# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1070/RM1990v045n03ABEH002355.  Google Scholar [2] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards,, Russian Math. Surveys, 46 (1991), 47.  doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [3] N. Chernov, Statistical properties of theperiodic Lorentz gas in Multidimensional case,, J. Stat. Physics, 74 (1994), 11.  doi: 10.1007/BF02186805.  Google Scholar [4] N. Chernov and C. Haskell, Nonuniformlyhyperbolic K-systems are Bernoulli,, Ergod. Th. Dynam. Sys., 16 (1996), 19.   Google Scholar [5] N. Chernov, Decay of correlations in dispersing billiards,, J. Statist. Phys., 94 (1999), 513.  doi: 10.1023/A:1004581304939.  Google Scholar [6] N. Chernov and D. Dolgopyat, Anomalous current in periodic Lorentz gases with infinite horizon,, Uspekhi Mat. Nauk, 64 (2009), 73.   Google Scholar [7] N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006).   Google Scholar [8] N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in: Hard Ball Systems and the Lorentz Gas, 101 (2000), 89.   Google Scholar [9] N. Chernov and H.-K. Zhang, Billiards with polynomial mixingrates,, Nonlineartity, 4 (2005), 1527.  doi: 10.1088/0951-7715/18/4/006.  Google Scholar [10] N. Chernov and H.-K. Zhang, A family of chaotic billiards withvariable mixing rates,, Stochastics and Dynamics, 5 (2005), 535.  doi: 10.1142/S0219493705001572.  Google Scholar [11] N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards,, Communications in Mathematical Physics, 277 (2008), 305.  doi: 10.1007/s00220-007-0360-x.  Google Scholar [12] D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz process,, Duke Math. J., 142 (2008), 241.  doi: 10.1215/00127094-2008-006.  Google Scholar [13] G. Gallavotti and D. Ornstein, Billiards and Bernoulli scheme,, Commun. Math. Phys. 38 (1974), 38 (1974), 83.  doi: 10.1007/BF01651505.  Google Scholar [14] A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Mapswith Singularities,", Lect. Notes. in Math. 1222, (1222).   Google Scholar [15] H. Lorentz, The motion of electrons in metallic bodies,, Proc. Amst. Acad., 7 (1905), 438.   Google Scholar [16] R. Markarian, Billiards with polynomial decayof correlations,, Er. Th. Dynam. Syst., 24 (2004), 177.  doi: 10.1017/S0143385703000270.  Google Scholar [17] D. Ornstein and B. Weiss, On the Bernoulli natureof systems with some hyperbolic structure,, Ergod. Th. Dynam. Sys., 18 (1998), 441.  doi: 10.1017/S0143385798100354.  Google Scholar [18] Ya. B. Pensin, Dynamical Systems With Generalized Hyperbolic Attractors: Hyperbolic, Ergodic and Topological Properties,, Ergod. Theory and Dyn. Syst., 12 (1992), 123.   Google Scholar [19] Ya. G. Sinai, Dynamical systems with elastic reflections.Ergodic properties of diepersing billiards,, Russian Math. Surveys, 25 (1970), 137.  doi: 10.1070/RM1970v025n02ABEH003794.  Google Scholar [20] Ya. G. Sinai and N. Chernov, Ergodicproperties of some systems of two-dimensional discs andthree-dimensional spheres,, Russian Math. Surveys, 42 (1987), 181.  doi: 10.1070/RM1987v042n03ABEH001421.  Google Scholar [21] M. Wojtkowski, Invariant families of cones and Lyapunov exponents,, Ergod. Th. Dynam. Syst., 5 (1985), 145.  doi: 10.1017/S0143385700002807.  Google Scholar [22] L.-S. Young, Statistical properties of systemswith some hyperbolicity including certain billiards,, Ann. Math., 147 (1998), 585.  doi: 10.2307/120960.  Google Scholar

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##### References:
