Absence of mixing in area-preserving flows on surfaces

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.

  • [AF:wea] Go to document A. Avila and G. Forni, "Weak mixing for interval exchange transformations and translation flows," Ann. of Math., vol. 165, iss. 2, pp. 637-664, 2007.
    @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.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {165},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {637--664},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A25 (28D05 37D40 37E35)},
      MRNUMBER = {2299743},
      MRREVIEWER = {Gabriela Schmith{ü}sen},
      DOI = {10.4007/annals.2007.165.637},
      ZBLNUMBER = {1136.37003},
      }
  • [AGY:exp] Go to document A. Avila, S. Gouëzel, and J. Yoccoz, "Exponential mixing for the Teichmüller flow," Publ. Math. Inst. Hautes Études Sci., iss. 104, pp. 143-211, 2006.
    @article {AGY:exp, MRKEY = {2264836},
      AUTHOR = {Avila, Artur and Gou{ë}zel, S{é}bastien and Yoccoz, Jean-Christophe},
      TITLE = {Exponential mixing for the {T}eichmüller flow},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {104},
      YEAR = {2006},
      PAGES = {143--211},
      ISSN = {0073-8301},
      MRCLASS = {37D40 (32G15 37A25)},
      MRNUMBER = {2264836},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.1007/s10240-006-0001-5},
      ZBLNUMBER = {pre05117096},
      }
  • [Ar:top] Go to document V. I. Arnol$’$d, "Topological and ergodic properties of closed $1$-forms with incommensurable periods," Funktsional. Anal. i Prilozhen., vol. 25, iss. 2, pp. 1-12, 96, 1991.
    @article {Ar:top, MRKEY = {1142204},
      AUTHOR = {Arnol{$'$}d, V. I.},
      TITLE = {Topological and ergodic properties of closed {$1$}-forms with incommensurable periods},
      JOURNAL = {Funktsional. Anal. i Prilozhen.},
      FJOURNAL = {Akademiya Nauk SSSR. Funktsional$'$ nyĭAnaliz i ego Prilozheniya},
      VOLUME = {25},
      YEAR = {1991},
      NUMBER = {2},
      PAGES = {1--12, 96},
      ISSN = {0374-1990},
      MRCLASS = {58F11 (58A10)},
      MRNUMBER = {1142204},
      DOI = {10.1007/BF01079587},
      }
  • [Ca:ani] E. Calabi, "An intrinsic characterization of harmonic one-forms," in Global Analysis (Papers in Honor of K. Kodaira), Tokyo: Univ. Tokyo Press, 1969, pp. 101-117.
    @incollection {Ca:ani, MRKEY = {0253370},
      AUTHOR = {Calabi, Eugenio},
      TITLE = {An intrinsic characterization of harmonic one-forms},
      BOOKTITLE = {Global {A}nalysis ({P}apers in {H}onor of {K}. {K}odaira)},
      PAGES = {101--117},
      PUBLISHER = {Univ. Tokyo Press},
      ADDRESS = {Tokyo},
      YEAR = {1969},
      MRCLASS = {57.50 (53.00)},
      MRNUMBER = {0253370},
      MRREVIEWER = {C. B. Allendoerfer},
      ZBLNUMBER = {0194.24701},
      }
  • [CFS:erg] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinaui, Ergodic Theory, New York: Springer-Verlag, 1982, vol. 245.
    @book {CFS:erg, MRKEY = {0832433},
      AUTHOR = {Cornfeld, I. P. and Fomin, S. V. and Sina{\u\i},
      Ya. G.},
      TITLE = {Ergodic Theory},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {245},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {x+486},
      ISBN = {0-387-90580-4},
      MRCLASS = {28D05 (54H20 58F11)},
      MRNUMBER = {0832433},
      MRREVIEWER = {D. Newton},
      ZBLNUMBER = {0493.28007},
      }
  • [FL:ons] Go to document K. Frcaczek and M. Lemańczyk, "On symmetric logarithm and some old examples in smooth ergodic theory," Fund. Math., vol. 180, iss. 3, pp. 241-255, 2003.
