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Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
1. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089 |
2. | Department of Mathematics, Harvard-Westlake School, Studio City, CA 91604, United States |
References:
[1] |
M. Abadi, Hitting, returning and the short correlation function, Bull. Braz. Math. Soc., 37 (2006), 593-609.
doi: 10.1007/s00574-006-0030-1. |
[2] |
M. Abadi, Poisson approximations via Chen-Stein for non-Markov processes, in In and Out of Equilibrium 2 (eds. V. Sidoravicius and M. E. Vares), Progr. Probab., 60, Birkhäuser, Basel, 2008, 1-19.
doi: 10.1007/978-3-7643-8786-0_1. |
[3] |
M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity, 21 (2008), 2871-2885.
doi: 10.1088/0951-7715/21/12/008. |
[4] |
J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structure, J. Stat. Phys., 131 (2008), 505-534.
doi: 10.1007/s10955-008-9482-6. |
[5] |
R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: The Chen-Stein method, Ann. Probab., 17 (1989), 9-25.
doi: 10.1214/aop/1176991491. |
[6] |
J.-R. Chazottes and P. Collet, Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 49-80.
doi: 10.1017/S0143385711000897. |
[7] |
P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergod. Th. & Dynam. Sys., 21 (2001), 401-420.
doi: 10.1017/S0143385701001201. |
[8] |
M. Denker, Remarks on weak limit laws for fractal sets, in Fractal Geometry and Stochastics (Finsterbergen, 1994), Progress in Probability, 37, Birkhäuser, Basel, 1995, 167-178.
doi: 10.1007/978-3-0348-7755-8_8. |
[9] |
M. Denker, M. Gordin and A. Sharova, A Poisson limit theorem for toral automorphisms, Illinois J. Math., 48 (2004), 1-20. |
[10] |
W. Doeblin, Remarques sur la théorie métrique des fraction continues, Compositio Mathematica, 7 (1940), 353-371. |
[11] |
D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
[12] |
J. M. Freitas, N. Haydn and M. Nicol, Convergence of rare event point processes to the Poisson process for planar billiards, Nonlinearity, 27 (2014), 1669-1687.
doi: 10.1088/0951-7715/27/7/1669. |
[13] |
A. Galves and B. Schmitt, Inequalities for hitting time in mixing dynamical systems, Random Comput. Dynam., 5 (1997), 337-347. |
[14] |
N. T. A. Haydn, Statistical properties of equilibrium states for rational maps, Ergod. Th. & Dynam. Sys., 20 (2000), 1371-1390.
doi: 10.1017/S0143385700000742. |
[15] |
N. T. A. Haydn, Entry and return times distribution, Dynamical Systems: An International Journal, 28 (2013), 333-353.
doi: 10.1080/14689367.2013.822459. |
[16] |
N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations, Nonlinearity, 27 (2014), 1323-1349.
doi: 10.1088/0951-7715/27/6/1323. |
[17] |
M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergod. Th. & Dynam. Syst., 13 (1993), 533-556.
doi: 10.1017/S0143385700007513. |
[18] |
M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications, Comm. Math. Phys., 206 (1999), 33-55.
doi: 10.1007/s002200050697. |
[19] |
M. Kač, On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc., 53 (1947), 1002-1010.
doi: 10.1090/S0002-9904-1947-08927-8. |
[20] |
M. Kupsa and Y. Lacroix, Asymptotics for hitting times, Ann. of Probab., 33 (2005), 610-619.
doi: 10.1214/009117904000000883. |
[21] |
Y. Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system, Israel J. Math., 132 (2002), 253-263.
doi: 10.1007/BF02784515. |
[22] |
F. Pène and B. Saussol, Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing, preprint, Université de Bretagne Occidentale. |
[23] |
B. Pitskel, Poisson law for Markov chains, Ergod. Th. & Dynam. Syst., 11 (1991), 501-513.
doi: 10.1017/S0143385700006301. |
[24] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3, Gauthiers-Villars, Paris 1899. |
[25] |
K. Wasilewska, Limiting Distribution and Error Terms for the Number of Visits to balls in Mixing Dynamical Systems, Ph.D. thesis, USC 2013. |
[26] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2), 147 (1998), 585-650.
