# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s00574-006-0030-1.  Google Scholar [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), (2008), 1.  doi: 10.1007/978-3-7643-8786-0_1.  Google Scholar [3] M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes,, Nonlinearity, 21 (2008), 2871.  doi: 10.1088/0951-7715/21/12/008.  Google Scholar [4] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structure,, J. Stat. Phys., 131 (2008), 505.  doi: 10.1007/s10955-008-9482-6.  Google Scholar [5] R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: The Chen-Stein method,, Ann. Probab., 17 (1989), 9.  doi: 10.1214/aop/1176991491.  Google Scholar [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.  doi: 10.1017/S0143385711000897.  Google Scholar [7] P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems,, Ergod. Th. & Dynam. Sys., 21 (2001), 401.  doi: 10.1017/S0143385701001201.  Google Scholar [8] M. Denker, Remarks on weak limit laws for fractal sets,, in Fractal Geometry and Stochastics (Finsterbergen, (1994), 167.  doi: 10.1007/978-3-0348-7755-8_8.  Google Scholar [9] M. Denker, M. Gordin and A. Sharova, A Poisson limit theorem for toral automorphisms,, Illinois J. Math., 48 (2004), 1.   Google Scholar [10] W. Doeblin, Remarques sur la théorie métrique des fraction continues,, Compositio Mathematica, 7 (1940), 353.   Google Scholar [11] D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Am. Math. Soc., 356 (2004), 1637.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar [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.  doi: 10.1088/0951-7715/27/7/1669.  Google Scholar [13] A. Galves and B. Schmitt, Inequalities for hitting time in mixing dynamical systems,, Random Comput. Dynam., 5 (1997), 337.   Google Scholar [14] N. T. A. Haydn, Statistical properties of equilibrium states for rational maps,, Ergod. Th. & Dynam. Sys., 20 (2000), 1371.  doi: 10.1017/S0143385700000742.  Google Scholar [15] N. T. A. Haydn, Entry and return times distribution,, Dynamical Systems: An International Journal, 28 (2013), 333.  doi: 10.1080/14689367.2013.822459.  Google Scholar [16] N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations,, Nonlinearity, 27 (2014), 1323.  doi: 10.1088/0951-7715/27/6/1323.  Google Scholar [17] M. Hirata, Poisson law for Axiom A diffeomorphisms,, Ergod. Th. & Dynam. Syst., 13 (1993), 533.  doi: 10.1017/S0143385700007513.  Google Scholar [18] M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications,, Comm. Math. Phys., 206 (1999), 33.  doi: 10.1007/s002200050697.  Google Scholar [19] M. Kač, On the notion of recurrence in discrete stochastic processes,, Bull. Amer. Math. Soc., 53 (1947), 1002.  doi: 10.1090/S0002-9904-1947-08927-8.  Google Scholar [20] M. Kupsa and Y. Lacroix, Asymptotics for hitting times,, Ann. of Probab., 33 (2005), 610.  doi: 10.1214/009117904000000883.  Google Scholar [21] Y. Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system,, Israel J. Math., 132 (2002), 253.  doi: 10.1007/BF02784515.  Google Scholar [22] F. Pène and B. Saussol, Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing,, preprint, ().   Google Scholar [23] B. Pitskel, Poisson law for Markov chains,, Ergod. Th. & Dynam. Syst., 11 (1991), 501.  doi: 10.1017/S0143385700006301.  Google Scholar [24] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste,, Vol. 3, (1899).   Google Scholar [25] K. Wasilewska, Limiting Distribution and Error Terms for the Number of Visits to balls in Mixing Dynamical Systems,, Ph.D. thesis, (2013).   Google Scholar [26] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. of Math. (2), 147 (1998), 585.  doi: 10.2307/120960.  Google Scholar [27] L.-S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar

show all references

##### References:
 [1] M. Abadi, Hitting, returning and the short correlation function,, Bull. Braz. Math. Soc., 37 (2006), 593.  doi: 10.1007/s00574-006-0030-1.  Google Scholar [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), (2008), 1.  doi: 10.1007/978-3-7643-8786-0_1.  Google Scholar [3] M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes,, Nonlinearity, 21 (2008), 2871.  doi: 10.1088/0951-7715/21/12/008.  Google Scholar [4] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structure,, J. Stat. Phys., 131 (2008), 505.  doi: 10.1007/s10955-008-9482-6.  Google Scholar [5] R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: The Chen-Stein method,, Ann. Probab., 17 (1989), 9.  doi: 10.1214/aop/1176991491.  Google Scholar [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.  doi: 10.1017/S0143385711000897.  Google Scholar [7] P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems,, Ergod. Th. & Dynam. Sys., 21 (2001), 401.  doi: 10.1017/S0143385701001201.  Google Scholar [8] M. Denker, Remarks on weak limit laws for fractal sets,, in Fractal Geometry and Stochastics (Finsterbergen, (1994), 167.  doi: 10.1007/978-3-0348-7755-8_8.  Google Scholar [9] M. Denker, M. Gordin and A. Sharova, A Poisson limit theorem for toral automorphisms,, Illinois J. Math., 48 (2004), 1.   Google Scholar [10] W. Doeblin, Remarques sur la théorie métrique des fraction continues,, Compositio Mathematica, 7 (1940), 353.   Google Scholar [11] D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Am. Math. Soc., 356 (2004), 1637.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar [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.  doi: 10.1088/0951-7715/27/7/1669.  Google Scholar [13] A. Galves and B. Schmitt, Inequalities for hitting time in mixing dynamical systems,, Random Comput. Dynam., 5 (1997), 337.   Google Scholar [14] N. T. A. Haydn, Statistical properties of equilibrium states for rational maps,, Ergod. Th. & Dynam. Sys., 20 (2000), 1371.  doi: 10.1017/S0143385700000742.  Google Scholar [15] N. T. A. Haydn, Entry and return times distribution,, Dynamical Systems: An International Journal, 28 (2013), 333.  doi: 10.1080/14689367.2013.822459.  Google Scholar [16] N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations,, Nonlinearity, 27 (2014), 1323.  doi: 10.1088/0951-7715/27/6/1323.  Google Scholar [17] M. Hirata, Poisson law for Axiom A diffeomorphisms,, Ergod. Th. & Dynam. Syst., 13 (1993), 533.  doi: 10.1017/S0143385700007513.  Google Scholar [18] M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications,, Comm. Math. Phys., 206 (1999), 33.  doi: 10.1007/s002200050697.  Google Scholar [19] M. Kač, On the notion of recurrence in discrete stochastic processes,, Bull. Amer. Math. Soc., 53 (1947), 1002.  doi: 10.1090/S0002-9904-1947-08927-8.  Google Scholar [20] M. Kupsa and Y. Lacroix, Asymptotics for hitting times,, Ann. of Probab., 33 (2005), 610.  doi: 10.1214/009117904000000883.  Google Scholar [21] Y. Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system,, Israel J. Math., 132 (2002), 253.  doi: 10.1007/BF02784515.  Google Scholar [22] F. Pène and B. Saussol, Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing,, preprint, ().   Google Scholar [23] B. Pitskel, Poisson law for Markov chains,, Ergod. Th. & Dynam. Syst., 11 (1991), 501.  doi: 10.1017/S0143385700006301.  Google Scholar [24] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste,, Vol. 3, (1899).   Google Scholar [25] K. Wasilewska, Limiting Distribution and Error Terms for the Number of Visits to balls in Mixing Dynamical Systems,, Ph.D. thesis, (2013).   Google Scholar [26] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. of Math. (2), 147 (1998), 585.  doi: 10.2307/120960.  Google Scholar [27] L.-S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar
 [1] Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445 [2] Nicolai Haydn, Sandro Vaienti. The limiting distribution and error terms for return times of dynamical systems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 589-616. doi: 10.3934/dcds.2004.10.589 [3] Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639 [4] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [5] Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635 [6] Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339 [7] V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73 [8] Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19 [9] Ioannis Konstantoulas. Effective decay of multiple correlations in semidirect product actions. Journal of Modern Dynamics, 2016, 10: 81-111. doi: 10.3934/jmd.2016.10.81 [10] Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327 [11] María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084 [12] Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185 [13] Maria José Pacifico, Fan Yang. Hitting times distribution and extreme value laws for semi-flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5861-5881. doi: 10.3934/dcds.2017255 [14] Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159 [15] Antonio Pumariño, Joan Carles Tatjer. Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 971-1005. doi: 10.3934/dcdsb.2007.8.971 [16] Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110 [17] Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34. [18] Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399 [19] Hans-Otto Walther. Contracting return maps for monotone delayed feedback. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 259-274. doi: 10.3934/dcds.2001.7.259 [20] Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems & Imaging, 2014, 8 (3) : 645-683. doi: 10.3934/ipi.2014.8.645

2018 Impact Factor: 1.143

## Metrics

• PDF downloads (15)
• HTML views (0)
• Cited by (3)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]