    @article {FL:ons, MRKEY = {2063628},
      AUTHOR = {Fr\c{a}czek, K. and Lema{ń}czyk, M.},
      TITLE = {On symmetric logarithm and some old examples in smooth ergodic theory},
      JOURNAL = {Fund. Math.},
      FJOURNAL = {Fundamenta Mathematicae},
      VOLUME = {180},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {241--255},
      ISSN = {0016-2736},
      MRCLASS = {37C40 (37A10 37A25 37E35)},
      MRNUMBER = {2063628},
      MRREVIEWER = {Miaohua Jiang},
      ZBLNUMBER = {1047.37027},
      DOI = {10.4064/fm180-3-3},
      }
  • [FU:erg] K. Frcaczek and C. Ulcigrai, Ergodic properties of infinite extension of area-preserving flows.
    @misc{FU:erg, SORTDATE={2011},
      AUTHOR = {Fr\c{a}czek, Krzysztof and Ulcigrai, C.},
      TITLE={Ergodic properties of infinite extension of area-preserving flows},
      NOTE={preprint, 2011},
      }
  • [FL:acl] Go to document K. Frcaczek and M. Lemańczyk, "On disjointness properties of some smooth flows," Fund. Math., vol. 185, iss. 2, pp. 117-142, 2005.
    @article {FL:acl, MRKEY = {2163107},
      AUTHOR = {Fr\c{a}czek, Krzysztof and Lema{ń}czyk, Mariusz},
      TITLE = {On disjointness properties of some smooth flows},
      JOURNAL = {Fund. Math.},
      FJOURNAL = {Fundamenta Mathematicae},
      VOLUME = {185},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {117--142},
      ISSN = {0016-2736},
      MRCLASS = {37A10 (28D05 28D10 37C40 37E35)},
      MRNUMBER = {2163107},
      MRREVIEWER = {Nathaniel F. G. Martin},
      DOI = {10.4064/fm185-2-2},
      ZBLNUMBER = {1093.37001},
     }
  • [Fo:dev] Go to document G. Forni, "Deviation of ergodic averages for area-preserving flows on surfaces of higher genus," Ann. of Math., vol. 155, iss. 1, pp. 1-103, 2002.
    @article {Fo:dev, MRKEY = {1888794},
      AUTHOR = {Forni, Giovanni},
      TITLE = {Deviation of ergodic averages for area-preserving flows on surfaces of higher genus},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {155},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--103},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A25 (32G15 37C40 37D25 37D40 37E35 53D25)},
      MRNUMBER = {1888794},
      MRREVIEWER = {Howard Masur},
      DOI = {10.2307/3062150},
      ZBLNUMBER = {1034.37003},
      }
  • [Ka:inv] A. B. Katok, "Invariant measures of flows on oriented surfaces," Soviet Math. Dokl., vol. 14, pp. 1104-1108, 1973.
    @article{Ka:inv,
      author = {Katok, Anatole B.},
      TITLE = {Invariant measures of flows on oriented surfaces},
      JOURNAL={Soviet Math. Dokl.},
      VOLUME={14},
      YEAR={1973},
      PAGES={1104--1108},
      MRNUMBER = {0331438},
      ZBLNUMBER = {0298.28013},
      }
  • [Ka:int] Go to document A. B. Katok, "Interval exchange transformations and some special flows are not mixing," Israel J. Math., vol. 35, iss. 4, pp. 301-310, 1980.
    @article {Ka:int, MRKEY = {0594335},
      AUTHOR = {Katok, Anatole B.},
      TITLE = {Interval exchange transformations and some special flows are not mixing},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {35},
      YEAR = {1980},
      NUMBER = {4},
      PAGES = {301--310},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {58F11 (28D20)},
      MRNUMBER = {0594335},
      MRREVIEWER = {Zbigniew Nitecki},
      DOI = {10.1007/BF02760655},
      ZBLNUMBER = {0437.28009},
      }
  • [Ke:int] Go to document M. Keane, "Interval exchange transformations," Math. Z., vol. 141, pp. 25-31, 1975.
