# 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] A. Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 41-56.   Google Scholar

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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] A. Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 41-56.   Google Scholar
The double cover
Combinatorial realization
New hexagon by gluing pieces of polygons
The parallelogram domain
 [1] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [2] Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320 [3] Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 [4] Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011 [5] Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386 [6] Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175 [7] Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016 [8] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [9] Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366 [10] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 [11] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [12] Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052 [13] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [14] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [15] Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011 [16] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [17] Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054 [18] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 [19] Björn Augner, Dieter Bothe. The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 533-574. doi: 10.3934/dcdss.2020406 [20] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

2019 Impact Factor: 1.338