Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow

Abstract

We consider the $\mathrm{SL}(2, \mathbb{R})$ action on moduli spaces of quadratic differentials. If $\mu$ is an $\mathrm{SL}(2, \mathbb{R})$-invariant probability measure, crucial information about the associated representation on $L^2(\mu)$ (and, in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $\!(0,1/4)\!$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$; i.e., for every $\delta\!>\!0$, there are only finitely many eigenvalues (counted with multiplicity) in $(0,1/4\!-\!\delta)$. In particular, all algebraic invariant measures have a spectral gap.

  • [athreya_et_al] Go to document J. Athreya, A. Bufetov, A. Eskin, and M. Mirzakhani, "Lattice point asymptotics and volume growth on Teichmüller space," Duke Math. J., vol. 161, iss. 6, pp. 1055-1111, 2012.
    @article {athreya_et_al, MRKEY = {2913101},
      AUTHOR = {Athreya, Jayadev and Bufetov, Alexander and Eskin, Alex and Mirzakhani, Maryam},
      TITLE = {Lattice point asymptotics and volume growth on {T}eichmüller space},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {161},
      YEAR = {2012},
      NUMBER = {6},
      PAGES = {1055--1111},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {32G15 (30F60 37D40)},
      MRNUMBER = {2913101},
      MRREVIEWER = {Lee-Peng Teo},
      DOI = {10.1215/00127094-1548443},
      ZBLNUMBER = {06029038},
     }
  • [athreya] Go to document J. S. Athreya, "Quantitative recurrence and large deviations for Teichmuller geodesic flow," Geom. Dedicata, vol. 119, pp. 121-140, 2006.
    @article {athreya, MRKEY = {2247652},
      AUTHOR = {Athreya, Jayadev S.},
      TITLE = {Quantitative recurrence and large deviations for {T}eichmuller geodesic flow},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {119},
      YEAR = {2006},
      PAGES = {121--140},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {32G15 (30F60 37A10 37D40)},
      MRNUMBER = {2247652},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1007/s10711-006-9058-z},
      ZBLNUMBER = {1108.32007},
     }
  • [AGY_teich] Go to document A. Avila, S. Gouëzel, and J. Yoccoz, "Exponential mixing for the Teichmüller flow," Publ. Math. Inst. Hautes Études Sci., vol. 104, pp. 143-211, 2006.
    @article {AGY_teich, 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},
      VOLUME = {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 = {05117096},
     }
  • [avila_resende] Go to document A. Avila and M. J. Resende, "Exponential mixing for the Teichmüller flow in the space of quadratic differentials," Comment. Math. Helv., vol. 87, iss. 3, pp. 589-638, 2012.
    @article {avila_resende, MRKEY = {2980521},
      AUTHOR = {Avila, Artur and Resende, Maria Jo{ã}o},
      TITLE = {Exponential mixing for the {T}eichmüller flow in the space of quadratic differentials},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      VOLUME = {87},
      YEAR = {2012},
      NUMBER = {3},
      PAGES = {589--638},
      ISSN = {0010-2571},
      MRCLASS = {37Dxx (32G15)},
      MRNUMBER = {2980521},
      DOI = {10.4171/CMH/263},
      ZBLNUMBER = {06067460},
     }
  • [babillot_ledrappier] M. Babillot and F. Ledrappier, "Geodesic paths and horocycle flow on abelian covers," in Lie Groups and Ergodic Theory, Bombay: Tata Inst. Fund. Res., 1998, vol. 14, pp. 1-32.
    @incollection {babillot_ledrappier, MRKEY = {1699356},
      AUTHOR = {Babillot, Martine and Ledrappier, Fran{ç}ois},
      TITLE = {Geodesic paths and horocycle flow on abelian covers},
      BOOKTITLE = {Lie Groups and Ergodic Theory},
      VENUE={{M}umbai, 1996},
      SERIES = {Tata Inst. Fund. Res. Stud. Math.},
      VOLUME = {14},
      PAGES = {1--32},
      PUBLISHER = {Tata Inst. Fund. Res.},
      ADDRESS = {Bombay},
      YEAR = {1998},
      MRCLASS = {37D40 (37C40)},
      MRNUMBER = {1699356},
      MRREVIEWER = {Gerhard Knieper},
      ZBLNUMBER = {0967.37020},
      }
  • [bt_aniso] Go to document V. Baladi and M. Tsujii, "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms," Ann. Inst. Fourier $($Grenoble$)$, vol. 57, iss. 1, pp. 127-154, 2007.