    @article {Ke:int, MRKEY = {0357739},
      AUTHOR = {Keane, Michael},
      TITLE = {Interval exchange transformations},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {141},
      YEAR = {1975},
      PAGES = {25--31},
      ISSN = {0025-5874},
      MRCLASS = {28A65},
      MRNUMBER = {0357739},
      MRREVIEWER = {M. Loeve},
      DOI = {10.1007/BF01236981},
      ZBLNUMBER = {0278.28010},
      }
  • [Ko:mix] A. V. Kovcergin, "On mixing in special flows over a rearrangement of segments and in smooth flows on surfaces," Mat. Sb., vol. 96(138), pp. 471-502, 1975.
    @article {Ko:mix, MRKEY = {0516507},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {On mixing in special flows over a rearrangement of segments and in smooth flows on surfaces},
      JOURNAL = {Mat. Sb.},
      VOLUME = {96(138)},
      YEAR = {1975},
      PAGES = {471--502},
      MRCLASS = {58F15 (28A65)},
      MRNUMBER = {0516507},
      MRREVIEWER = {D. Newton},
      ZBLNUMBER={0321.28012},
      NOTE={in Russian; translated in {\it Math. USSR Sbornik} \textbf{25} (1975), 441 },
      }
  • [Ko:non] A. V. Kovcergin, "Nonsingular saddle points and the absence of mixing," Mat. Zametki, vol. 19, pp. 453-468, 1976.
    @article{Ko:non,
      author = {Ko{\v{c}}ergin, A. V.},
      TITLE={Nonsingular saddle points and the absence of mixing},
      JOURNAL={Mat. Zametki},
      VOLUME={19},
      YEAR={1976},
      PAGES={453--468},
      NOTE={in Russian; translated in {\it Math. Notes} {\bf 19},
      277--286},
      MRNUMBER = {0415681},
      ZBLNUMBER = {0344.28008},
      }
  • [Ko:nonI] Go to document A. V. Kovcergin, "Nondegenerate fixed points and mixing in flows on a two-dimensional torus," Mat. Sb., vol. 194, iss. 8, pp. 83-112, 2003.
    @article {Ko:nonI, MRKEY = {2034533},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {Nondegenerate fixed points and mixing in flows on a two-dimensional torus},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. MatematicheskiĭSbornik},
      VOLUME = {194},
      YEAR = {2003},
      NUMBER = {8},
      PAGES = {83--112},
      ISSN = {0368-8666},
      MRCLASS = {37A25 (28D05 37A05)},
      MRNUMBER = {2034533},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.1070/SM2003v194n08ABEH000762},
      ZBLNUMBER = {1077.37005},
      }
  • [Ko:som] Go to document A. V. Kovcergin, "Some generalizations of theorems on mixing flows with nondegenerate saddles on a two-dimensional torus," Mat. Sb., vol. 195, iss. 9, pp. 19-36, 2004.
    @article {Ko:som, MRKEY = {2122367},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {Some generalizations of theorems on mixing flows with nondegenerate saddles on a two-dimensional torus},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. MatematicheskiĭSbornik},
      VOLUME = {195},
      YEAR = {2004},
      NUMBER = {9},
      PAGES = {19--36},
      ISSN = {0368-8666},
      MRCLASS = {37A25 (28D05 37A05 37E35)},
      MRNUMBER = {2122367},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.1070/SM2004v195n09ABEH000843},
      ZBLNUMBER = {1082.37008},
      }
  • [Ko:nonII] Go to document A. V. Kovcergin, "Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II," Mat. Sb., vol. 195, iss. 3, pp. 15-46, 2004.
    @article {Ko:nonII, MRKEY = {2068956},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {Nondegenerate fixed points and mixing in flows on a two-dimensional torus. {II}},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. MatematicheskiĭSbornik},
      VOLUME = {195},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {15--46},
      ISSN = {0368-8666},
      MRCLASS = {37A25 (28D05 37A05)},
      MRNUMBER = {2068956},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.1070/SM2004v195n03ABEH000807},
      ZBLNUMBER = {1077.37006},
      }
  • [Ko:wel] Go to document A. V. Kovcergin, "Well-approximable angles and mixing for flows on $\Bbb T^2$ with nonsingular fixed points," Electron. Res. Announc. Amer. Math. Soc., vol. 10, pp. 113-121, 2004.
