2017, 11: 563-588. doi: 10.3934/jmd.2017022

Logarithmic laws and unique ergodicity

1. 

Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA

2. 

Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210-2889, USA

Received  June 07, 2017 Revised  August 24, 2017 Published  November 2017

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.

Citation: Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Chaika, H. Masur and M. Wolf, Limits in PMF of Teichmüller geodesics, arXiv: 1406.0564, 2014. Google Scholar

[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.  Google Scholar

[5]

A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.   Google Scholar

[6]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  doi: 10.2307/1971341.  Google Scholar

[11]

______, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387–442.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.   Google Scholar

[17] K. Strebel, Quadratic Differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verla, Berlin, 1984.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

______, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441–530. doi: 10.2307/2007091.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Chaika, H. Masur and M. Wolf, Limits in PMF of Teichmüller geodesics, arXiv: 1406.0564, 2014. Google Scholar

[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.  Google Scholar

[5]

A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.   Google Scholar

[6]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  doi: 10.2307/1971341.  Google Scholar

[11]

______, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387–442.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.   Google Scholar

[17] K. Strebel, Quadratic Differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verla, Berlin, 1984.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

______, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441–530. doi: 10.2307/2007091.  Google Scholar

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.
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