Limit theorems for translation flows

Abstract

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

  • [adam] Go to document B. Adamczewski, "Symbolic discrepancy and self-similar dynamics," Ann. Inst. Fourier $($Grenoble$)$, vol. 54, iss. 7, pp. 2201-2234 (2005), 2004.
    @article {adam, MRKEY = {2139693},
      AUTHOR = {Adamczewski, Boris},
      TITLE = {Symbolic discrepancy and self-similar dynamics},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {54},
      YEAR = {2004},
      NUMBER = {7},
      PAGES = {2201--2234 (2005)},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {11K38 (37A30 37A45)},
      MRNUMBER = {2139693},
      MRREVIEWER = {Jean-Paul Allouche},
      DOI = {10.5802/aif.2079},
      ZBLNUMBER = {1066.11032},
     }
  • [arnouxfisher] Go to document P. Arnoux and A. M. Fisher, "Anosov families, renormalization and non-stationary subshifts," Ergodic Theory Dynam. Systems, vol. 25, iss. 3, pp. 661-709, 2005.
    @article {arnouxfisher, MRKEY = {2142941},
      AUTHOR = {Arnoux, Pierre and Fisher, Albert M.},
      TITLE = {Anosov families, renormalization and non-stationary subshifts},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {25},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {661--709},
      ISSN = {0143-3857},
      MRCLASS = {37D20 (37B55)},
      MRNUMBER = {2142941},
      MRREVIEWER = {Vadim A. Ka{\u\i}manovich},
      DOI = {10.1017/S0143385704000641},
      ZBLNUMBER = {1140.37314},
     }
  • [bufetov-susp] A. I. Bufetov, Suspension flows over Vershik’s automorphisms.
    @misc{bufetov-susp,
      author={Bufetov, Alexander I.},
      TITLE={Suspension flows over {V}ershik's automorphisms},
      ARXIV={0804.3970},
      SORTYEAR={2009},
     }
  • [bufetov-finadd] A. I. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum.
    @misc{bufetov-finadd,
      author={Bufetov, Alexander I.},
      TITLE={Finitely-additive measures on the asymptotic foliations of a {M}arkov compactum},
      ARXIV={0902.3303},
      SORTYEAR={2010},
     }
  • [fisher] Go to document A. M. Fisher, "Nonstationary mixing and the unique ergodicity of adic transformations," Stoch. Dyn., vol. 9, iss. 3, pp. 335-391, 2009.
    @article {fisher, MRKEY = {2566907},
      AUTHOR = {Fisher, Albert M.},
      TITLE = {Nonstationary mixing and the unique ergodicity of adic transformations},
      JOURNAL = {Stoch. Dyn.},
      FJOURNAL = {Stochastics and Dynamics},
      VOLUME = {9},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {335--391},
      ISSN = {0219-4937},
      MRCLASS = {37B10 (28D05 37A05)},
      MRNUMBER = {2566907},
      MRREVIEWER = {Jean-Ren{é} Chazottes},
      DOI = {10.1142/S0219493709002701},
      ZBLNUMBER = {1231.28014},
     }
  • [gjerde] R. Gjerde and &. Johansen, "Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations," Math. Scand., vol. 90, iss. 1, pp. 87-100, 2002.
    @article {gjerde, MRKEY = {1887096},
      AUTHOR = {Gjerde, Richard and Johansen, {\O}rjan},
      TITLE = {Bratteli-{V}ershik models for {C}antor minimal systems associated to interval exchange transformations},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {90},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {87--100},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {37B10 (37A45)},
      MRNUMBER = {1887096},
      MRREVIEWER = {Peter G. Danilaev},
      ZBLNUMBER = {1019.37006},
      }
  • [sinai] Go to document . Y. G. Sinai and C. Ulcigrai, "Weak mixing in interval exchange transformations of periodic type," Lett. Math. Phys., vol. 74, iss. 2, pp. 111-133, 2005.