    @article {bt_aniso, MRKEY = {2313087},
      AUTHOR = {Baladi, Viviane and Tsujii, Masato},
      TITLE = {Anisotropic {H}ölder and {S}obolev spaces for hyperbolic diffeomorphisms},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {57},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {127--154},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {37C30 (37D20 46E35)},
      MRNUMBER = {2313087},
      MRREVIEWER = {M. L. Blank},
      DOI = {10.5802/aif.2253},
      ZBLNUMBER = {1138.37011},
      }
  • [CR_book] S. M. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings, Princeton, NJ: Princeton Univ. Press, 1999, vol. 47.
    @book {CR_book, MRKEY = {1668103},
      AUTHOR = {Baouendi, M. Salah and Ebenfelt, Peter and Rothschild, Linda Preiss},
      TITLE = {Real Submanifolds in Complex Space and Their Mappings},
      SERIES = {Princeton Mathematical Series},
      VOLUME = {47},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1999},
      PAGES = {xii+404},
      ISBN = {0-691-00498-6},
      MRCLASS = {32Vxx},
      MRNUMBER = {1668103},
      MRREVIEWER = {Elisabetta Barletta},
      ZBLNUMBER = {0944.32040},
      }
  • [BGK_coupling] Go to document J. Bardet, S. Gouëzel, and G. Keller, "Limit theorems for coupled interval maps," Stoch. Dyn., vol. 7, iss. 1, pp. 17-36, 2007.
    @article {BGK_coupling, MRKEY = {2293068},
      AUTHOR = {Bardet, Jean-Baptiste and Gou{ë}zel, S{é}bastien and Keller, Gerhard},
      TITLE = {Limit theorems for coupled interval maps},
      JOURNAL = {Stoch. Dyn.},
      FJOURNAL = {Stochastics and Dynamics},
      VOLUME = {7},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {17--36},
      ISSN = {0219-4937},
      MRCLASS = {37L60 (37A50 37C40 37D20 37D35 60F05)},
      MRNUMBER = {2293068},
      MRREVIEWER = {Miaohua Jiang},
      DOI = {10.1142/S0219493707001913},
      ZBLNUMBER = {1114.37006},
      }
  • [butterley_liverani] Go to document O. Butterley and C. Liverani, "Smooth Anosov flows: correlation spectra and stability," J. Mod. Dyn., vol. 1, iss. 2, pp. 301-322, 2007.
    @article {butterley_liverani, MRKEY = {2285731},
      AUTHOR = {Butterley, Oliver and Liverani, Carlangelo},
      TITLE = {Smooth {A}nosov flows: correlation spectra and stability},
      JOURNAL = {J. Mod. Dyn.},
      FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {1},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {301--322},
      ISSN = {1930-5311},
      MRCLASS = {37D30 (37C30 37M25)},
      MRNUMBER = {2285731},
      MRREVIEWER = {Hans Henrik Rugh},
      DOI = {10.3934/jmd.2007.1.301},
      ZBLNUMBER = {1144.37011},
      }
  • [casselman_milicic] Go to document W. Casselman and D. Milivcić, "Asymptotic behavior of matrix coefficients of admissible representations," Duke Math. J., vol. 49, iss. 4, pp. 869-930, 1982.
    @article {casselman_milicic, MRKEY = {0683007},
      AUTHOR = {Casselman, William and Mili{\v{c}}i{ć},
      Dragan},
      TITLE = {Asymptotic behavior of matrix coefficients of admissible representations},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {49},
      YEAR = {1982},
      NUMBER = {4},
      PAGES = {869--930},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E46 (22E45)},
      MRNUMBER = {0683007},
      MRREVIEWER = {Juan Tirao},
      DOI = {10.1215/S0012-7094-82-04943-2},
      ZBLNUMBER = {0524.22014},
      }
  • [dixmier] J. Dixmier, Les $C^{\ast} $-Algèbres et leurs Représentations, Gauthier-Villars Éditeur, Paris, 1969, vol. XXIX.
