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 and 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).

[2]

Miguel Abadi and Nicolas Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity, 21 (2008), 2871-2885. doi: 10.1088/0951-7715/21/12/008.

[3]

Ray V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562. doi: 10.1090/S0002-9939-1969-0247028-5.

[4]

V. Chaumoitre and M. Kupsa, Asympotics for return times of rank-one systems, Stochastics and Dynamics, 5 (2005), 65-73. doi: 10.1142/S0219493705001298.

[5]

V. Chaumoitre and M. Kupsa, $k$-limit laws of return and hitting times, Discrete and Continuous Dynamical Systems A, 15 (2006), 73-86. doi: 10.3934/dcds.2006.15.73.

[6]

Tomasz Downarowicz and Yves Lacroix, Law of series, Ergodic Theory and Dynam. Systems, 31 (2011), 351-367. doi: 10.1017/S0143385709001217.

[7]

Sébastien Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65.

[8]

Paulina Grzegorek and Michal Kupsa, Return times in a process generated by a typical partition, Nonlinearity, 22 (2009), 371-379. doi: 10.1088/0951-7715/22/2/007.

[9]

A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems, Random Comput. Dynam., 5 (1997), 337-347.

[10]

N. Haydn, Y. Lacroix and S. Vaienti, Hitting and return times in ergodic dynamical systems, Annals of Probability, 33 (2005), 2043-2050. doi: 10.1214/009117905000000242.

[11]

M. Kupsa and Y. Lacroix, Asymptotics for hitting times, Annals of Probability, 33 (2005), 610-619. doi: 10.1214/009117904000000883.

[12]

Yves Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system, Israel J. Math., 132 (2002), 1253-264. doi: 10.1007/BF02784515.

[13]

B. Pitskel, Poisson limit law for markov chains, Ergodic Theory Dynam. Systems, 11 (1991), 501-513. doi: 10.1017/S0143385700006301.

show all references

References:
[1]

Miguel Abadi, Exponential approximation for hitting times in mixing processes, Math. Phys. Electron. J., 7 (2001).

[2]

Miguel Abadi and Nicolas Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity, 21 (2008), 2871-2885. doi: 10.1088/0951-7715/21/12/008.

[3]

Ray V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562. doi: 10.1090/S0002-9939-1969-0247028-5.

[4]

V. Chaumoitre and M. Kupsa, Asympotics for return times of rank-one systems, Stochastics and Dynamics, 5 (2005), 65-73. doi: 10.1142/S0219493705001298.

[5]

V. Chaumoitre and M. Kupsa, $k$-limit laws of return and hitting times, Discrete and Continuous Dynamical Systems A, 15 (2006), 73-86. doi: 10.3934/dcds.2006.15.73.

[6]

Tomasz Downarowicz and Yves Lacroix, Law of series, Ergodic Theory and Dynam. Systems, 31 (2011), 351-367. doi: 10.1017/S0143385709001217.

[7]

Sébastien Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65.

[8]

Paulina Grzegorek and Michal Kupsa, Return times in a process generated by a typical partition, Nonlinearity, 22 (2009), 371-379. doi: 10.1088/0951-7715/22/2/007.

[9]

A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems, Random Comput. Dynam., 5 (1997), 337-347.

[10]

N. Haydn, Y. Lacroix and S. Vaienti, Hitting and return times in ergodic dynamical systems, Annals of Probability, 33 (2005), 2043-2050. doi: 10.1214/009117905000000242.

[11]

M. Kupsa and Y. Lacroix, Asymptotics for hitting times, Annals of Probability, 33 (2005), 610-619. doi: 10.1214/009117904000000883.

[12]

Yves Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system, Israel J. Math., 132 (2002), 1253-264. doi: 10.1007/BF02784515.

[13]

B. Pitskel, Poisson limit law for markov chains, Ergodic Theory Dynam. Systems, 11 (1991), 501-513. doi: 10.1017/S0143385700006301.

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