    @article {sinai, MRKEY = {2191950},
      AUTHOR = {Sinai, {\relax Ya} G. and Ulcigrai, C.},
      TITLE = {Weak mixing in interval exchange transformations of periodic type},
      JOURNAL = {Lett. Math. Phys.},
      FJOURNAL = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
      VOLUME = {74},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {111--133},
      ISSN = {0377-9017},
      CODEN = {LMPHDY},
      MRCLASS = {37A05 (37A25 37A30 37E05)},
      MRNUMBER = {2191950},
      MRREVIEWER = {Maria E. Saprykina},
      DOI = {10.1007/s11005-005-0011-0},
      ZBLNUMBER = {1105.37002},
     }
  • [solomyak] B. Solomyak, "On the spectral theory of adic transformations," in Representation Theory and Dynamical Systems, Providence, RI: Amer. Math. Soc., 1992, vol. 9, pp. 217-230.
    @incollection {solomyak, MRKEY = {1166205},
      AUTHOR = {Solomyak, B.},
      TITLE = {On the spectral theory of adic transformations},
      BOOKTITLE = {Representation Theory and Dynamical Systems},
      SERIES = {Adv. Soviet Math.},
      VOLUME = {9},
      PAGES = {217--230},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1992},
      MRCLASS = {28D05 (47A35)},
      MRNUMBER = {1166205},
      MRREVIEWER = {Nathaniel Friedman},
      ZBLNUMBER = {0770.28012},
     }
  • [veechamj3] Go to document W. A. Veech, "The metric theory of interval exchange transformations. III. The Sah-Arnoux-Fathi invariant," Amer. J. Math., vol. 106, iss. 6, pp. 1389-1422, 1984.
    @article {veechamj3, MRKEY = {0765584},
      AUTHOR = {Veech, William A.},
      TITLE = {The metric theory of interval exchange transformations. {III}. {T}he {S}ah-{A}rnoux-{F}athi invariant},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {106},
      YEAR = {1984},
      NUMBER = {6},
      PAGES = {1389--1422},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {28D05 (58D20 58F11)},
      MRNUMBER = {0765584},
      MRREVIEWER = {Andr{é}s del Junco},
      DOI = {10.2307/2374398},
      ZBLNUMBER = {0631.28008},
     }
  • [Vershik1] A. M. Vershik, "A theorem on Markov periodic approximation in ergodic theory," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$, vol. 115, pp. 72-82, 306, 1982.
    @article {Vershik1, MRKEY = {0660072},
      AUTHOR = {Vershik, A. M.},
      TITLE = {A theorem on {M}arkov periodic approximation in ergodic theory},
      JOURNAL = {Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$},
      FJOURNAL = {Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)},
      VOLUME = {115},
      YEAR = {1982},
      PAGES = {72--82, 306},
      ISSN = {0206-8540},
      MRCLASS = {28D05 (47A35)},
      MRNUMBER = {0660072},
      MRREVIEWER = {L{á}szl{ó} Zsid{ó}},
      ZBLNUMBER = {0517.47005},
     }
  • [Vershik2] Go to document A. M. Vershik, "The adic realizations of the ergodic actions with the homeomorphisms of the Markov compact and the ordered Bratteli diagrams," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$, vol. 223, iss. Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I, pp. 120-126, 338, 1995.
    @article {Vershik2, MRKEY = {1374314},
      AUTHOR = {Vershik, A. M.},
      TITLE = {The adic realizations of the ergodic actions with the homeomorphisms of the {M}arkov compact and the ordered {B}ratteli diagrams},
      JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. MatematicheskiĭInstitut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
      VOLUME = {223},
      YEAR = {1995},
      NUMBER = {Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I},
      PAGES = {120--126, 338},
      ISSN = {0373-2703},
      MRCLASS = {28D05 (46L55 46L80)},
      MRNUMBER = {1374314},
      MRREVIEWER = {A. I. Danilenko},
      DOI = {10.1007/BF02355799},
      ZBLNUMBER = {0884.28013},
     }
  • [VL] A. M. Vershik and A. N. Livshits, "Adic models of ergodic transformations, spectral theory, substitutions, and related topics," in Representation Theory and Dynamical Systems, Providence, RI: Amer. Math. Soc., 1992, vol. 9, pp. 185-204.
    @incollection {VL, MRKEY = {1166202},
      AUTHOR = {Vershik, A. M. and Livshits, A. N.},
      TITLE = {Adic models of ergodic transformations, spectral theory, substitutions, and related topics},
      BOOKTITLE = {Representation Theory and Dynamical Systems},
      SERIES = {Adv. Soviet Math.},
      VOLUME = {9},
      PAGES = {185--204},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1992},
      MRCLASS = {46L89 (28D05 47A35)},
      MRNUMBER = {1166202},
      ZBLNUMBER = {0770.28013},
     }
  • [AV] Go to document A. Avila and M. Viana, "Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture," Acta Math., vol. 198, iss. 1, pp. 1-56, 2007.
