# American Institute of Mathematical Sciences

May  2012, 11(3): 1339-1361. doi: 10.3934/cpaa.2012.11.1339

## Exponential return times in a zero-entropy process

 1 Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wysńskiego 27, 50-370 Wrocław, Poland 2 Institute of Information Theory and Automation, The Academy of Sciences of the Czech Republic, Prague 8, CZ-18208, Czech Republic

Received  December 2010 Revised  April 2011 Published  December 2011

We construct a zero-entropy weakly mixing finite-valued process with the exponential limit law for return resp. hitting times. This limit law is obtained in almost every point, taking the limit along the full sequence of cylinders around the point. Till now, the exponential limit law for return resp. hitting times, taking the limit along the full sequence of cylinders, have been obtained only in positive-entropy processes satisfying some strong mixing conditions of Rosenblatt type.
Citation: 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
##### References:
 [1] Miguel Abadi, Exponential approximation for hitting times in mixing processes,, Math. Phys. Electron. J., 7 (2001).   Google Scholar [2] Miguel Abadi and Nicolas 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 [3] Ray V. Chacon, Weakly mixing transformations which are not strongly mixing,, Proc. Amer. Math. Soc., 22 (1969), 559.  doi: 10.1090/S0002-9939-1969-0247028-5.  Google Scholar [4] V. Chaumoitre and M. Kupsa, Asympotics for return times of rank-one systems,, Stochastics and Dynamics, 5 (2005), 65.  doi: 10.1142/S0219493705001298.  Google Scholar [5] V. Chaumoitre and M. Kupsa, $k$-limit laws of return and hitting times,, Discrete and Continuous Dynamical Systems A, 15 (2006), 73.  doi: 10.3934/dcds.2006.15.73.  Google Scholar [6] Tomasz Downarowicz and Yves Lacroix, Law of series,, Ergodic Theory and Dynam. Systems, 31 (2011), 351.  doi: 10.1017/S0143385709001217.  Google Scholar [7] Sébastien Ferenczi, Systems of finite rank,, Colloq. Math., 73 (1997), 35.   Google Scholar [8] Paulina Grzegorek and Michal Kupsa, Return times in a process generated by a typical partition,, Nonlinearity, 22 (2009), 371.  doi: 10.1088/0951-7715/22/2/007.  Google Scholar [9] A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems,, Random Comput. Dynam., 5 (1997), 337.   Google Scholar [10] N. Haydn, Y. Lacroix and S. Vaienti, Hitting and return times in ergodic dynamical systems,, Annals of Probability, 33 (2005), 2043.  doi: 10.1214/009117905000000242.  Google Scholar [11] M. Kupsa and Y. Lacroix, Asymptotics for hitting times,, Annals of Probability, 33 (2005), 610.  doi: 10.1214/009117904000000883.  Google Scholar [12] Yves Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system,, Israel J. Math., 132 (2002), 1253.  doi: 10.1007/BF02784515.  Google Scholar [13] B. Pitskel, Poisson limit law for markov chains,, Ergodic Theory Dynam. Systems, 11 (1991), 501.  doi: 10.1017/S0143385700006301.  Google Scholar

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##### References:
 [1] Miguel Abadi, Exponential approximation for hitting times in mixing processes,, Math. Phys. Electron. J., 7 (2001).   Google Scholar [2] Miguel Abadi and Nicolas 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 [3] Ray V. Chacon, Weakly mixing transformations which are not strongly mixing,, Proc. Amer. Math. Soc., 22 (1969), 559.  doi: 10.1090/S0002-9939-1969-0247028-5.  Google Scholar [4] V. Chaumoitre and M. Kupsa, Asympotics for return times of rank-one systems,, Stochastics and Dynamics, 5 (2005), 65.  doi: 10.1142/S0219493705001298.  Google Scholar [5] V. Chaumoitre and M. Kupsa, $k$-limit laws of return and hitting times,, Discrete and Continuous Dynamical Systems A, 15 (2006), 73.  doi: 10.3934/dcds.2006.15.73.  Google Scholar [6] Tomasz Downarowicz and Yves Lacroix, Law of series,, Ergodic Theory and Dynam. Systems, 31 (2011), 351.  doi: 10.1017/S0143385709001217.  Google Scholar [7] Sébastien Ferenczi, Systems of finite rank,, Colloq. Math., 73 (1997), 35.   Google Scholar [8] Paulina Grzegorek and Michal Kupsa, Return times in a process generated by a typical partition,, Nonlinearity, 22 (2009), 371.  doi: 10.1088/0951-7715/22/2/007.  Google Scholar [9] A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems,, Random Comput. Dynam., 5 (1997), 337.   Google Scholar [10] N. Haydn, Y. Lacroix and S. Vaienti, Hitting and return times in ergodic dynamical systems,, Annals of Probability, 33 (2005), 2043.  doi: 10.1214/009117905000000242.  Google Scholar [11] M. Kupsa and Y. Lacroix, Asymptotics for hitting times,, Annals of Probability, 33 (2005), 610.  doi: 10.1214/009117904000000883.  Google Scholar [12] Yves Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system,, Israel J. Math., 132 (2002), 1253.  doi: 10.1007/BF02784515.  Google Scholar [13] B. Pitskel, Poisson limit law for markov chains,, Ergodic Theory Dynam. Systems, 11 (1991), 501.  doi: 10.1017/S0143385700006301.  Google Scholar
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