Abstract
We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus $g\geq2$ 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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@article {AF:wea, MRKEY = {2299743},
AUTHOR = {Avila, Artur and Forni, Giovanni},
TITLE = {Weak mixing for interval exchange transformations and translation flows},
JOURNAL = {Ann. of Math.},
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AUTHOR = {Sinai, Yakov G. and Ulcigrai, Corinna},
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} -
[Ul:erg]
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AUTHOR = {Ulcigrai, Corinna},
TITLE = {Ergodic Properties of Some Area-Preserving Flows},
NOTE = {{Ph.D}. thesis, {P}rinceton {U}niversity},
PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
YEAR = {2007},
PAGES = {125},
MRCLASS = {Thesis},
MRNUMBER = {2625338},
URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3256588},
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}