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Statistical properties of type D dispersing billiards
1. | Penn State University, State College, PA, 16802, USA |
2. | Yeshiva University, New York, NY, 10016, USA |
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 [
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. |
show all references
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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