Absence of mixing in area-preserving flows on surfaces

Corinna Ulcigrai

doi:http://doi.org/10.4007/annals.2011.173.3.10

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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[Ve:gau] $Go to document$ W. A. Veech, "Gauss measures for transformations on the space of interval exchange maps," Ann. of Math., vol. 115, iss. 1, pp. 201-242, 1982.

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@article {Ve:gau, MRKEY = {0644019}, AUTHOR = {Veech, William A.}, TITLE = {Gauss measures for transformations on the space of interval exchange maps}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {115}, YEAR = {1982}, NUMBER = {1}, PAGES = {201--242}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {28D05 (58F11)}, MRNUMBER = {0644019}, MRREVIEWER = {Michael Keane}, DOI = {10.2307/1971391}, ZBLNUMBER = {0486.28014}, }
[Vi:IET] $Go to document$ M. Viana, Dynamics of interval exchange maps and Teichmüller flows.

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@misc{Vi:IET, author={Viana, M.}, TITLE={Dynamics of interval exchange maps and {T}eichm{ü}ller flows}, NOTE={lecture notes}, URL={http://w3.impa.br/~viana/}, }
[Yo:con] $Go to document$ J. Yoccoz, "Continued fraction algorithms for interval exchange maps: an introduction," in Frontiers in Number Theory, Physics, and Geometry. I, New York: Springer-Verlag, 2006, pp. 401-435.

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@incollection {Yo:con, MRKEY = {2261103}, AUTHOR = {Yoccoz, Jean-Christophe}, TITLE = {Continued fraction algorithms for interval exchange maps: an introduction}, BOOKTITLE = {Frontiers in Number Theory, Physics, and Geometry. {I}}, PAGES = {401--435}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2006}, MRCLASS = {37A45 (11A55)}, MRNUMBER = {2261103}, MRREVIEWER = {Habib Marzougui}, DOI = {10.1007/978-3-540-31347-2_12}, ZBLNUMBER = {1127.28011}, }
[Zo:fin] $Go to document$ A. Zorich, "Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents," Ann. Inst. Fourier $($Grenoble$)$, vol. 46, iss. 2, pp. 325-370, 1996.

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@article {Zo:fin, MRKEY = {1393518}, AUTHOR = {Zorich, Anton}, TITLE = {Finite {G}auss measure on the space of interval exchange transformations. {L}yapunov exponents}, JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$}, FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier}, VOLUME = {46}, YEAR = {1996}, NUMBER = {2}, PAGES = {325--370}, ISSN = {0373-0956}, CODEN = {AIFUA7}, MRCLASS = {58F11 (28D05)}, MRNUMBER = {1393518}, MRREVIEWER = {Vadim A. Ka{\u\i}manovich}, URL = {http://www.numdam.org/item?id=AIF_1996__46_2_325_0}, ZBLNUMBER = {0853.28007}, }
[Zo:dev] $Go to document$ A. Zorich, "Deviation for interval exchange transformations," Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1477-1499, 1997.

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@article {Zo:dev, MRKEY = {1488330}, AUTHOR = {Zorich, Anton}, TITLE = {Deviation for interval exchange transformations}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {17}, YEAR = {1997}, NUMBER = {6}, PAGES = {1477--1499}, ISSN = {0143-3857}, MRCLASS = {58F11}, MRNUMBER = {1488330}, MRREVIEWER = {Vadim A. Ka{\u\i}manovich}, DOI = {10.1017/S0143385797086215}, ZBLNUMBER = {0958.37002}, }
[Zo:how] A. Zorich, "How do the leaves of a closed $1$-form wind around a surface?," in Pseudoperiodic Topology, Providence, RI: Amer. Math. Soc., 1999, vol. 197, pp. 135-178.

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@incollection {Zo:how, MRKEY = {1733872}, AUTHOR = {Zorich, Anton}, TITLE = {How do the leaves of a closed {$1$}-form wind around a surface?}, BOOKTITLE = {Pseudoperiodic Topology}, SERIES = {Amer. Math. Soc. Transl. Ser. 2}, VOLUME = {197}, PAGES = {135--178}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1999}, MRCLASS = {57M50 (37C10 37C85 37E35)}, MRNUMBER = {1733872}, MRREVIEWER = {Boris Hasselblatt}, ZBLNUMBER = {0976.37012}, }

Absence of mixing in area-preserving flows on surfaces

Abstract

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