces which are minimal and have only simple non-degenerate saddles are typically not mixing.

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数学年刊(Annals of Mathematics)

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