Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions
Recent progresses in boundary layer theory
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
Nicolai T. A. Haydn 1, and Kasia Wasilewska 2,
Department of Mathematics, University of Southern California, Los Angeles, CA 90089
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.
Keywords: attractors., polynomial decay, correlations, Return times distribution.
Mathematics Subject Classification: Primary: 37D25; Secondary: 60G1.
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
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
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
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
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
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
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
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
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
M. Denker, M. Gordin and A. Sharova, A Poisson limit theorem for toral automorphisms,, Illinois J. Math., 48 (2004), 1. Google Scholar
W. Doeblin, Remarques sur la théorie métrique des fraction continues,, Compositio Mathematica, 7 (1940), 353. Google Scholar
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
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
A. Galves and B. Schmitt, Inequalities for hitting time in mixing dynamical systems,, Random Comput. Dynam., 5 (1997), 337. Google Scholar
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
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
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
M. Hirata, Poisson law for Axiom A diffeomorphisms,, Ergod. Th. & Dynam. Syst., 13 (1993), 533. doi: 10.1017/S0143385700007513. Google Scholar
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
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
M. Kupsa and Y. Lacroix, Asymptotics for hitting times,, Ann. of Probab., 33 (2005), 610. doi: 10.1214/009117904000000883. Google Scholar
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
F. Pène and B. Saussol, Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing,, preprint, (). Google Scholar
B. Pitskel, Poisson law for Markov chains,, Ergod. Th. & Dynam. Syst., 11 (1991), 501. doi: 10.1017/S0143385700006301. Google Scholar
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste,, Vol. 3, (1899). Google Scholar
K. Wasilewska, Limiting Distribution and Error Terms for the Number of Visits to balls in Mixing Dynamical Systems,, Ph.D. thesis, (2013). Google Scholar
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. of Math. (2), 147 (1998), 585. doi: 10.2307/120960. Google Scholar
L.-S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153. doi: 10.1007/BF02808180. Google Scholar
Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020125
Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133
Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004
Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041
Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020456
Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021002
Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $ 2\times 2 $ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020358
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
Nicolai T. A. Haydn Kasia Wasilewska