    @book {dixmier, MRKEY = {0246136},
      AUTHOR = {Dixmier, Jacques},
      TITLE = {Les {$C^{\ast} $}-Algèbres et leurs Représentations},
      SERIES = {Deuxième édition. Cahiers Scientifiques},
      VOLUME={XXIX},
      PUBLISHER = {Gauthier-Villars Éditeur, Paris},
      YEAR = {1969},
      PAGES = {xv+390},
      MRCLASS = {46.65},
      MRNUMBER = {0246136},
      ZBLNUMBER = {0174.18601},
      }
  • [every_curve_is_teich] Go to document J. S. Ellenberg and D. B. McReynolds, "Arithmetic Veech sublattices of ${ SL}(2,\Bbb Z)$," Duke Math. J., vol. 161, iss. 3, pp. 415-429, 2012.
    @article {every_curve_is_teich, MRKEY = {2881227},
      AUTHOR = {Ellenberg, Jordan S. and McReynolds, D. B.},
      TITLE = {Arithmetic {V}eech sublattices of {${\rm SL}(2,\bold Z)$}},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {161},
      YEAR = {2012},
      NUMBER = {3},
      PAGES = {415--429},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {37D40 (32G15)},
      MRNUMBER = {2881227},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1215/00127094-1507412},
      ZBLNUMBER = {1244.32009},
      }
  • [eskin_masur] Go to document A. Eskin and H. Masur, "Asymptotic formulas on flat surfaces," Ergodic Theory Dynam. Systems, vol. 21, iss. 2, pp. 443-478, 2001.
    @article {eskin_masur, MRKEY = {1827113},
      AUTHOR = {Eskin, Alex and Masur, Howard},
      TITLE = {Asymptotic formulas on flat surfaces},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {21},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {443--478},
      ISSN = {0143-3857},
      MRCLASS = {37F30 (37C35)},
      MRNUMBER = {1827113},
      MRREVIEWER = {Douglas P. Dokken},
      DOI = {10.1017/S0143385701001225},
      ZBLNUMBER = {1096.37501},
      }
  • [eskin_mirzakhani] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the ${ SL}(2,\mathbb{R})$ action on moduli space, 2013.
    @misc{eskin_mirzakhani,
      author = {Eskin, Alex and Mirzakhani, M.},
      TITLE={Invariant and stationary measures for the {${\rm SL}(2,\mathbb{R})$} action on moduli space},
      YEAR={2013},
      ARXIV= {1302.3320},
     }
  • [forni_deviation] 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 {forni_deviation, 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},
      }
  • [fried_analytique] Go to document D. Fried, "The zeta functions of Ruelle and Selberg. I," Ann. Sci. École Norm. Sup., vol. 19, iss. 4, pp. 491-517, 1986.
    @article {fried_analytique, MRKEY = {0875085},
      AUTHOR = {Fried, David},
      TITLE = {The zeta functions of {R}uelle and {S}elberg. {I}},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {19},
      YEAR = {1986},
      NUMBER = {4},
      PAGES = {491--517},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {58F20},
      MRNUMBER = {0875085},
      MRREVIEWER = {M. Rees},
      URL = {http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0},
      ZBLNUMBER = {0609.58033},
      }
  • [gouezel_liverani] Go to document S. Gouëzel and C. Liverani, "Banach spaces adapted to Anosov systems," Ergodic Theory Dynam. Systems, vol. 26, iss. 1, pp. 189-217, 2006.
    @article {gouezel_liverani, MRKEY = {2201945},
      AUTHOR = {Gou{ë}zel, S{é}bastien and Liverani, Carlangelo},
      TITLE = {Banach spaces adapted to {A}nosov systems},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {26},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {189--217},
      ISSN = {0143-3857},
      MRCLASS = {37D20 (37C30 46F05 47B38)},
      MRNUMBER = {2201945},
      MRREVIEWER = {Masato Tsujii},
      DOI = {10.1017/S0143385705000374},
      ZBLNUMBER = {1088.37010},
      }
  • [hejhal_2] D. A. Hejhal, The Selberg Trace Formula for ${ PSL}(2,\,{\bf R})$. Vol. 2, New York: Springer-Verlag, 1983, vol. 1001.
