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

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

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.

Citation: Dong Han Kim, Luca Marchese, Stefano Marmi. Long hitting time for translation flows and L-shaped billiards. Journal of Modern Dynamics, 2019, 14: 291-353. doi: 10.3934/jmd.2019011
References:
[1]

M. Artigiani, L. Marchese and C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Geometry, Groups and Dynamics, 10 (2016), no. 4, 1287–1337. doi: 10.4171/GGD/384.

[2]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Commun. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427.

[3]

V. Beresnevich and S. Velani, Ubiquity and a general logarithmic law for geodesics, in Dynamical Systems and Diophantine Approximations, Sémin. Congr., 19, Soc. Math. France, Paris, 2009, 21–36.

[4]

M. Boshernitzan and J. Chaika, Diophantine properties of IETs and general systems: Quantitative proximality and connectivity, Invent. Math., 192 (2013), 375-412.  doi: 10.1007/s00222-012-0413-4.

[5]

G. H. Choe and B. K. Seo, Recurrence speed of multiples of an irrational number, Proc. Japan Acad. Ser. A, 77 (2001), 134-137.  doi: 10.3792/pjaa.77.134.

[6]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\text{SL}(2, \mathbb{R})$ action on moduli space, arXiv: 1302.3320.

[7]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $ \text{SL}(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673–721. doi: 10.4007/annals.2015.182.2.7.

[8]

K. Falconer, Fractal Geometry, Wiley, 2003. doi: 10.1002/0470013850.

[9]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.

[10]

S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386.  doi: 10.4310/MRL.2005.v12.n3.a8.

[11]

E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403.  doi: 10.4310/MRL.1996.v3.n3.a8.

[12]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller disc in $ {\mathscr{H} }(2)$, Israel J. Math., 151 (2006), 281-321.  doi: 10.1007/BF02777365.

[13]

P. HubertL. Marchese and C. Ulcigrai, Lagrange Spectra in Teichmüller Dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255.  doi: 10.1007/s00039-015-0321-z.

[14]

V. Jarník, Diophantischen Approximationen und Hausdorffsches Mass, Mat. Sbornik, 36 (1929), 371-382. 

[15] A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, 1964. 
[16]

D. H. Kim, Diophantine type of interval exchange maps, Ergodic Theory Dynam. Systems, 34 (2014), 1990-2017.  doi: 10.1017/etds.2013.22.

[17]

D. H. Kim and S. Marmi, The recurrence time for interval exchange maps, Nonlinearity, 21 (2008), 2201-2210.  doi: 10.1088/0951-7715/21/9/016.

[18]

D. H. Kim and S. Marmi, Bounded type interval exchange maps, Nonlinearity, 27 (2014), 637-645.  doi: 10.1088/0951-7715/27/4/637.

[19]

D. H. Kim and B. K. Seo, The waiting time for irrational rotations, Nonlinearity, 16 (2003), 1861-1868.  doi: 10.1088/0951-7715/16/5/318.

[20]

L. MarcheseR. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, Comment. Math. Helv., 93 (2018), 225-289.  doi: 10.4171/CMH/434.

[21]

S. MarmiP. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872.  doi: 10.1090/S0894-0347-05-00490-X.

[22]

S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Commun. Math. Phys., 344 (2016), 117-139.  doi: 10.1007/s00220-016-2624-9.

[23]

C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Math. Ann., 333 (2005), 87-130.  doi: 10.1007/s00208-005-0666-y.

[24]

J. Smillie and B. Weiss, Finiteness results for flat surfaces: a survey and problem list, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007,125–137.

[25]

W. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on generic translation surfaces, Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,205–258. doi: 10.1090/conm/385/07199.

[27]

P. Walters, Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[28]

J. C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69.

[29]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,437–583. doi: 10.1007/978-3-540-31347-2_13.

show all references

References:
[1]

M. Artigiani, L. Marchese and C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Geometry, Groups and Dynamics, 10 (2016), no. 4, 1287–1337. doi: 10.4171/GGD/384.

[2]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Commun. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427.

[3]

V. Beresnevich and S. Velani, Ubiquity and a general logarithmic law for geodesics, in Dynamical Systems and Diophantine Approximations, Sémin. Congr., 19, Soc. Math. France, Paris, 2009, 21–36.

[4]

M. Boshernitzan and J. Chaika, Diophantine properties of IETs and general systems: Quantitative proximality and connectivity, Invent. Math., 192 (2013), 375-412.  doi: 10.1007/s00222-012-0413-4.

[5]

G. H. Choe and B. K. Seo, Recurrence speed of multiples of an irrational number, Proc. Japan Acad. Ser. A, 77 (2001), 134-137.  doi: 10.3792/pjaa.77.134.

[6]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\text{SL}(2, \mathbb{R})$ action on moduli space, arXiv: 1302.3320.

[7]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $ \text{SL}(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673–721. doi: 10.4007/annals.2015.182.2.7.

[8]

K. Falconer, Fractal Geometry, Wiley, 2003. doi: 10.1002/0470013850.

[9]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.

[10]

S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386.  doi: 10.4310/MRL.2005.v12.n3.a8.

[11]

E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403.  doi: 10.4310/MRL.1996.v3.n3.a8.

[12]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller disc in $ {\mathscr{H} }(2)$, Israel J. Math., 151 (2006), 281-321.  doi: 10.1007/BF02777365.

[13]

P. HubertL. Marchese and C. Ulcigrai, Lagrange Spectra in Teichmüller Dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255.  doi: 10.1007/s00039-015-0321-z.

[14]

V. Jarník, Diophantischen Approximationen und Hausdorffsches Mass, Mat. Sbornik, 36 (1929), 371-382. 

[15] A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, 1964. 
[16]

D. H. Kim, Diophantine type of interval exchange maps, Ergodic Theory Dynam. Systems, 34 (2014), 1990-2017.  doi: 10.1017/etds.2013.22.

[17]

D. H. Kim and S. Marmi, The recurrence time for interval exchange maps, Nonlinearity, 21 (2008), 2201-2210.  doi: 10.1088/0951-7715/21/9/016.

[18]

D. H. Kim and S. Marmi, Bounded type interval exchange maps, Nonlinearity, 27 (2014), 637-645.  doi: 10.1088/0951-7715/27/4/637.

[19]

D. H. Kim and B. K. Seo, The waiting time for irrational rotations, Nonlinearity, 16 (2003), 1861-1868.  doi: 10.1088/0951-7715/16/5/318.

[20]

L. MarcheseR. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, Comment. Math. Helv., 93 (2018), 225-289.  doi: 10.4171/CMH/434.

[21]

S. MarmiP. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872.  doi: 10.1090/S0894-0347-05-00490-X.

[22]

S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Commun. Math. Phys., 344 (2016), 117-139.  doi: 10.1007/s00220-016-2624-9.

[23]

C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Math. Ann., 333 (2005), 87-130.  doi: 10.1007/s00208-005-0666-y.

[24]

J. Smillie and B. Weiss, Finiteness results for flat surfaces: a survey and problem list, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007,125–137.

[25]

W. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on generic translation surfaces, Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,205–258. doi: 10.1090/conm/385/07199.

[27]

P. Walters, Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[28]

J. C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69.

[29]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,437–583. doi: 10.1007/978-3-540-31347-2_13.

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 $|{{\rm {Re}}} (\gamma, \theta)|$
Figure 2.  The saddle connection $\gamma$ appearing in Case (3) in the proof of Proposition 5.1, when $\pi_b(D) = 3$
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 $C_0'$
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
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
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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