Abstract
We consider a ϕ-mixing shift T on a sequence space Ω and study the number of returns {Tkω ∈ U} to a union U of cylinders of length n until the first return {Tkω ∈ V} to another union V of cylinder sets of length m. It turns out that if probabilities of the sets U and V are small and of the same order, then the above number of returns has approximately geometric distribution. Under appropriate conditions we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymptotical behavior of numbers of returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a “hole”.
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References
M. Abadi, Exponential approximation for hitting times in mixing processes, Mathematical Physics Electronic Journal 7 (2001), Article no. 2.
M. Abadi and B. Saussol, Hitting and returning into rare events for all alpha-mixing processes, Stochastic Processes and their Applications 121 (2011), 314–323.
M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity 21 (2008), 2871–2885.
R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: the Chen-Stein method, Annals of Probability 17 (1989), 9–25.
P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1995.
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975.
R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, UT, 2007.
S. M. Buckley, Is the maximal function of a Lipschitz function continuous? Annales AcademiæScientiarum Fennicæ. Mathematica 24 (1999), 519–528.
M. Demers, P. Wright and L.-S. Young, Entropy, Lyapunov exponents and escape rates in open systems, Ergodic Theory and Dynamical Systems 32 (2012), 1270–1301.
S. Galatolo, J. Rousseau and B. Saussol, Skew products, quantitative recurrence, shrinking targets and decay of correlations, Ergodic Theory and Dynamical Systems 35 (2015), 1814–1845.
A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems, Random & Computational Dynamics 5 (1997), 337–348.
S. Gouëzel, Decay of correlations for nonuniformly expanding systems, Bulletin de la Société Mathématique de France 134 (2006), 1–31.
N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations, Nonlinearity 27 (2014), 1323–1349.
N. Haydn and F. Yang, Local Escape Rates for φ-mixing Dynamical Systems, Ergodic Theory and Dynamical Systems 40 (2020), 2854–280.
L. Heinrich, Mixing properties and central limit theorem for a class of non-identical piecewise monotonic C2-transformations, Mathematische Nachrichten 181 (1996), 185–214.
Yu. Kifer and A. Rapaport, Poisson and compound Poisson approximations in conventional and nonconventional setups, Probability Theory and Related Fields 160 (2014), 797–831.
Yu. Kifer and A. Rapaport, Geometric distribution for multiple returnes until a hazard, Nonlinearity 32 (2019), 1525–1545.
I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Communications in Mathematical Physics 260 (2005), 131–146.
F. Pène and B. Saussol, Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing, Ergodic Theory and Dynamical Systems 36 (2016), 2602–2626.
Y. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformai expanding maps and Moran-like geometric constructions, Journal of Statistical Physics 86 (1997), 233–275.
K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, Vol. 2, Cambridge University Press, Cambridge, 1983.
J. Rousseau and B. Saussol, Poincare recurrence for observations, Transactions of the American Mathematical Society 362 (2010), 5845–5859.
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics 7 (1998), 585–650.
L.-S. Young, Recurrence time and rate of mixing, Israel Journal of Mathematics 110 (1999), 153–188.
R. Zweimüller, Mixing limit theorems for ergodic transformations, Journal of Theoretical Probability 20 (2007), 1059–1071.
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Kifer, Y., Yang, F. Geometric law for numbers of returns until a hazard under ϕ-mixing. Isr. J. Math. 244, 319–357 (2021). https://doi.org/10.1007/s11856-021-2182-5
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DOI: https://doi.org/10.1007/s11856-021-2182-5