    @article {AV, MRKEY = {2316268},
      AUTHOR = {Avila, Artur and Viana, Marcelo},
      TITLE = {Simplicity of {L}yapunov spectra: proof of the {Z}orich-{K}ontsevich conjecture},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {198},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {1--56},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {37A20 (28D05 32G15 37B99 37E99)},
      MRNUMBER = {2316268},
      MRREVIEWER = {Athanase Papadopoulos},
      DOI = {10.1007/s11511-007-0012-1},
      ZBLNUMBER = {1143.37001},
     }
  • [aranson] Go to document S. H. Aranson and V. Z. Grines, "Certain invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive systems)," Mat. Sb., vol. 90(132), pp. 372-402, 479, 1973.
    @article {aranson, MRKEY = {0339275},
      AUTHOR = {Aranson, S. H. and Grines, V. Z.},
      TITLE = {Certain invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive systems)},
      JOURNAL = {Mat. Sb.},
      VOLUME = {90(132)},
      YEAR = {1973},
      PAGES = {372--402, 479},
      MRCLASS = {58F10 (54H20)},
      MRNUMBER = {0339275},
      MRREVIEWER = {N. G. Markley},
      ZBLNUMBER = {0252.54027},
      DOI = {10.1070/SM1973v019n03ABEH001784},
      }
  • [pesinbarreira] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge: Cambridge Univ. Press, 2007, vol. 115.
    @book {pesinbarreira, MRKEY = {2348606},
      AUTHOR = {Barreira, Luis and Pesin, Yakov},
      TITLE = {Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents},
      SERIES = {Encyclopedia Math. Appl.},
      VOLUME = {115},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      PAGES = {xiv+513},
      ISBN = {978-0-521-83258-8; 0-521-83258-6},
      MRCLASS = {37D25 (34D08 37A25 37C40 37D10)},
      MRNUMBER = {2348606},
      MRREVIEWER = {Ian Melbourne},
      ZBLNUMBER = {1144.37002},
     }
  • [billingsley] Go to document P. Billingsley, Convergence of Probability Measures, Second ed., New York: John Wiley & Sons, 1999.
    @book {billingsley, MRKEY = {1700749},
      AUTHOR = {Billingsley, Patrick},
      TITLE = {Convergence of Probability Measures},
      SERIES = {Wiley Series Prob. Stat.},
      EDITION = {Second},
      NOTE = {A Wiley-Interscience Publication},
      PUBLISHER = {John Wiley \& Sons},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {x+277},
      ISBN = {0-471-19745-9},
      MRCLASS = {60B10 (28A33 60F17)},
      MRNUMBER = {1700749},
      DOI = {10.1002/9780470316962},
      ZBLNUMBER = {0944.60003},
     }
  • [bogachev] Go to document V. I. Bogachev, Measure Theory. Vol. I, II, New York: Springer-Verlag, 2007.
    @book {bogachev, MRKEY = {2267655},
      AUTHOR = {Bogachev, V. I.},
      TITLE = {Measure Theory. {V}ol. {I},
      {II}},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2007},
      PAGES = {Vol. I: xviii+500 pp., Vol. II: xiv+575},
      ISBN = {978-3-540-34513-8; 3-540-34513-2},
      MRCLASS = {28-02 (28Axx 28Cxx 46G12 60G42 60G44)},
      MRNUMBER = {2267655},
      MRREVIEWER = {Ren{é} L. Schilling},
      DOI = {10.1007/978-3-540-34514-5},
      ZBLNUMBER = {1120.28001},
     }
  • [bonahon1] Go to document F. Bonahon, "Transverse Hölder distributions for geodesic laminations," Topology, vol. 36, iss. 1, pp. 103-122, 1997.
