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 and Continuous Dynamical Systems, 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. CheungY. 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. EskinH. 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. KerckhoffH. 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. CheungY. 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. EskinH. 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. KerckhoffH. 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. 

Figure 1.  The double cover
Figure 2.  Combinatorial realization
Figure 3.  New hexagon by gluing pieces of polygons
Figure 4.  The parallelogram domain
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