Long hitting time for translation flows and L-shaped billiards

Dong Han Kim; Luca Marchese; Stefano Marmi

doi:10.3934/jmd.2019011

Article Contents

2019, Volume 14: 291-353. Doi: 10.3934/jmd.2019011

This volume Previous Article Möbius disjointness for topological models of ergodic systems with discrete spectrum Next Article Global rigidity of conjugations for locally non-discrete subgroups of $ {\rm {Diff}}^{\omega} (S^1) $

Long hitting time for translation flows and L-shaped billiards

1.
Department of Mathematics Education, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, 04620 Seoul, Korea
2.
Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539, 99 Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
3.
Scuola Normale Superiore and C.N.R.S. UMI 3483 Laboratorio Fibonacci, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Received: December 16, 2017

Revised: August 04, 2018

Published: March 2019

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Abstract

We consider the flow in direction $ \theta $ on a translation surface and we study the asymptotic behavior for $ r\to 0 $ of the time needed by orbits to hit the $ r $-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction $ \theta $. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction $ \theta $ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.

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Mathematics Subject Classification: Primary: 37E35, 11K60; Secondary: 11K55.

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Figure 1. In red, a billiard trajectory in a L-shaped billiard $L(a, b, a', b')$. The green path $\gamma$ is a generalized diagonal, the vector on the left of the figure its planar development Hol$(\gamma)$. The deviation of the direction $\theta$ from $\gamma$ is $|{{\rm {Re}}} (\gamma, \theta)|$

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Figure 2. The saddle connection $\gamma$ appearing in Case (3) in the proof of Proposition 5.1, when $\pi_b(D) = 3$

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Figure 3. A vertical splitting pair $(\sigma_0, \gamma_0)$, with two cylinders $C_0$ and $C_0'$ around $\sigma_0$ and $\sigma_0'$ respectively, where $W_0 = \Delta_0 = 1$ and $|\sigma_0| = 2$. For $\alpha\sim 0, 1925$ we have $\alpha^{-1}\sim5, 1948$, so that $ [W_0/\alpha] = 5 = 1\bmod 2\cdot\mathbb{Z} $. A trajectory with slope $\alpha$ travels in $C_0$, then it crosses $\gamma_0$ and repeats the path inside $C_0$, modulo a vertical translation by $ \delta_V = G(\alpha)\cdot W_0\sim0, 1948 $. As long as it re-enters inside $C_0$ such trajectory does not visit the cylinder $C_0'$

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Figure 4. The Eierlegende Wollmilchsau surface $X_{EW}$. On the left its vertical cylinder decomposition, while the horizontal cylinder decomposition appears on the upper part of the right side of the picture. In both figures it is represented the same path. On the lower part of the right side of the picture are represented the three intersection criteria stated in Lemma 8.2, where the line segment $S$ with slope $-\infty\leq \alpha(S) <-1$ is represented in green and the line segment $I$ with slope $0 <\alpha(I) <1$ is represented in red

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Figure 5. The four separatrix diagrams at a conical point of angle $6\pi$, with the corresponding $f\in S_3$ and the return angles represented respectively above and below each of them. The second and third diagrams can be realized by surfaces in $\mathscr{H}(2)$, which are represented below

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Figure 6. For each integer $N\geq 4$ and any value of the invariant $\textrm{IWP}$ the pair $(\sigma_2, \gamma_2)$ is a horizontal splitting pair. One Weierstrass points is always given by the conical point, which has integer coordinates. In each of the four figures, the other five Weierstrass points are represented by a black dot

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