doi: 10.2307/120960. |
[27] |
L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
show all references
References:
[1] |
M. Abadi, Hitting, returning and the short correlation function, Bull. Braz. Math. Soc., 37 (2006), 593-609.
doi: 10.1007/s00574-006-0030-1. |
[2] |
M. Abadi, Poisson approximations via Chen-Stein for non-Markov processes, in In and Out of Equilibrium 2 (eds. V. Sidoravicius and M. E. Vares), Progr. Probab., 60, Birkhäuser, Basel, 2008, 1-19.
doi: 10.1007/978-3-7643-8786-0_1. |
[3] |
M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity, 21 (2008), 2871-2885.
doi: 10.1088/0951-7715/21/12/008. |
[4] |
J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structure, J. Stat. Phys., 131 (2008), 505-534.
doi: 10.1007/s10955-008-9482-6. |
[5] |
R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: The Chen-Stein method, Ann. Probab., 17 (1989), 9-25.
doi: 10.1214/aop/1176991491. |
[6] |
J.-R. Chazottes and P. Collet, Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 49-80.
doi: 10.1017/S0143385711000897. |
[7] |
P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergod. Th. & Dynam. Sys., 21 (2001), 401-420.
doi: 10.1017/S0143385701001201. |
[8] |
M. Denker, Remarks on weak limit laws for fractal sets, in Fractal Geometry and Stochastics (Finsterbergen, 1994), Progress in Probability, 37, Birkhäuser, Basel, 1995, 167-178.
doi: 10.1007/978-3-0348-7755-8_8. |
[9] |
M. Denker, M. Gordin and A. Sharova, A Poisson limit theorem for toral automorphisms, Illinois J. Math., 48 (2004), 1-20. |
[10] |
W. Doeblin, Remarques sur la théorie métrique des fraction continues, Compositio Mathematica, 7 (1940), 353-371. |
[11] |
D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
[12] |
J. M. Freitas, N. Haydn and M. Nicol, Convergence of rare event point processes to the Poisson process for planar billiards, Nonlinearity, 27 (2014), 1669-1687.
doi: 10.1088/0951-7715/27/7/1669. |
[13] |
A. Galves and B. Schmitt, Inequalities for hitting time in mixing dynamical systems, Random Comput. Dynam., 5 (1997), 337-347. |
[14] |
N. T. A. Haydn, Statistical properties of equilibrium states for rational maps, Ergod. Th. & Dynam. Sys., 20 (2000), 1371-1390.
doi: 10.1017/S0143385700000742. |
[15] |
N. T. A. Haydn, Entry and return times distribution, Dynamical Systems: An International Journal, 28 (2013), 333-353.
doi: 10.1080/14689367.2013.822459. |
[16] |
N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations, Nonlinearity, 27 (2014), 1323-1349.
doi: 10.1088/0951-7715/27/6/1323. |
[17] |
M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergod. Th. & Dynam. Syst., 13 (1993), 533-556.
doi: 10.1017/S0143385700007513. |
[18] |
M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications, Comm. Math. Phys., 206 (1999), 33-55.
doi: 10.1007/s002200050697. |
[19] |
M. Kač, On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc., 53 (1947), 1002-1010.
doi: 10.1090/S0002-9904-1947-08927-8. |
[20] |
M. Kupsa and Y. Lacroix, Asymptotics for hitting times, Ann. of Probab., 33 (2005), 610-619.
doi: 10.1214/009117904000000883. |
[21] |
Y. Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system, Israel J. Math., 132 (2002), 253-263.
doi: 10.1007/BF02784515. |
[22] |
F. Pène and B. Saussol, Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing, preprint, Université de Bretagne Occidentale. |
[23] |
B. Pitskel, Poisson law for Markov chains, Ergod. Th. & Dynam. Syst., 11 (1991), 501-513.
doi: 10.1017/S0143385700006301. |
[24] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3, Gauthiers-Villars, Paris 1899. |
[25] |
K. Wasilewska, Limiting Distribution and Error Terms for the Number of Visits to balls in Mixing Dynamical Systems, Ph.D. thesis, USC 2013. |
[26] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2), 147 (1998), 585-650.
doi: 10.2307/120960. |
[27] |
L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
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