    @article {bonahon1, MRKEY = {1410466},
      AUTHOR = {Bonahon, Francis},
      TITLE = {Transverse {H}ölder distributions for geodesic laminations},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {36},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {103--122},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {57M50 (46F99 57N05)},
      MRNUMBER = {1410466},
      MRREVIEWER = {Mark Brittenham},
      DOI = {10.1016/0040-9383(96)00001-8},
      ZBLNUMBER = {0871.57027},
     }
  • [bonahon2] Go to document F. Bonahon, "Geodesic laminations with transverse Hölder distributions," Ann. Sci. École Norm. Sup., vol. 30, iss. 2, pp. 205-240, 1997.
    @article {bonahon2, MRKEY = {1432054},
      AUTHOR = {Bonahon, Francis},
      TITLE = {Geodesic laminations with transverse {H}ölder distributions},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {30},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {205--240},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {57N05 (57R30)},
      MRNUMBER = {1432054},
      MRREVIEWER = {Athanase Papadopoulos},
      DOI = {10.1016/S0012-9593(97)89919-3},
      ZBLNUMBER = {0887.57018},
     }
  • [bhm] Go to document X. Bressaud, P. Hubert, and A. Maass, "Persistence of wandering intervals in self-similar affine interval exchange transformations," Ergodic Theory Dynam. Systems, vol. 30, iss. 3, pp. 665-686, 2010.
    @article {bhm, MRKEY = {2643707},
      AUTHOR = {Bressaud, Xavier and Hubert, Pascal and Maass, Alejandro},
      TITLE = {Persistence of wandering intervals in self-similar affine interval exchange transformations},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {30},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {665--686},
      ISSN = {0143-3857},
      MRCLASS = {37A05 (26A18 28D05 37E05)},
      MRNUMBER = {2643707},
      MRREVIEWER = {Jan Kwiatkowski},
      DOI = {10.1017/S0143385709000418},
      ZBLNUMBER = {1200.37002},
     }
  • [bufetov] Go to document A. I. Bufetov, "Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials," J. Amer. Math. Soc., vol. 19, iss. 3, pp. 579-623, 2006.
    @article {bufetov, MRKEY = {2220100},
      AUTHOR = {Bufetov, Alexander I.},
      TITLE = {Decay of correlations for the {R}auzy-{V}eech-{Z}orich induction map on the space of interval exchange transformations and the central limit theorem for the {T}eichmüller flow on the moduli space of abelian differentials},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {19},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {579--623},
      ISSN = {0894-0347},
      MRCLASS = {37A25 (32G15 37D40 37E05 37F30 60F05)},
      MRNUMBER = {2220100},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.1090/S0894-0347-06-00528-5},
      ZBLNUMBER = {1100.37002},
     }
  • [bufetov-lthm] A. I. Bufetov, Limit theorems for translation flows.
    @misc{bufetov-lthm,
      author={Bufetov, Alexander I.},
      TITLE={Limit theorems for translation flows},
      ARXIV={0804.3970},
      SORTYEAR={2011},
     }
  • [bufgur] Go to document A. I. Bufetov and B. M. Gurevich, "Existence and uniqueness of a measure with maximal entropy for the Teichmüller flow on the moduli space of abelian differentials," Mat. Sb., vol. 202, iss. 7, pp. 3-42, 2011.
    @article {bufgur, MRKEY = {2857792},
      AUTHOR = {Bufetov, Alexander I. and Gurevich, B. M.},
      TITLE = {Existence and uniqueness of a measure with maximal entropy for the {T}eichmüller flow on the moduli space of abelian differentials},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. MatematicheskiĭSbornik},
      VOLUME = {202},
      YEAR = {2011},
      NUMBER = {7},
      PAGES = {3--42},
      ISSN = {0368-8666},
      MRCLASS = {37D35 (28D20 37D40 37E35)},
      MRNUMBER = {2857792},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1070/SM2011v202n07ABEH004172},
      ZBLNUMBER = {1241.28011},
     }
  • [dumontkamae] Go to document J. Dumont, T. Kamae, and S. Takahashi, "Minimal cocycles with the scaling property and substitutions," Israel J. Math., vol. 95, pp. 393-410, 1996.
