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January  2018, 38(1): 343-361. doi: 10.3934/dcds.2018017

Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one

Nasab Yassine ,

Université de Bretagne Occidentale, LMBA, CNRS UMR 6205, Institut des sciences et Techniques, 29238 Brest Cedex 3, France

* Corresponding author: Nasab Yassine

Received  September 2016 Revised  July 2017 Published  September 2017

We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of $\mathbb{Z}$-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.

Keywords: Return time, quantitative recurrence, local limit theorem.
Mathematics Subject Classification: Primary:37B20;Secondary:37A50, 60F05.
Citation: Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
References:
[1]

M. Abadi and A. Galves, Inequalities for the occurrence times of rare events in mixing processes, Markov Process. Related Fields, 7 (2001), 97-112. 

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.

[3]

X. Bressaud and R. Zweimüller, Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure, Ann. Henri Poincaré, 2 (2001), 501-512.  doi: 10.1007/PL00001042.

[4]

W. Feller, An Introduction to Probability Theory and its Application 2 2nd edition, Wiley, New york, 1971. doi: 10.2307/3029053.

[5]

Y. Givarc'h and J. Hardy, Théorémes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Annales Inst. H. Poincaré(B), Probabilités et Statistiques, 24 (1988), 73-98. 

[6]

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness Lecture Notes in Mathematics, 1766 Springer, Berlin, 2001. doi: 10.1007/b87874.

[7]

M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 533-556.  doi: 10.1017/S0143385700007513.

[8]

S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab. Appl., 2 (1957), 378-406. 

[9]

S. V. Nagaev, More exact statement of limit theores of homogeneous Markov chains, Theor. Probab. Appl., 6 (1961), 62-81. 

[10]

F. Péne and B. Saussol, Back to balls in billiards, Comm. Math. Phys., 293 (2010), 837-866.  doi: 10.1007/s00220-009-0911-4.

[11]

F. Péne and B. Saussol, Quantitative recurrence in two-dimensioinal extended processes, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 1065-1084.  doi: 10.1214/08-AIHP195.

[12]

F. Péne, B. Saussol and R. Zweimüller, Recurrence rates and hitting-time distributions for random walks on the line, The Annals of Probability, 41 (2013), 619-635.  doi: 10.1214/11-AOP698.

[13]

F. Péne, B. Saussol and R. Zweimüller, Return and hitting time limits for rare events of null-recurrent Markov maps, Ergod. Th. Dynam. Sys., 37 (2017), 244-276.  doi: 10.1017/etds.2015.38.

[14]

B. Saussol, An introduction to quantitative poincaré recurrence in dynamical systems, Reviews in Mathematical Physics, 21 (2009), 949-979.  doi: 10.1142/S0129055X09003785.

[15]

B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete and Continuous Dynamical Systems, 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259.

show all references

References:
[1]

M. Abadi and A. Galves, Inequalities for the occurrence times of rare events in mixing processes, Markov Process. Related Fields, 7 (2001), 97-112. 

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.

[3]

X. Bressaud and R. Zweimüller, Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure, Ann. Henri Poincaré, 2 (2001), 501-512.  doi: 10.1007/PL00001042.

[4]

W. Feller, An Introduction to Probability Theory and its Application 2 2nd edition, Wiley, New york, 1971. doi: 10.2307/3029053.

[5]

Y. Givarc'h and J. Hardy, Théorémes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Annales Inst. H. Poincaré(B), Probabilités et Statistiques, 24 (1988), 73-98. 

[6]

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness Lecture Notes in Mathematics, 1766 Springer, Berlin, 2001. doi: 10.1007/b87874.

[7]

M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 533-556.  doi: 10.1017/S0143385700007513.

[8]

S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab. Appl., 2 (1957), 378-406. 

[9]

S. V. Nagaev, More exact statement of limit theores of homogeneous Markov chains, Theor. Probab. Appl., 6 (1961), 62-81. 

[10]

F. Péne and B. Saussol, Back to balls in billiards, Comm. Math. Phys., 293 (2010), 837-866.  doi: 10.1007/s00220-009-0911-4.

[11]

F. Péne and B. