# 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, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585
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##### References:
 [1] Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems, 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, 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, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639 [4] Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19 [5] 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 [6] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [7] Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635 [8] 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 [9] V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73 [10] Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185 [11] Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327 [12] María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084 [13] Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159 [14] Maria José Pacifico, Fan Yang. Hitting times distribution and extreme value laws for semi-flows. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5861-5881. doi: 10.3934/dcds.2017255 [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] Yunqing Zou, Zhengkui Lin, Dongya Han, T. C. Edwin Cheng, Chin-Chia Wu. Two-agent integrated scheduling of production and distribution operations with fixed departure times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021005 [17] 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 [18] Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34. [19] Josselin Garnier, Knut Solna. Filtered Kirchhoff migration of cross correlations of ambient noise signals. Inverse Problems & Imaging, 2011, 5 (2) : 371-390. doi: 10.3934/ipi.2011.5.371 [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

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