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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

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