May  2016, 36(5): 2585-2611. doi: 10.3934/dcds.2016.36.2585

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

Received  December 2014 Revised  September 2015 Published  October 2015

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors.
Citation: Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585
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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