    @book {hejhal_2, MRKEY = {0711197},
      AUTHOR = {Hejhal, Dennis A.},
      TITLE = {The {S}elberg Trace Formula for {${\rm PSL}(2,\,{\bf R})$}. {V}ol. 2},
      SERIES = {Lecture Notes in Mathematics},
      VOLUME = {1001},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {viii+806},
      ISBN = {3-540-12323-7},
      MRCLASS = {11F72 (58G25)},
      MRNUMBER = {0711197},
      MRREVIEWER = {Audrey A. Terras},
      ZBLNUMBER = {0543.10020},
      }
  • [helgason2] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Providence, RI: Amer. Math. Soc., 2000, vol. 83.
    @book {helgason2, MRKEY = {1790156},
      AUTHOR = {Helgason, Sigurdur},
      TITLE = {Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {83},
      NOTE = {corrected reprint of the 1984 original},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2000},
      PAGES = {xxii+667},
      ISBN = {0-8218-2673-5},
      MRCLASS = {22-02 (22E30 22E46 43A85 43A90 44A12 53-02 58-02)},
      MRNUMBER = {1790156},
      ZBLNUMBER = {0965.43007},
      }
  • [hennion] Go to document H. Hennion, "Sur un théorème spectral et son application aux noyaux lipchitziens," Proc. Amer. Math. Soc., vol. 118, iss. 2, pp. 627-634, 1993.
    @article {hennion, MRKEY = {1129880},
      AUTHOR = {Hennion, Hubert},
      TITLE = {Sur un théorème spectral et son application aux noyaux lipchitziens},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {118},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {627--634},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {60J10 (28D99 47A35 47N30)},
      MRNUMBER = {1129880},
      MRREVIEWER = {Yves Derriennic},
      DOI = {10.2307/2160348},
      ZBLNUMBER = {0772.60049},
     }
  • [hormander] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Berlin: Springer-Verlag, 2003.
    @book {hormander, MRKEY = {MR1996773},
      AUTHOR = {H{ö}rmander, Lars},
      TITLE= {The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis},
      SERIES = {Classics in Mathematics},
      NOTE = {reprint of the second (1990) edition},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {Berlin},
      YEAR = {2003},
      PAGES = {x+440},
      ISBN = {3-540-00662-1},
      MRCLASS = {35-02},
      MRNUMBER = {1996773},
      ZBLNUMBER = {1028.35001},
     }
  • [howe_moore] Go to document R. E. Howe and C. C. Moore, "Asymptotic properties of unitary representations," J. Funct. Anal., vol. 32, iss. 1, pp. 72-96, 1979.
    @article {howe_moore, MRKEY = {0533220},
      AUTHOR = {Howe, Roger E. and Moore, Calvin C.},
      TITLE = {Asymptotic properties of unitary representations},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {32},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {72--96},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {22E50 (22D10 22E50 43A30 43A80)},
      MRNUMBER = {0533220},
      MRREVIEWER = {Jonathan M. Rosenberg},
      DOI = {10.1016/0022-1236(79)90078-8},
      ZBLNUMBER = {0404.22015},
      }
  • [Iwaniec] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Madrid: Revista Matemática Iberoamericana, 1995.
    @book {Iwaniec, MRKEY = {1325466},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Introduction to the Spectral Theory of Automorphic Forms},
      SERIES = {Bibl. Rev. Mat. Iberoamericana},
      PUBLISHER = {Revista Matemática Iberoamericana},
      ADDRESS = {Madrid},
      YEAR = {1995},
      PAGES = {xiv+247},
      MRCLASS = {11F72 (11-02 11F12 11F37 35Pxx 58G25)},
      MRNUMBER = {1325466},
      MRREVIEWER = {Solomon Friedberg},
      ZBLNUMBER = {0847.11028},
      }
  • [kato_pe] T. Kato, Perturbation Theory for Linear Operators, New York: Springer-Verlag, 1966, vol. 132.