    @article {Ko:wel, MRKEY = {2119032},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {Well-approximable angles and mixing for flows on {$\Bbb T\sp 2$} with nonsingular fixed points},
      JOURNAL = {Electron. Res. Announc. Amer. Math. Soc.},
      FJOURNAL = {Electronic Research Announcements of the American Mathematical Society},
      VOLUME = {10},
      YEAR = {2004},
      PAGES = {113--121},
      ISSN = {1079-6762},
      MRCLASS = {37E25 (37A05 37A25 37C10)},
      MRNUMBER = {2119032},
      MRREVIEWER = {Miaohua Jiang},
      DOI = {10.1090/S1079-6762-04-00136-2},
      ZBLNUMBER = {1068.37027},
      }
  • [Ko:non2] A. V. Kovcergin, "Nondegenerate saddles and the absence of mixing in flows on surfaces," Tr. Mat. Inst. Steklova, vol. 256, pp. 252-266, 2007.
    @article {Ko:non2, MRKEY = {2336903},
      AUTHOR = {Ko{\v{c}}ergin, A. V.},
      TITLE = {Nondegenerate saddles and the absence of mixing in flows on surfaces},
      JOURNAL = {Tr. Mat. Inst. Steklova},
      FJOURNAL = {Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossiĭskaya Akademiya Nauk},
      VOLUME = {256},
      YEAR = {2007},
      NUMBERBAD = {Din. Sist. i Optim.},
      PAGES = {252--266},
      ISSN = {0371-9685},
      MRCLASS = {37A25 (28D05 37A05 37E35)},
      MRNUMBER = {2336903},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      ZBLNUMBER = {1153.37303},
     }
  • [KT:spe] A. B. Katok and J. -P. Thouvenot, "Spectral properties and combinatorial constructions in ergodic theory," in Handbook of Dynamical Systems, Vol. 1B, Amsterdam: Elsevier B.V., 2006, pp. 649-743.
    @incollection{KT:spe,
      author={Katok, Anatole B. and Thouvenot, J.-P.},
      TITLE={Spectral properties and combinatorial constructions in ergodic theory},
      BOOKTITLE={Handbook of Dynamical Systems, Vol. {\rm 1B}},
      PAGES={649--743},
      PUBLISHER={Elsevier B.V.},
      ADDRESS={Amsterdam},
      YEAR={2006},
      ZBLNUMBER={1130.37304},
      MRNUMBER={2186251},
      }
  • [Le:sur] M. Lemańczyk, "Sur l’absence de mélange pour des flots spéciaux au-dessus d’une rotation irrationnelle," Colloq. Math., vol. 84/85, pp. 29-41, 2000.
    @article {Le:sur, MRKEY = {1778837},
      AUTHOR = {Lema{ń}czyk, M.},
      TITLE = {Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle},
      JOURNAL = {Colloq. Math.},
      FJOURNAL = {Colloquium Mathematicum},
      VOLUME = {84/85},
      YEAR = {2000},
      NUMBERBAD = {, part 1},
      PAGES = {29--41},
      ISSN = {0010-1354},
      CODEN = {CQMAAQ},
      MRCLASS = {37A25 (37A05 37A20 37A45)},
      MRNUMBER = {1778837},
      MRREVIEWER = {Nathaniel F. G. Martin},
      ZBLNUMBER = {0983.37004},
      }
  • [Le:feu] Go to document G. Levitt, "Feuilletages des surfaces," Ann. Inst. Fourier $($Grenoble$)$, vol. 32, iss. 2, pp. 179-217, 1982.
    @article {Le:feu, MRKEY = {0662443},
      AUTHOR = {Levitt, Gilbert},
      TITLE = {Feuilletages des surfaces},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {32},
      YEAR = {1982},
      NUMBER = {2},
      PAGES = {179--217},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {57R30},
      MRNUMBER = {0662443},
      MRREVIEWER = {Ulrich Hirsch},
      URL = {http://www.numdam.org/item?id=AIF_1982__32_2_179_0},
      ZBLNUMBER = {0454.57015},
     }
  • [Ma:tra] A. Mayer, "Trajectories on the closed orientable surfaces," Rec. Math. [Mat. Sbornik] N.S., vol. 12(54), pp. 71-84, 1943.
