# American Institute of Mathematical Sciences

May 2018, 38(5): 2395-2409. doi: 10.3934/dcds.2018099

## On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori

 School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

Received  February 2017 Published  March 2018

Fund Project: The author is supported by NSFC under grant No. 11401167

Cheung, Hubert and Masur [Invent. Math., 183(2011), no.2, pp. 337-383] proved that the Hausdorff dimension of the set of nonergodic directions of billiards in a kind of rectangle with barrier is either 0 or $\frac{1}{2}$. As an application of their argument, we prove that there exist the third-kind two-genus double covers of tori in which the set of minimal and non-ergodic directions have Hausdorff dimension $\frac{1}{2}$.

Citation: Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099
##### References:
 [1] J. S. Athreya and J. Chaika, The Hausdorff dimension of non-uniquely ergodic directions in $H(2)$ is almost everywhere $\frac{1}{2}$, Geom. Topol., 19 (2015), 3537-3563. [2] Y. Cheung, Hausdorff dimension of the set of nonergodic directions. With an appendix by M. Boshernitzan, Ann. of Math., 158 (2003), 661-678. doi: 10.4007/annals.2003.158.661. [3] Y. Cheung, Y. P. Hubert and H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. Math., 183 (2011), 337-383. doi: 10.1007/s00222-010-0279-2. [4] Y. Cheung and H. Masur, Minimal non-ergodic directions on genus-2 translation surfaces, Ergodic Theory Dynam. Systems, 26 (2006), 341-351. doi: 10.1017/S0143385705000465. [5] A. Eskin, H. Masur and M. Schmoll, Billiards in rectangles with barriers, Duke Math. J., 118 (2003), 427-463. doi: 10.1215/S0012-7094-03-11832-3. [6] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990. [7] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and qudratic differentials, Ann. of Math., 124 (1986), 293-311. doi: 10.2307/1971280. [8] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math., 134 (1991), 455-543. doi: 10.2307/2944356. [9] C.T. McMullen, Dynamics of SL$_2(\mathbb{R})$ over moduli space in genus two, Ann. of Math., 165 (2007), 397-456. doi: 10.4007/annals.2007.165.397. [10] A. Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 41-56.

show all references

##### References:
 [1] J. S. Athreya and J. Chaika, The Hausdorff dimension of non-uniquely ergodic directions in $H(2)$ is almost everywhere $\frac{1}{2}$, Geom. Topol., 19 (2015), 3537-3563. [2] Y. Cheung, Hausdorff dimension of the set of nonergodic directions. With an appendix by M. Boshernitzan, Ann. of Math., 158 (2003), 661-678. doi: 10.4007/annals.2003.158.661. [3] Y. Cheung, Y. P. Hubert and H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. Math., 183 (2011), 337-383. doi: 10.1007/s00222-010-0279-2. [4] Y. Cheung and H. Masur, Minimal non-ergodic directions on genus-2 translation surfaces, Ergodic Theory Dynam. Systems, 26 (2006), 341-351. doi: 10.1017/S0143385705000465. [5] A. Eskin, H. Masur and M. Schmoll, Billiards in rectangles with barriers, Duke Math. J., 118 (2003), 427-463. doi: 10.1215/S0012-7094-03-11832-3. [6] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990. [7] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and qudratic differentials, Ann. of Math., 124 (1986), 293-311. doi: 10.2307/1971280. [8] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math., 134 (1991), 455-543. doi: 10.2307/2944356. [9] C.T. McMullen, Dynamics of SL$_2(\mathbb{R})$ over moduli space in genus two, Ann. of Math., 165 (2007), 397-456. doi: 10.4007/annals.2007.165.397. [10] A. Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 41-56.
The double cover
Combinatorial realization
New hexagon by gluing pieces of polygons
The parallelogram domain
 [1] E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401 [2] 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 [3] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [4] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [5] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [6] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [7] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [8] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [9] Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098 [10] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 [11] Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125 [12] Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 [13] Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417 [14] Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 [15] Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699 [16] Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 [17] Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006 [18] Manuel Fernández-Martínez, Miguel Ángel López Guerrero. Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1129-1137. doi: 10.3934/dcdss.2015.8.1129 [19] Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110 [20] Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2018303

2017 Impact Factor: 1.179