American Institute of Mathematical Sciences

October  2022, 42(10): 4823-4851. doi: 10.3934/dcds.2022073

Statistical properties of type D dispersing billiards

 1 Penn State University, State College, PA, 16802, USA 2 Yeshiva University, New York, NY, 10016, USA

*Corresponding author: Péter Nándori

Received  August 2020 Revised  December 2021 Published  October 2022 Early access  June 2022

Fund Project: MB was partially supported by NSF DMS 1800811 and PN was partially supported by NSF DMS 1800811 and NSF DMS 1952876

We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian [9]. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties.

Citation: Margaret Brown, Péter Nándori. Statistical properties of type D dispersing billiards. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4823-4851. doi: 10.3934/dcds.2022073
References:
 [1] H. Attarchi, M. Bolding and L. A. Bunimovich, Ehrenfests' wind-tree model is dynamically Richer than the Lorentz gas, Journal of Statistical Physics, 180 (2020), 440-458.  doi: 10.1007/s10955-019-02460-8. [2] P. M. Bleher, Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J. Stat. Phys., 66 (1992), 315-373.  doi: 10.1007/BF01060071. [3] L. A. Bunimovich and Ya. G. Sina${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Statistical properties of Lorentz gas with periodic configuration of scatterers,, Comm. Math. Phys., 78 (1981), 479-497.  doi: 10.1007/BF02046760. [4] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markov partitions for two dimensional hyperbolic billiard, Uspekhi Mat. Nauk, 45 (1990), 97-134.  doi: 10.1070/RM1990v045n03ABEH002355. [5] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of twodimensional hyperbolic billiards, Uspekhi Mat. Nauk, 46 (1991), 43-92.  doi: 10.1070/RM1991v046n04ABEH002827. [6] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939. [7] N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094.  doi: 10.1007/s10955-006-9036-8. [8] N. Chernov and D. Dolgopyat, Brownian Brownian Motion - I, Memoirs of American Mathematical Society, 198 (2009), 927. doi: 10.1090/memo/0927. [9] N. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys & Monographs, 127 AMS, Providence, RI, 2006. xii+316 pp. doi: 10.1090/surv/127. [10] N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity, 18 (2005), 1527-1553.  doi: 10.1088/0951-7715/18/4/006. [11] N. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3. [12] J. De Simoi and D. Dolgopyat, Dispersing Fermi-Ulam models, arXiv: 2003.00053, (2020). [13] J. De Simoi and I. P. Tóth, An expansion estimate for dispersing planar billiards with corner points, Annals Henri Poincaré, 15 (2014), 1223-1243.  doi: 10.1007/s00023-013-0272-6. [14] M. F. Demers and H.-K Zhang, A Functional Analytic Approach to Perturbations of the Lorentz Gas, Communications in Mathematical Physics, 324 (2013), 767–830. doi: 10.1007/s00220-013-1820-0. [15] M. F. Demers and H.-K Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379. [16] F. Pène and D. Terhesiu, Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon, Comm. Math. Phys., 382 (2021), 1625-1689.  doi: 10.1007/s00220-021-03984-5. [17] W. Philipp and W. Stout, Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, Memoir. Amer.Math. Soc., 161 (1975). doi: 10.1090/memo/0161. [18] L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dyn. Syst., 28 (2008), 587-612.  doi: 10.1017/S0143385707000478. [19] Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards,, Math. Surv., 25 (1970), 137-189.  doi: 10.1070/RM1970v025n02ABEH003794. [20] D. Szász and T. Varjú, Limit laws and recurrence for the Lorentz process with infinite horizon, J. Stat. Phys., 129 (2007), 59-80.  doi: 10.1007/s10955-007-9367-0. [21] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.

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References:
 [1] H. Attarchi, M. Bolding and L. A. Bunimovich, Ehrenfests' wind-tree model is dynamically Richer than the Lorentz gas, Journal of Statistical Physics, 180 (2020), 440-458.  doi: 10.1007/s10955-019-02460-8. [2] P. M. Bleher, Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J. Stat. Phys., 66 (1992), 315-373.  doi: 10.1007/BF01060071. [3] L. A. Bunimovich and Ya. G. Sina${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Statistical properties of Lorentz gas with periodic configuration of scatterers,, Comm. Math. Phys., 78 (1981), 479-497.  doi: 10.1007/BF02046760. [4] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markov partitions for two dimensional hyperbolic billiard, Uspekhi Mat. Nauk, 45 (1990), 97-134.  doi: 10.1070/RM1990v045n03ABEH002355. [5] L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of twodimensional hyperbolic billiards, Uspekhi Mat. Nauk, 46 (1991), 43-92.  doi: 10.1070/RM1991v046n04ABEH002827. [6] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939. [7] N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094.  doi: 10.1007/s10955-006-9036-8. [8] N. Chernov and D. Dolgopyat, Brownian Brownian Motion - I, Memoirs of American Mathematical Society, 198 (2009), 927. doi: 10.1090/memo/0927. [9] N. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys & Monographs, 127 AMS, Providence, RI, 2006. xii+316 pp. doi: 10.1090/surv/127. [10] N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity, 18 (2005), 1527-1553.  doi: 10.1088/0951-7715/18/4/006. [11] N. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3. [12] J. De Simoi and D. Dolgopyat, Dispersing Fermi-Ulam models, arXiv: 2003.00053, (2020). [13] J. De Simoi and I. P. Tóth, An expansion estimate for dispersing planar billiards with corner points, Annals Henri Poincaré, 15 (2014), 1223-1243.  doi: 10.1007/s00023-013-0272-6. [14] M. F. Demers and H.-K Zhang, A Functional Analytic Approach to Perturbations of the Lorentz Gas, Communications in Mathematical Physics, 324 (2013), 767–830. doi: 10.1007/s00220-013-1820-0. [15] M. F. Demers and H.-K Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379. [16] F. Pène and D. Terhesiu, Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon, Comm. Math. Phys., 382 (2021), 1625-1689.  doi: 10.1007/s00220-021-03984-5. [17] W. Philipp and W. Stout, Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, Memoir. Amer.Math. Soc., 161 (1975). doi: 10.1090/memo/0161. [18] L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dyn. Syst., 28 (2008), 587-612.  doi: 10.1017/S0143385707000478. [19] Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards,, Math. Surv., 25 (1970), 137-189.  doi: 10.1070/RM1970v025n02ABEH003794. [20] D. Szász and T. Varjú, Limit laws and recurrence for the Lorentz process with infinite horizon, J. Stat. Phys., 129 (2007), 59-80.  doi: 10.1007/s10955-007-9367-0. [21] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.
A scatterer with a convex corner point (left) and a concave corner point (right)
A simple corridor bounded by two corner points
Singular trajectories after a long flight. The trajectory on the top panel is tangent to the scatterer on the left and the trajectory on the bottom panel is tangent to the scatterer on the right. A neighborhood of the first collision point is magnified for better visibility
Singularity structure near type 3 and type 1 boundary points Similar figures can be found in [4,Figure 11]. An unstable curve is indicated with bold on both panels
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