    @article {dumontkamae, MRKEY = {1418302},
      AUTHOR = {Dumont, Jean-Marie and Kamae, Teturo and Takahashi, Satoshi},
      TITLE = {Minimal cocycles with the scaling property and substitutions},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {95},
      YEAR = {1996},
      PAGES = {393--410},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {58F25 (11A67 11B85 28A80 54H20 58F03)},
      MRNUMBER = {1418302},
      MRREVIEWER = {Karl Petersen},
      DOI = {10.1007/BF02761048},
      ZBLNUMBER = {0866.54033},
     }
  • [EKZ] Go to document A. Eskin, M. Kontsevich, and A. Zorich, "Lyapunov spectrum of square-tiled cyclic covers," J. Mod. Dyn., vol. 5, iss. 2, pp. 319-353, 2011.
    @article {EKZ, MRKEY = {2820564},
      AUTHOR = {Eskin, Alex and Kontsevich, Maxim and Zorich, Anton},
      TITLE = {Lyapunov spectrum of square-tiled cyclic covers},
      JOURNAL = {J. Mod. Dyn.},
      FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {5},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {319--353},
      ISSN = {1930-5311},
      MRCLASS = {37D40 (14H30 30F30 32G15 32G20 37D25)},
      MRNUMBER = {2820564},
      MRREVIEWER = {Lee-Peng Teo},
      DOI = {10.3934/jmd.2011.5.319},
      ZBLNUMBER = {1254.32019},
     }
  • [flafo] Go to document L. Flaminio and G. Forni, "Invariant distributions and time averages for horocycle flows," Duke Math. J., vol. 119, iss. 3, pp. 465-526, 2003.
    @article {flafo, MRKEY = {2003124},
      AUTHOR = {Flaminio, Livio and Forni, Giovanni},
      TITLE = {Invariant distributions and time averages for horocycle flows},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {119},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {465--526},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {37D40 (22E46 37A20 58J50)},
      MRNUMBER = {2003124},
      MRREVIEWER = {Boris Hasselblatt},
      DOI = {10.1215/S0012-7094-03-11932-8},
      ZBLNUMBER = {1044.37017},
     }
  • [F1] Go to document G. Forni, "Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus," Ann. of Math., vol. 146, iss. 2, pp. 295-344, 1997.
    @article {F1, MRKEY = {1477760},
      AUTHOR = {Forni, Giovanni},
      TITLE = {Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {146},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {295--344},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F11 (58F17)},
      MRNUMBER = {1477760},
      MRREVIEWER = {Boris Hasselblatt},
      DOI = {10.2307/2952464},
      ZBLNUMBER = {0893.58037},
     }
  • [F2] 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 {F2, 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},
     }
  • [forniarxiv] G. Forni, Sobolev regularity of solutions of the cohomological equation.
    @misc{forniarxiv,
      author={Forni, Giovanni},
      TITLE = {Sobolev regularity of solutions of the cohomological equation},
      ARXIV={0707.0940},
      SORTYEAR={2015},
     }
  • [forni-nonunif] Go to document G. Forni, "A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle," J. Mod. Dyn., vol. 5, iss. 2, pp. 355-395, 2011.
    @article {forni-nonunif, MRKEY = {2820565},
      AUTHOR = {Forni, Giovanni},
      TITLE = {A geometric criterion for the nonuniform hyperbolicity of the {K}ontsevich-{Z}orich cocycle},
      JOURNAL = {J. Mod. Dyn.},
      FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {5},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {355--395},
      ISSN = {1930-5311},
      MRCLASS = {37D25 (14H30 30F30 32G15 32G20)},
      MRNUMBER = {2820565},
      MRREVIEWER = {Lee-Peng Teo},
      DOI = {10.3934/jmd.2011.5.355},
      ZBLNUMBER = {1238.30031},
     }
  • [FMZ] G. Forni, C. Matheus, and A. Zorich, Zero Lyapunov exponents of the Hodge bundle.
    @misc{FMZ,
      author={Forni, Giovanni and Matheus, C. and Zorich, A.},
      TITLE={Zero {L}yapunov exponents of the {H}odge bundle},
      ARXIV= {1201.6075},
     }
  • [ito] Go to document S. Ito, "A construction of transversal flows for maximal Markov automorphisms," Tokyo J. Math., vol. 1, iss. 2, pp. 305-324, 1978.
