December  2012, 32(12): 4445-4466. doi: 10.3934/dcds.2012.32.4445

Free path of billiards with flat points

1. 

Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003

Received  March 2011 Revised  May 2012 Published  August 2012

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
Citation: Hong-Kun Zhang. Free path of billiards with flat points. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445
References:
[1]

L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markovpartitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152. doi: 10.1070/RM1990v045n03ABEH002355.

[2]

L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys, 46 (1991), 47-106. doi: 10.1070/RM1991v046n04ABEH002827.

[3]

N. Chernov, Statistical properties of theperiodic Lorentz gas in Multidimensional case, J. Stat. Physics, 74 (1994), 11-53. doi: 10.1007/BF02186805.

[4]

N. Chernov and C. Haskell, Nonuniformlyhyperbolic K-systems are Bernoulli, Ergod. Th. Dynam. Sys., 16 (1996), 19-44.

[5]

N. Chernov, Decay of correlations in dispersing billiards, J. Statist. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.

[6]

N. Chernov and D. Dolgopyat, Anomalous current in periodic Lorentz gases with infinite horizon, Uspekhi Mat. Nauk, 64 (2009), 73-124.

[7]

N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006.

[8]

N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, in: Hard Ball Systems and the Lorentz Gas, Ed. by D. Szasz, Encyclopaedia of Mathematical Sciences 101, 89-120, Springer, 2000.

[9]

N. Chernov and H.-K. Zhang, Billiards with polynomial mixingrates, Nonlineartity, 4 (2005), 1527-1553 doi: 10.1088/0951-7715/18/4/006.

[10]

N. Chernov and H.-K. Zhang, A family of chaotic billiards withvariable mixing rates, Stochastics and Dynamics, 5 (2005), 535-553 doi: 10.1142/S0219493705001572.

[11]

N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Communications in Mathematical Physics, 277 (2008), 305-321. doi: 10.1007/s00220-007-0360-x.

[12]

D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz process, Duke Math. J., 142 (2008), 241-281. doi: 10.1215/00127094-2008-006.

[13]

G. Gallavotti and D. Ornstein, Billiards and Bernoulli scheme, Commun. Math. Phys. 38 (1974), 83-101. doi: 10.1007/BF01651505.

[14]

A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Mapswith Singularities," Lect. Notes. in Math. 1222, Springer, 1986.

[15]

H. Lorentz, The motion of electrons in metallic bodies, Proc. Amst. Acad., 7 (1905), 438-453.

[16]

R. Markarian, Billiards with polynomial decayof correlations, Er. Th. Dynam. Syst., 24 (2004), 177-197. doi: 10.1017/S0143385703000270.

[17]

D. Ornstein and B. Weiss, On the Bernoulli natureof systems with some hyperbolic structure, Ergod. Th. Dynam. Sys., 18 (1998), 441-456. doi: 10.1017/S0143385798100354.

[18]

Ya. B. Pensin, Dynamical Systems With Generalized Hyperbolic Attractors: Hyperbolic, Ergodic and Topological Properties, Ergod. Theory and Dyn. Syst., 12 (1992), 123-152.

[19]

Ya. G. Sinai, Dynamical systems with elastic reflections.Ergodic properties of diepersing billiards, Russian Math. Surveys, 25 (1970), 137-189. doi: 10.1070/RM1970v025n02ABEH003794.

[20]

Ya. G. Sinai and N. Chernov, Ergodicproperties of some systems of two-dimensional discs andthree-dimensional spheres, Russian Math. Surveys, 42 (1987), 181-207. doi: 10.1070/RM1987v042n03ABEH001421.

[21]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. Dynam. Syst., 5 (1985), 145-161. doi: 10.1017/S0143385700002807.

[22]

L.-S. Young, Statistical properties of systemswith some hyperbolicity including certain billiards, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.

show all references

References:
[1]

L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markovpartitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152. doi: 10.1070/RM1990v045n03ABEH002355.

[2]

L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys, 46 (1991), 47-106. doi: 10.1070/RM1991v046n04ABEH002827.

[3]

N. Chernov, Statistical properties of theperiodic Lorentz gas in Multidimensional case, J. Stat. Physics, 74 (1994), 11-53. doi: 10.1007/BF02186805.