    @book {kato_pe, MRKEY = {0203473},
      AUTHOR = {Kato, Tosio},
      TITLE = {Perturbation Theory for Linear Operators},
      SERIES = {Grundlehren Math. Wiss.},
      VOLUME={132},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1966},
      PAGES = {xix+592},
      MRCLASS = {47.00 (47.48)},
      MRNUMBER = {0203473},
      MRREVIEWER = {L. de Branges},
      ADDRESS = {New York},
      ZBLNUMBER = {0148.12601},
      }
  • [sarnak_strong_spectral_gap] Go to document D. Kelmer and P. Sarnak, "Strong spectral gaps for compact quotients of products of ${ PSL}(2,\Bbb R)$," J. Eur. Math. Soc. $($JEMS$)$, vol. 11, iss. 2, pp. 283-313, 2009.
    @article {sarnak_strong_spectral_gap, MRKEY = {2486935},
      AUTHOR = {Kelmer, Dubi and Sarnak, Peter},
      TITLE = {Strong spectral gaps for compact quotients of products of {${\rm PSL}(2,\Bbb R)$}},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {11},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {283--313},
      ISSN = {1435-9855},
      MRCLASS = {11F72 (11M36 22E45)},
      MRNUMBER = {2486935},
      MRREVIEWER = {Anton Deitmar},
      DOI = {10.4171/JEMS/151},
      ZBLNUMBER = {1254.11050},
      }
  • [knapp] A. W. Knapp, Representation Theory of Semisimple Groups, Princeton, NJ: Princeton Univ. Press, 2001.
    @book {knapp, MRKEY = {1880691},
      AUTHOR = {Knapp, Anthony W.},
      TITLE = {Representation Theory of Semisimple Groups},
      SERIES = {Princeton Landmarks in Math.},
      NOTE = {an overview based on examples, reprint of the 1986 original},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {2001},
      PAGES = {xx+773},
      ISBN = {0-691-09089-0},
      MRCLASS = {22E46 (22-01 22E30)},
      MRNUMBER = {1880691},
      ZBLNUMBER = {0993.22001},
      }
  • [Kontsevich_Zorich] 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 {Kontsevich_Zorich, 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},
      }
  • [Lanneau_connected] Go to document E. Lanneau, "Connected components of the strata of the moduli spaces of quadratic differentials," Ann. Sci. Éc. Norm. Supér., vol. 41, iss. 1, pp. 1-56, 2008.
    @article {Lanneau_connected, MRKEY = {2423309},
      AUTHOR = {Lanneau, Erwan},
      TITLE = {Connected components of the strata of the moduli spaces of quadratic differentials},
      JOURNAL = {Ann. Sci. Éc. Norm. Supér.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {41},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {1--56},
      ISSN = {0012-9593},
      MRCLASS = {30F30 (30F60 32G15 37D40 37F30 57M60)},
      MRNUMBER = {2423309},
      MRREVIEWER = {Athanase Papadopoulos},
      ZBLNUMBER = {1161.30033},
      URL = {http://smf4.emath.fr/Publications/AnnalesENS/4_41/html/ens_ann-sc_41_1-56.html}
    }
  • [liverani_contact] Go to document C. Liverani, "On contact Anosov flows," Ann. of Math., vol. 159, iss. 3, pp. 1275-1312, 2004.
    @article {liverani_contact, MRKEY = {2113022},
      AUTHOR = {Liverani, Carlangelo},
      TITLE = {On contact {A}nosov flows},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {159},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {1275--1312},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37D20 (37A99 37D40 37J60)},
      MRNUMBER = {2113022},
      MRREVIEWER = {M. L. Blank},
      DOI = {10.4007/annals.2004.159.1275},
      ZBLNUMBER = {1067.37031},
      }
  • [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},
      }
  • [matheus_schmithusen] C. Matheus and G. Weitze-Schmithüsen, Explicit Teichmüller curves with complementary series, 2011.
    @misc{matheus_schmithusen,
      author={Matheus, C. and Weitze-Schmith{ü}sen, G.},
      TITLE={Explicit {T}eichmüller curves with complementary series},
      NOTE={to appear in \emph{Bulletin de la SMF}},
      YEAR={2011},
      ARXIV={1109.0517},
     }
  • [mcmullen] Go to document C. T. McMullen, "Dynamics of ${ SL}_2(\Bbb R)$ over moduli space in genus two," Ann. of Math., vol. 165, iss. 2, pp. 397-456, 2007.
