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Logarithmic laws and unique ergodicity

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  • We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

    Mathematics Subject Classification: Primary: 30F60; Secondary: 37A25.

    Citation:

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  • Figure 1.  The times $s(\delta)$ and $s(\delta')$ are both smaller than the return times of $I_a$ and $I_c$, respectively. Since $|I_b|\leq \frac{\mathrm{Area(subcomplex)}}{\text{return time to }I_b}$, and so, if the return time to $I_b$ is much greater than $\max\{s(\delta), s(\delta')\}$, we have a saddle connection between $p$ and $p'$ that is made small under $g_t$.

    Figure 2.  On the left, an illustration of the definition of shadowing. On the right, the first step in the recursive procedure used in the Proof of Lemma 24.

  • [1] J. ChaikaY. Cheung and H. Masur, Winning games for bounded geodesics in moduli spaces of quadratic differentials, J. Mod. Dyn., 7 (2013), 395-427.  doi: 10.3934/jmd.2013.7.395.
    [2] Y. Cheung and A. Eskin, Unique ergodicity of translation flows, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007,213–221.
    [3] J. Chaika, H. Masur and M. Wolf, Limits in PMF of Teichmüller geodesics, arXiv: 1406.0564, 2014.
    [4] Y.-E. ChoiK. Rafi and C. Series, Lines of minima and Teichmüller geodesics, Geom. Funct. Anal., 18 (2008), 698-754.  doi: 10.1007/s00039-008-0675-6.
    [5] A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778. 
    [6] S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.
    [7] A. Ya. Khinchin, Continued Fractions, Russian ed., with a preface by B. V. Gnedenko, Reprint of the 1964 translation, Dover Publications, Inc., Mineola, NY, 1997.
    [8] S. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311.  doi: 10.2307/1971280.
    [9] H. B. Keynes and D. Newton, A "minimal", non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), 101-105.  doi: 10.1007/BF01214699.
    [10] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  doi: 10.2307/1971341.
    [11] ______, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387–442.
    [12] ______, Logarithmic law for geodesics in moduli space, in Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math., 150, Amer. Math. Soc., Providence, RI, 1993,229–245.
    [13] ______, Geometry of Teichmüller space with the Teichmüller metric, in Surveys in Differential Geometry. Vol. ⅪⅤ. Geometry of Riemann Surfaces and Their Moduli Spaces, Surv. Differ. Geom., 14, Int. Press, Somerville, MA, 2009,295–313.
    [14] C. T. McMullen, Dynamics of SL2(R) over moduli space in genus two, Ann. of Math. (2), 165 (2007), 397-456.  doi: 10.4007/annals.2007.165.397.
    [15] K. Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol., 9 (2005), 179-202.  doi: 10.2140/gt.2005.9.179.
    [16] E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878. 
    [17] K. StrebelQuadratic Differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verla, Berlin, 1984. 
    [18] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.
    [19] R. Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal., 24 (2014), 360-386.  doi: 10.1007/s00039-014-0269-4.
    [20] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the KroneckerWeyl theorem mod 2, Trans. Amer. Math. Soc., 140 (1969), 1-33. 
    [21] ______, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441–530. doi: 10.2307/2007091.
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