[4]

N. Chernov and C. Haskell, Nonuniformlyhyperbolic K-systems are Bernoulli, Ergod. Th. Dynam. Sys., 16 (1996), 19-44.

[5]

N. Chernov, Decay of correlations in dispersing billiards, J. Statist. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.

[6]

N. Chernov and D. Dolgopyat, Anomalous current in periodic Lorentz gases with infinite horizon, Uspekhi Mat. Nauk, 64 (2009), 73-124.

[7]

N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006.

[8]

N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, in: Hard Ball Systems and the Lorentz Gas, Ed. by D. Szasz, Encyclopaedia of Mathematical Sciences 101, 89-120, Springer, 2000.

[9]

N. Chernov and H.-K. Zhang, Billiards with polynomial mixingrates, Nonlineartity, 4 (2005), 1527-1553 doi: 10.1088/0951-7715/18/4/006.

[10]

N. Chernov and H.-K. Zhang, A family of chaotic billiards withvariable mixing rates, Stochastics and Dynamics, 5 (2005), 535-553 doi: 10.1142/S0219493705001572.

[11]

N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Communications in Mathematical Physics, 277 (2008), 305-321. doi: 10.1007/s00220-007-0360-x.

[12]

D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz process, Duke Math. J., 142 (2008), 241-281. doi: 10.1215/00127094-2008-006.

[13]

G. Gallavotti and D. Ornstein, Billiards and Bernoulli scheme, Commun. Math. Phys. 38 (1974), 83-101. doi: 10.1007/BF01651505.

[14]

A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Mapswith Singularities," Lect. Notes. in Math. 1222, Springer, 1986.

[15]

H. Lorentz, The motion of electrons in metallic bodies, Proc. Amst. Acad., 7 (1905), 438-453.

[16]

R. Markarian, Billiards with polynomial decayof correlations, Er. Th. Dynam. Syst., 24 (2004), 177-197. doi: 10.1017/S0143385703000270.

[17]

D. Ornstein and B. Weiss, On the Bernoulli natureof systems with some hyperbolic structure, Ergod. Th. Dynam. Sys., 18 (1998), 441-456. doi: 10.1017/S0143385798100354.

[18]

Ya. B. Pensin, Dynamical Systems With Generalized Hyperbolic Attractors: Hyperbolic, Ergodic and Topological Properties, Ergod. Theory and Dyn. Syst., 12 (1992), 123-152.

[19]

Ya. G. Sinai, Dynamical systems with elastic reflections.Ergodic properties of diepersing billiards, Russian Math. Surveys, 25 (1970), 137-189. doi: 10.1070/RM1970v025n02ABEH003794.

[20]

Ya. G. Sinai and N. Chernov, Ergodicproperties of some systems of two-dimensional discs andthree-dimensional spheres, Russian Math. Surveys, 42 (1987), 181-207. doi: 10.1070/RM1987v042n03ABEH001421.

[21]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. Dynam. Syst., 5 (1985), 145-161. doi: 10.1017/S0143385700002807.

[22]

L.-S. Young, Statistical properties of systemswith some hyperbolicity including certain billiards, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.

[1]

Margaret Brown, Péter Nándori. Statistical properties of type D dispersing billiards. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022073

[2]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[3]

Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893

[4]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085

[5]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307

[6]

Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043

[7]

Senda Ounaies, Jean-Marc Bonnisseau, Souhail Chebbi, Halil Mete Soner. Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1323-1331. doi: 10.3934/jimo.2016.12.1323

[8]

Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control and Related Fields, 2016, 6 (4) : 629-651. doi: 10.3934/mcrf.2016018

[9]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks and Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021

[10]

Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132

[11]

Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903

[12]

Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15

[13]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

[14]

Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257

[15]

Fabio Bagagiolo. An infinite horizon optimal control problem for some switching systems. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 443-462. doi: 10.3934/dcdsb.2001.1.443

[16]

Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623

[17]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control and Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022

[18]

Nobusumi Sagara. Recursive variational problems in nonreflexive Banach spaces with an infinite horizon: An existence result. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1219-1232. doi: 10.3934/dcdss.2018069

[19]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145

[20]

Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]