    @article {mcmullen, MRKEY = {2299738},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Dynamics of {${\rm SL}\sb 2(\Bbb R)$} over moduli space in genus two},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {165},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {397--456},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {32G15 (37C85 37F99)},
      MRNUMBER = {2299738},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.4007/annals.2007.165.397},
      ZBLNUMBER = {1131.14027},
      }
  • [moller_linear] Go to document M. Möller, "Linear manifolds in the moduli space of one-forms," Duke Math. J., vol. 144, pp. 447-487, 2008.
    @article {moller_linear, MRKEY = {2444303},
      AUTHOR = {M{ö}ller, M.},
      TITLE = {Linear manifolds in the moduli space of one-forms},
      JOURNAL = {Duke Math. J.},
      VOLUME = {144},
      YEAR = {2008},
      PAGES = {447--487},
      MRNUMBER = {2444303},
      DOI = {10.1215/00127094-2008-041},
      ZBLNUMBER = {1148.32007},
     }
  • [ratner] Go to document M. Ratner, "The rate of mixing for geodesic and horocycle flows," Ergodic Theory Dynam. Systems, vol. 7, iss. 2, pp. 267-288, 1987.
    @article {ratner, MRKEY = {0896798},
      AUTHOR = {Ratner, Marina},
      TITLE = {The rate of mixing for geodesic and horocycle flows},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {7},
      YEAR = {1987},
      NUMBER = {2},
      PAGES = {267--288},
      ISSN = {0143-3857},
      MRCLASS = {58F17 (22E40)},
      MRNUMBER = {0896798},
      MRREVIEWER = {M. Rees},
      DOI = {10.1017/S0143385700004004},
      ZBLNUMBER = {0623.22008},
      }
  • [Ratner_sl] Go to document M. Ratner, "Raghunathan’s conjectures for ${ SL}(2,\Bbb R)$," Israel J. Math., vol. 80, iss. 1-2, pp. 1-31, 1992.
    @article {Ratner_sl, MRKEY = {1248925},
      AUTHOR = {Ratner, Marina},
      TITLE = {Raghunathan's conjectures for {${\rm SL}(2,\bold R)$}},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {80},
      YEAR = {1992},
      NUMBER = {1-2},
      PAGES = {1--31},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {22E40 (28D05 58F17)},
      MRNUMBER = {1248925},
      MRREVIEWER = {Nimish A. Shah},
      DOI = {10.1007/BF02808152},
      ZBLNUMBER = {0785.22013},
      }
  • [Selberg] Go to document A. Selberg, "On the estimation of Fourier coefficients of modular forms," in Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., 1965, vol. VIII, pp. 1-15.
    @incollection {Selberg, MRKEY = {0182610},
      AUTHOR = {Selberg, Atle},
      TITLE = {On the estimation of {F}ourier coefficients of modular forms},
      BOOKTITLE = {Proc. {S}ympos. {P}ure {M}ath.},
      VOLUME={{VIII}},
      PAGES = {1--15},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1965},
      MRCLASS = {10.20},
      MRNUMBER = {0182610},
      MRREVIEWER = {J. R. Smart},
      ZBLNUMBER = {0142.33903},
      DOI = {10.1090/pspum/008/0182610},
     }
  • [Veech_flow] 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_flow, 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},
      }
  • [wall] H. S. Wall, Analytic Theory of Continued Fractions, New York, N. Y.: D. Van Nostrand Company, 1948.
    @book {wall, MRKEY = {0025596},
      AUTHOR = {Wall, H. S.},
      TITLE = {Analytic {T}heory of {C}ontinued {F}ractions},
      PUBLISHER = {D. Van Nostrand Company},
      ADDRESS={New York, N. Y.},
      YEAR = {1948},
      PAGES = {xiii+433},
      MRCLASS = {40.0X},
      MRNUMBER = {0025596},
      MRREVIEWER = {R. C. Buck},
      ZBLNUMBER = {0035.03601},
      }

Authors

Artur Avila

CNRS UMR 7586, Institut de Mathématiques de Jussieu, Paris, France and IMPA, Rio de Janeiro, Brazil

Sébastien Gouëzel

IRMAR, Université de Rennes 1, Rennes, France