    @article {ito, MRKEY = {0519199},
      AUTHOR = {Ito, Shunji},
      TITLE = {A construction of transversal flows for maximal {M}arkov automorphisms},
      JOURNAL = {Tokyo J. Math.},
      FJOURNAL = {Tokyo Journal of Mathematics},
      VOLUME = {1},
      YEAR = {1978},
      NUMBER = {2},
      PAGES = {305--324},
      ISSN = {0387-3870},
      MRCLASS = {28D05 (28D20 58F25)},
      MRNUMBER = {0519199},
      MRREVIEWER = {R. L. Adler},
      DOI = {10.3836/tjm/1270216501},
      ZBLNUMBER = {0446.28017},
     }
  • [kamae] Go to document T. Kamae, "Linear expansions, strictly ergodic homogeneous cocycles and fractals," Israel J. Math., vol. 106, pp. 313-337, 1998.
    @article {kamae, MRKEY = {1656897},
      AUTHOR = {Kamae, Teturo},
      TITLE = {Linear expansions, strictly ergodic homogeneous cocycles and fractals},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {106},
      YEAR = {1998},
      PAGES = {313--337},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {28D15 (28A80)},
      MRNUMBER = {1656897},
      MRREVIEWER = {Run Sheng Yang},
      DOI = {10.1007/BF02773474},
      ZBLNUMBER = {0914.28014},
     }
  • [katok] A. B. Katok, "Invariant measures of flows on orientable surfaces," Dokl. Akad. Nauk SSSR, vol. 211, pp. 775-778, 1973.
    @article {katok, MRKEY = {0331438},
      AUTHOR = {Katok, A. B.},
      TITLE = {Invariant measures of flows on orientable surfaces},
      JOURNAL = {Dokl. Akad. Nauk SSSR},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {211},
      YEAR = {1973},
      PAGES = {775--778},
      ISSN = {0002-3264},
      MRCLASS = {58F99},
      MRNUMBER = {0331438},
      MRREVIEWER = {D. Newton},
      ZBLNUMBER = {0298.28013},
     }
  • [kenyon] Go to document R. Kenyon, "A group of paths in ${\bf R}^2$," Trans. Amer. Math. Soc., vol. 348, iss. 8, pp. 3155-3172, 1996.
    @article {kenyon, MRKEY = {1340179},
      AUTHOR = {Kenyon, Richard},
      TITLE = {A group of paths in {${\bf R}\sp 2$}},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {348},
      YEAR = {1996},
      NUMBER = {8},
      PAGES = {3155--3172},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {20F34 (20E08 58F03)},
      MRNUMBER = {1340179},
      MRREVIEWER = {Lee Mosher},
      DOI = {10.1090/S0002-9947-96-01562-0},
      ZBLNUMBER = {0896.52024},
     }
  • [kz] Go to document M. Kontsevich and A. Zorich, "Connected components of the moduli spaces of Abelian differentials with prescribed singularities," Invent. Math., vol. 153, iss. 3, pp. 631-678, 2003.
    @article {kz, MRKEY = {2000471},
      AUTHOR = {Kontsevich, Maxim and Zorich, Anton},
      TITLE = {Connected components of the moduli spaces of {A}belian differentials with prescribed singularities},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {153},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {631--678},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32G15 (37D40 37D50 37F99)},
      MRNUMBER = {2000471},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.1007/s00222-003-0303-x},
      ZBLNUMBER = {1087.32010},
     }
  • [kz1] M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, 2007.
    @misc{kz1,
      author = {Kontsevich, Maxim and Zorich, Anton},
      TITLE = {Lyapunov exponents and {H}odge theory},
      YEAR={2007},
      ARXIV={hep-th/9701164},
     }
  • [MMY] Go to document S. Marmi, P. Moussa, and J. -C. Yoccoz, "The cohomological equation for Roth-type interval exchange maps," J. Amer. Math. Soc., vol. 18, iss. 4, pp. 823-872, 2005.
    @article {MMY, MRKEY = {2163864},
      AUTHOR = {Marmi, S. and Moussa, P. and Yoccoz, J.-C.},
      TITLE = {The cohomological equation for {R}oth-type interval exchange maps},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {18},
      YEAR = {2005},
      NUMBER = {4},
      PAGES = {823--872},
      ISSN = {0894-0347},
      MRCLASS = {37A20 (11K50 37A45 37C15 37E05 37E35)},
      MRNUMBER = {2163864},
      MRREVIEWER = {Giovanni Forni},
      DOI = {10.1090/S0894-0347-05-00490-X},
      ZBLNUMBER = {1112.37002},
     }
  • [MMY2] Go to document S. Marmi, P. Moussa, and J. -C. Yoccoz, "Affine interval exchange maps with a wandering interval," Proc. Lond. Math. Soc., vol. 100, iss. 3, pp. 639-669, 2010.
