# American Institute of Mathematical Sciences

November 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. Chaika, Y. 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. Choi, K. 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. Kerckhoff, H. 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. Strebel, Quadratic 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.

show all references

##### References:
 [1] J. Chaika, Y. 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. Choi, K. 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. Kerckhoff, H. 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. Strebel, Quadratic 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.
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 $g_t$.
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. S. Athreya, Anish Ghosh, Amritanshu Prasad. Ultrametric logarithm laws I. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 337-348. doi: 10.3934/dcdss.2009.2.337 [2] Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, Ⅱ. Journal of Modern Dynamics, 2017, 11: 1-16. doi: 10.3934/jmd.2017001 [3] Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209 [4] Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845 [5] Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, I. Journal of Modern Dynamics, 2009, 3 (3) : 359-378. doi: 10.3934/jmd.2009.3.359 [6] Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133 [7] Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018 [8] Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715 [9] Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147 [10] François Ledrappier, Omri Sarig. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 411-433. doi: 10.3934/dcds.2006.16.411 [11] David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477 [12] Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34 [13] Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367 [14] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 [15] José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269 [16] Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119 [17] Santos González, Llorenç Huguet, Consuelo Martínez, Hugo Villafañe. Discrete logarithm like problems and linear recurring sequences. Advances in Mathematics of Communications, 2013, 7 (2) : 187-195. doi: 10.3934/amc.2013.7.187 [18] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [19] Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187 [20] Karl Grill, Christian Tutschka. Ergodicity of two particles with attractive interaction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4831-4838. doi: 10.3934/dcds.2015.35.4831

2017 Impact Factor: 0.425