Closest distance between iterates of typical points

Boyuan Zhao

doi:10.3934/dcds.2024026

Article Contents

2024, Volume 44, Issue 8: 2252-2279. Doi: 10.3934/dcds.2024026

This issue Previous Article Classification of radial solutions for fully nonlinear equations with Hardy potential Next Article Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Closest distance between iterates of typical points

Boyuan Zhao

University of St Andrews, United Kingdom

Received: September 2023

Revised: March 2024

Early access: March 2024

Published: August 2024

BZ is supported by the China Scholarship Council.

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Abstract

The shortest distance between the first $ n $ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi entropy. For interval maps with a Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.

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Mathematics Subject Classification: Primary: 37B20, 37B10; Secondary: 60F10.

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References

[1]	J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237. doi: 10.1142/S0219493701000114.
[2]	R. Arratia and M. S. Waterman, An Erdös-Rényi law with shifts, Adv. Math., 55 (1985), 13-23. doi: 10.1016/0001-8708(85)90003-9.
[3]	V. Barros, L. Liao and J. Rousseau, On the shortest distance between orbits and the longest common substring problem, Adv. Math., 344 (2019), 311-339. doi: 10.1016/j.aim.2019.01.001.
[4]	V. Barros and J. Rousseau, Shortest distance between multiple orbits and generalized fractal dimensions, Ann. Henri Poincaré, 22 (2021), 1853-1885. doi: 10.1007/s00023-021-01039-y.
[5]	R. Bowen, Equilibrium States and The Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470. Springer, Berlin, 2008.
[6]	P. Collet, C. Giardina and F. Redig, Matching with shift for one-Dimensional Gibbs measures, Ann. Appl. Probab., 19 (2009), 1581-1602. doi: 10.1214/08-AAP588.
[7]	A. Dembo, S. Karlin and O. Zeitouni, Critical phenomena for sequence matching with scoring, Ann. Probab., 22 (1994), 1993-2021.
[8]	A. Dembo, S. Karlin and O. Zeitouni, Limit distribution of maximal non-aligned two-sequence segmental score, Ann. Probab., 22 (1994), 2022-2039.
[9]	C. Gabriela, V. Girardin and L. Loïck, Computation and estimation of generalized entropy rates for denumerable Markov chains, IEEE Trans. Inform. Theory, 57 (2011), 4026-4034. doi: 10.1109/TIT.2011.2133710.
[10]	A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems, Random Comput. Dynam., 5 (1997), 337-347.
[11]	S. Gouëzel, J. Rousseau and M. Stadlbauer, Minimal distance between random orbits, preprint, 2022, arXiv: 2209.13240.
[12]	N. Haydn and S. Vaienti, The Rényi entropy function and the large deviation of short return times, Ergodic Theory and Dynam. Systems, 30 (2010), 159-179. doi: 10.1017/S0143385709000030.
[13]	M. Holland, M. Nicol and A. Török, Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc., 364 (2012), 661-688. doi: 10.1090/S0002-9947-2011-05271-2.
[14]	G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193. doi: 10.1007/BF01240219.
[15]	C. Liverani, B. Sassoul and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory and Dynam. Systems, 19 (1999), 671-685. doi: 10.1017/S0143385799133856.
[16]	R. E. A. C. Paley and A. Zygmund, On some series of functions, (3), Math. Proc. Cambridge Philos. Soc., 28 (1932), 190-205. doi: 10.1017/S0305004100010860.
[17]	O. M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.
[18]	O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758. doi: 10.1090/S0002-9939-03-06927-2.