 [1] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markovpartitions for two-dimensional hyperbolic billiards,, Russian Math. Surveys, 45 (1990), 105.  doi: 10.1070/RM1990v045n03ABEH002355.  Google Scholar [2] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards,, Russian Math. Surveys, 46 (1991), 47.  doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [3] N. Chernov, Statistical properties of theperiodic Lorentz gas in Multidimensional case,, J. Stat. Physics, 74 (1994), 11.  doi: 10.1007/BF02186805.  Google Scholar [4] N. Chernov and C. Haskell, Nonuniformlyhyperbolic K-systems are Bernoulli,, Ergod. Th. Dynam. Sys., 16 (1996), 19.   Google Scholar [5] N. Chernov, Decay of correlations in dispersing billiards,, J. Statist. Phys., 94 (1999), 513.  doi: 10.1023/A:1004581304939.  Google Scholar [6] N. Chernov and D. Dolgopyat, Anomalous current in periodic Lorentz gases with infinite horizon,, Uspekhi Mat. Nauk, 64 (2009), 73.   Google Scholar [7] N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006).   Google Scholar [8] N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in: Hard Ball Systems and the Lorentz Gas, 101 (2000), 89.   Google Scholar [9] N. Chernov and H.-K. Zhang, Billiards with polynomial mixingrates,, Nonlineartity, 4 (2005), 1527.  doi: 10.1088/0951-7715/18/4/006.  Google Scholar [10] N. Chernov and H.-K. Zhang, A family of chaotic billiards withvariable mixing rates,, Stochastics and Dynamics, 5 (2005), 535.  doi: 10.1142/S0219493705001572.  Google Scholar [11] N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards,, Communications in Mathematical Physics, 277 (2008), 305.  doi: 10.1007/s00220-007-0360-x.  Google Scholar [12] D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz process,, Duke Math. J., 142 (2008), 241.  doi: 10.1215/00127094-2008-006.  Google Scholar [13] G. Gallavotti and D. Ornstein, Billiards and Bernoulli scheme,, Commun. Math. Phys. 38 (1974), 38 (1974), 83.  doi: 10.1007/BF01651505.  Google Scholar [14] A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Mapswith Singularities,", Lect. Notes. in Math. 1222, (1222).   Google Scholar [15] H. Lorentz, The motion of electrons in metallic bodies,, Proc. Amst. Acad., 7 (1905), 438.   Google Scholar [16] R. Markarian, Billiards with polynomial decayof correlations,, Er. Th. Dynam. Syst., 24 (2004), 177.  doi: 10.1017/S0143385703000270.  Google Scholar [17] D. Ornstein and B. Weiss, On the Bernoulli natureof systems with some hyperbolic structure,, Ergod. Th. Dynam. Sys., 18 (1998), 441.  doi: 10.1017/S0143385798100354.  Google Scholar [18] Ya. B. Pensin, Dynamical Systems With Generalized Hyperbolic Attractors: Hyperbolic, Ergodic and Topological Properties,, Ergod. Theory and Dyn. Syst., 12 (1992), 123.   Google Scholar [19] Ya. G. Sinai, Dynamical systems with elastic reflections.Ergodic properties of diepersing billiards,, Russian Math. Surveys, 25 (1970), 137.  doi: 10.1070/RM1970v025n02ABEH003794.  Google Scholar [20] Ya. G. Sinai and N. Chernov, Ergodicproperties of some systems of two-dimensional discs andthree-dimensional spheres,, Russian Math. Surveys, 42 (1987), 181.  doi: 10.1070/RM1987v042n03ABEH001421.  Google Scholar [21] M. Wojtkowski, Invariant families of cones and Lyapunov exponents,, Ergod. Th. Dynam. Syst., 5 (1985), 145.  doi: 10.1017/S0143385700002807.  Google Scholar [22] L.-S. Young, Statistical properties of systemswith some hyperbolicity including certain billiards,, Ann. Math., 147 (1998), 585.  doi: 10.2307/120960.  Google Scholar
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