    @article {Ma:tra, MRKEY = {0009485},
      AUTHOR = {Mayer, A.},
      TITLE = {Trajectories on the closed orientable surfaces},
      JOURNAL = {Rec. Math. [Mat. Sbornik] N.S.},
      VOLUME = {12(54)},
      YEAR = {1943},
      PAGES = {71--84},
      MRCLASS = {53.0X},
      MRNUMBER = {0009485},
      MRREVIEWER = {W. Hurewicz},
      ZBLNUMBER = {0063.03856},
      }
  • [Ma:int] Go to document H. Masur, "Interval exchange transformations and measured foliations," Ann. of Math., vol. 115, iss. 1, pp. 169-200, 1982.
    @article {Ma:int, MRKEY = {0644018},
      AUTHOR = {Masur, Howard},
      TITLE = {Interval exchange transformations and measured foliations},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {115},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {169--200},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {28D05 (10K10 58F11 58F18)},
      MRNUMBER = {0644018},
      MRREVIEWER = {D. Newton},
      DOI = {10.2307/1971341},
      ZBLNUMBER = {0497.28012},
      }
  • [No:the] S. P. Novikov, "The Hamiltonian formalism and a multivalued analogue of Morse theory," Uspekhi Mat. Nauk, vol. 37, iss. 5, pp. 3-49, 248, 1982.
    @article {No:the, MRKEY = {0676612},
      AUTHOR = {Novikov, S. P.},
      TITLE = {The {H}amiltonian formalism and a multivalued analogue of {M}orse theory},
      JOURNAL = {Uspekhi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {37},
      YEAR = {1982},
      NUMBER = {5},
      NOTE={in Russian; translated in {\it Russian Math. Surveys} {\bf 37} (1982), 1--56},
      PAGES = {3--49, 248},
      ISSN = {0042-1316},
      MRCLASS = {58E05 (58E30 58F05)},
      MRNUMBER = {0676612},
      MRREVIEWER = {P. Ver Eecke},
      ZBLNUMBER = {0571.58011},
     }
  • [NZ:flo] I. Nikolaev and E. Zhuzhoma, Flows on 2-Dimensional Manifolds. An Overview, New York: Springer-Verlag, 1999, vol. 1705.
    @book {NZ:flo, MRKEY = {1707298},
      AUTHOR = {Nikolaev, Igor and Zhuzhoma, Evgeny},
      TITLE = {Flows on 2-Dimensional Manifolds. An Overview},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1705},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {xx+294},
      ISBN = {3-540-66080-1},
      MRCLASS = {37E35 (37-02 37C10)},
      MRNUMBER = {1707298},
      MRREVIEWER = {Robert Ghrist},
      ZBLNUMBER = {1022.37027},
      }
  • [Ra:ech] G. Rauzy, "Échanges d’intervalles et transformations induites," Acta Arith., vol. 34, iss. 4, pp. 315-328, 1979.
    @article {Ra:ech, MRKEY = {0543205},
      AUTHOR = {Rauzy, G{é}rard},
      TITLE = {\'{E}changes d'intervalles et transformations induites},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {34},
      YEAR = {1979},
      NUMBER = {4},
      PAGES = {315--328},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {10K10 (28D05 58F11)},
      MRNUMBER = {0543205},
      MRREVIEWER = {F. M. Dekking},
      ZBLNUMBER = {0414.28018},
      }
  • [Sch:abs] Go to document D. Scheglov, "Absence of mixing for smooth flows on genus two surfaces," J. Mod. Dyn., vol. 3, iss. 1, pp. 13-34, 2009.
    @article {Sch:abs, MRKEY = {2481330},
      AUTHOR = {Scheglov, Dmitri},
      TITLE = {Absence of mixing for smooth flows on genus two surfaces},
      JOURNAL = {J. Mod. Dyn.},
      FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {3},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {13--34},
      ISSN = {1930-5311},
      MRCLASS = {37A25 (37A10 37C40 37E35)},
      MRNUMBER = {2481330},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.3934/jmd.2009.3.13},
      ZBLNUMBER = {1183.37080},
      }
  • [SK:mix] Go to document Y. G. Sinaui and K. M. Khanin, "Mixing of some classes of special flows over rotations of the circle," Funktsional. Anal. i Prilozhen., vol. 26, iss. 3, pp. 1-21, 1992.
