American Institute of Mathematical Sciences

Advanced
January  2011, 5(1): 33-48. doi: 10.3934/jmd.2011.5.33

Perfect retroreflectors and billiard dynamics

Pavel Bachurin 1, , Konstantin Khanin 2, , Jens Marklof 3, and  Alexander Plakhov 4,

1. 

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, United States
2. 

Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario M5S 2E4, Canada
3. 

School of Mathematics, University of Bristol, Bristol BS8 1TW
4. 

Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal

Received  December 2009 Revised  November 2010 Published  April 2011

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.
Keywords: billiards, dynamical renormalization, retroreflectors., homogeneous flow, Recurrence, circle rotation.
Mathematics Subject Classification: Primary: 37A50; Secondary: 37A17, 11K0.
Citation: Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov. Perfect retroreflectors and billiard dynamics. Journal of Modern Dynamics, 2011, 5 (1) : 33-48. doi: 10.3934/jmd.2011.5.33
References:
[1]

M. Boshernitzan, A condition for minimal interval-exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.  Google Scholar

[2]

M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, Ergodic Theory Dynam. Systems, 12 (1992), 425-428. doi: 10.1017/S0143385700006866.  Google Scholar

[3]

M. Boshernitzan and A. Nogueira, Generalized functions of interval-exchange maps, Ergodic Theory Dynam. Systems, 24 (2004), 697-705. doi: 10.1017/S0143385704000021.  Google Scholar

[4]

J. E. Eaton, On spherically symmetric lenses, Trans. IRE Antennas Propag., 4 (1952) 66-71. Google Scholar

[5]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, \arXiv{0912.2891}., ().   Google Scholar

[6]

A. Katok and A. Stepin, Approximations in ergodic theory, (Russian) Uspehi Mat. Nauk, 22 (1967), 81-106.  Google Scholar

[7]

M. Loeve, "Probability Theory I," Fourth edition, Graduate Texts in Mathematics, Vol. 45, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[8]

J. Marklof, Distribution modulo one and Ratner's theorem, Equidistribution in Number Theory, An Introduction, 217-244, NATO Sci. Ser. II Math. Phys. Chem., 237, Springer, Dordrecht, 2007.  Google Scholar

[9]

J. Marklof, The $n$-point correlations between values of a linear form, With an appendix by Zeév Rudnick, Ergodic Theory Dynam. Systems, 20 (2000), 1127-1172. doi: 10.1017/S0143385700000626.  Google Scholar

[10]

J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Annals of Math., 172 (2010), 1949-2033. doi: 10.4007/annals.2010.172.1949.  Google Scholar

[11]

A. E. Mazel and Y. G. Sinai, A limiting distribution connected with fractional parts of linear forms, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), 220-229, Cambridge Univ. Press, Cambridge, 1992.  Google Scholar

[12]

A. Plakhov and P. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271-2287. doi: 10.1088/0951-7715/20/9/013.  Google Scholar

[13]

C. L. Siegel, "Lectures on the Geometry of Numbers," Notes by B. Friedman, Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter, With a preface by Chandrasekharan. Springer-Verlag, Berlin, 1989.  Google Scholar

[14]

T. Tyc, U. Leonhardt, Transmutation of singularities in optical instruments, New J. Physics, 10 (2008), 115038 (8pp). Google Scholar

[15]

W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval-exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153. doi: 10.1017/S0143385700003862.  Google Scholar

show all references

References:
[1]

M. Boshernitzan, A condition for minimal interval-exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.  Google Scholar

[2]

M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, Ergodic Theory Dynam. Systems, 12 (1992), 425-428. doi: 10.1017/S0143385700006866.  Google Scholar

[3]

M. Boshernitzan and A. Nogueira, Generalized functions of interval-exchange maps, Ergodic Theory Dynam. Systems, 24 (2004), 697-705. doi: 10.1017/S0143385704000021.  Google Scholar

[4]

J. E. Eaton, On spherically symmetric lenses, Trans. IRE Antennas Propag., 4 (1952) 66-71. Google Scholar

[5]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, \arXiv{0912.2891}., ().   Google Scholar

[6]

A. Katok and A. Stepin, Approximations in ergodic theory, (Russian) Uspehi Mat. Nauk, 22 (1967), 81-106.  Google Scholar

[7]

M. Loeve, "Probability Theory I," Fourth edition, Graduate Texts in Mathematics, Vol. 45, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[8]

J. Marklof, Distribution modulo one and Ratner's theorem, Equidistribution in Number Theory, An Introduction, 217-244, NATO Sci. Ser. II Math. Phys. Chem., 237, Springer, Dordrecht, 2007.  Google Scholar

[9]

J. Marklof, The $n$-point correlations between values of a linear form, With an appendix by Zeév Rudnick, Ergodic Theory Dynam. Systems, 20 (2000), 1127-1172. doi: 10.1017/S0143385700000626.  Google Scholar

[10]

J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Annals of Math., 172 (2010), 1949-2033. doi: 10.4007/annals.2010.172.1949.  Google Scholar

[11]

A. E. Mazel and Y. G. Sinai, A limiting distribution connected with fractional parts of linear forms, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), 220-229, Cambridge Univ. Press, Cambridge, 1992.  Google Scholar

[12]

A. Plakhov and P. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271-2287. doi: 10.1088/0951-7715/20/9/013.  Google Scholar

[13]

C. L. Siegel, "Lectures on the Geometry of Numbers," Notes by B. Friedman, Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter, With a preface by Chandrasekharan. Springer-Verlag, Berlin, 1989.  Google Scholar

[14]

T. Tyc, U. Leonhardt, Transmutation of singularities in optical instruments, New J. Physics, 10 (2008), 115038 (8pp). Google Scholar

[15]

W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval-exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153. doi: 10.1017/S0143385700003862.  Google Scholar

[1]

Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
[2]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[3]

Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399
[4]

Vincent Penné, Benoît Saussol, Sandro Vaienti. Dimensions for recurrence times: topological and dynamical properties. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 783-798. doi: 10.3934/dcds.1999.5.783
[5]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[6]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[7]

Alexander Plakhov. Mathematical retroreflectors. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1211-1235. doi: 10.3934/dcds.2011.30.1211
[8]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
[9]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
[10]

Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039
[11]

Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225
[12]

Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787
[13]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[14]

Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1389-1409. doi: 10.3934/dcds.2017057
[15]

Matthieu Hillairet, Ayman Moussa, Franck Sueur. On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow. Kinetic & Related Models, 2019, 12 (4) : 681-701. doi: 10.3934/krm.2019026
[16]

A. A. Pinto, D. Sullivan. The circle and the solenoid. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 463-504. doi: 10.3934/dcds.2006.16.463
[17]

De Tang. Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4913-4928. doi: 10.3934/dcdsb.2019037
[18]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
[19]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[20]

Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences