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

Yan Huang

doi:10.3934/dcds.2018099

Article Contents

2018, Volume 38, Issue 5: 2395-2409. Doi: 10.3934/dcds.2018099

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

Yan Huang

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

Received: January 31, 2017

Published: June 2018

The author is supported by NSFC under grant No. 11401167

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Abstract

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 $\frac{1}{2}$.

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Mathematics Subject Classification: Primary: 32G15.

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Figure 1. The double cover

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Figure 2. Combinatorial realization

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Figure 3. New hexagon by gluing pieces of polygons

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Figure 4. The parallelogram domain

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References

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