    @article {SK:mix, MRKEY = {1189019},
      AUTHOR = {Sina{\u\i},
      Ya. G. and Khanin, K. M.},
      TITLE = {Mixing of some classes of special flows over rotations of the circle},
      JOURNAL = {Funktsional. Anal. i Prilozhen.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional$'$ nyĭAnaliz i ego Prilozheniya},
      VOLUME = {26},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {1--21},
      ISSN = {0374-1990},
      MRCLASS = {58F11 (28D05 28D15)},
      MRNUMBER = {1189019},
      MRREVIEWER = {M. L. Blank},
      DOI = {10.1007/BF01075628},
      ZBLNUMBER = {0797.58045},
      }
  • [SU:lim] Y. G. Sinai and C. Ulcigrai, "A limit theorem for Birkhoff sums of non-integrable functions over rotations," in Geometric and Probabilistic Structures in Dynamics, Providence, RI: Amer. Math. Soc., 2008, vol. 469, pp. 317-340.
    @incollection {SU:lim, MRKEY = {2478478},
      AUTHOR = {Sinai, Yakov G. and Ulcigrai, Corinna},
      TITLE = {A limit theorem for {B}irkhoff sums of non-integrable functions over rotations},
      BOOKTITLE = {Geometric and Probabilistic Structures in Dynamics},
      SERIES = {Contemp. Math.},
      VOLUME = {469},
      PAGES = {317--340},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2008},
      MRCLASS = {37A30 (37E10 60B10)},
      MRNUMBER = {2478478},
      MRREVIEWER = {Marta Tyran-Kami{ń}ska},
      ZBLNUMBER = {1154.37307},
      }
  • [Ul:erg] Go to document C. Ulcigrai, Ergodic Properties of Some Area-Preserving Flows, ProQuest LLC, Ann Arbor, MI, 2007.
    @book {Ul:erg, MRKEY = {2625338},
      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},
      }
  • [Ul:mix] Go to document C. Ulcigrai, "Mixing of asymmetric logarithmic suspension flows over interval exchange transformations," Ergodic Theory Dynam. Systems, vol. 27, iss. 3, pp. 991-1035, 2007.
    @article {Ul:mix, MRKEY = {2322189},
      AUTHOR = {Ulcigrai, Corinna},
      TITLE = {Mixing of asymmetric logarithmic suspension flows over interval exchange transformations},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {27},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {991--1035},
      ISSN = {0143-3857},
      MRCLASS = {37A25 (28D05 37A05 37E35)},
      MRNUMBER = {2322189},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.1017/S0143385706000836},
      ZBLNUMBER = {1135.37004},
      }
  • [Ul:wm] Go to document C. Ulcigrai, "Weak mixing for logarithmic flows over interval exchange transformations," J. Mod. Dyn., vol. 3, iss. 1, pp. 35-49, 2009.
    @article {Ul:wm, MRKEY = {2481331},
      AUTHOR = {Ulcigrai, Corinna},
      TITLE = {Weak mixing for logarithmic flows over interval exchange transformations},
      JOURNAL = {J. Mod. Dyn.},
      FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {3},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {35--49},
      ISSN = {1930-5311},
      MRCLASS = {37A25 (37A10 37C40 37E35)},
      MRNUMBER = {2481331},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      DOI = {10.3934/jmd.2009.3.35},
      ZBLNUMBER = {1183.37081},
      }
  • [Ve:int] W. A. Veech, "Interval exchange transformations," J. Analyse Math., vol. 33, pp. 222-272, 1978.
    @article {Ve:int, MRKEY = {0516048},
      AUTHOR = {Veech, William A.},
      TITLE = {Interval exchange transformations},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {33},
      YEAR = {1978},
      PAGES = {222--272},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {28D99 (54H20)},
      MRNUMBER = {0516048},
      MRREVIEWER = {Michael Keane},
      ZBLNUMBER = {0455.28006},
      }
  • [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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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},
      }

Authors

Corinna Ulcigrai

Department of Mathematics
University of Bristol
Bristol BS8 1TW
United Kingdom