    @article {MMY2, MRKEY = {2640286},
      AUTHOR = {Marmi, S. and Moussa, P. and Yoccoz, J.-C.},
      TITLE = {Affine interval exchange maps with a wandering interval},
      JOURNAL = {Proc. Lond. Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {100},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {639--669},
      ISSN = {0024-6115},
      MRCLASS = {37C15 (11K50 28D05 37A25 37A45 37E05)},
      MRNUMBER = {2640286},
      MRREVIEWER = {Jan Kwiatkowski},
      DOI = {10.1112/plms/pdp037},
      ZBLNUMBER = {1196.37041},
     }
  • [masur] Go to document H. Masur, "Interval exchange transformations and measured foliations," Ann. of Math., vol. 115, iss. 1, pp. 169-200, 1982.
    @article {masur, 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},
     }
  • [veech] 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 {veech, 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},
     }
  • [veechamj] Go to document W. A. Veech, "The metric theory of interval exchange transformations. I. Generic spectral properties," Amer. J. Math., vol. 106, iss. 6, pp. 1331-1359, 1984.
    @article {veechamj, MRKEY = {0765582},
      AUTHOR = {Veech, William A.},
      TITLE = {The metric theory of interval exchange transformations. {I}. {G}eneric spectral properties},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {106},
      YEAR = {1984},
      NUMBER = {6},
      PAGES = {1331--1359},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {28D05 (28D20 58F11)},
      MRNUMBER = {0765582},
      MRREVIEWER = {Andr{é}s del Junco},
      DOI = {10.2307/2374396},
      ZBLNUMBER = {0631.28006},
     }
  • [veechamj2] Go to document W. A. Veech, "The metric theory of interval exchange transformations. II. Approximation by primitive interval exchanges," Amer. J. Math., vol. 106, iss. 6, pp. 1361-1387, 1984.
    @article {veechamj2, MRKEY = {0765583},
      AUTHOR = {Veech, William A.},
      TITLE = {The metric theory of interval exchange transformations. {II}. {A}pproximation by primitive interval exchanges},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {106},
      YEAR = {1984},
      NUMBER = {6},
      PAGES = {1361--1387},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {28D05 (58D20 58F11)},
      MRNUMBER = {0765583},
      MRREVIEWER = {Andr{é}s del Junco},
      DOI = {10.2307/2374397},
      ZBLNUMBER = {0631.28007},
     }
  • [V2] Go to document W. A. Veech, "The Teichmüller geodesic flow," Ann. of Math., vol. 124, iss. 3, pp. 441-530, 1986.
    @article {V2, MRKEY = {0866707},
      AUTHOR = {Veech, William A.},
      TITLE = {The {T}eichmüller geodesic flow},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {124},
      YEAR = {1986},
      NUMBER = {3},
      PAGES = {441--530},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F17 (32G15 58F11)},
      MRNUMBER = {0866707},
      MRREVIEWER = {M. Rees},
      DOI = {10.2307/2007091},
      ZBLNUMBER = {0658.32016},
     }
  • [V3] Go to document W. A. Veech, "Moduli spaces of quadratic differentials," J. Analyse Math., vol. 55, pp. 117-171, 1990.
    @article {V3, MRKEY = {1094714},
      AUTHOR = {Veech, William A.},
      TITLE = {Moduli spaces of quadratic differentials},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {55},
      YEAR = {1990},
      PAGES = {117--171},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {32G15 (30F60)},
      MRNUMBER = {1094714},
      MRREVIEWER = {Subhashis Nag},
      DOI = {10.1007/BF02789200},
      ZBLNUMBER = {0722.30032},
     }
  • [Z] Go to document A. Zorich, "Deviation for interval exchange transformations," Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1477-1499, 1997.
    @article {Z, 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},
     }
  • [zorichwind] 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 {zorichwind, 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

Alexander I. Bufetov

Laboratoire d'Analyse, Topologie, Probabilités, Aix-Marseille Université, CNRS, the Steklov Institute of Mathematics, the Institute for Information Transmission Problems, National Research University Higher School of Economics, the Independent University of Moscow, Rice University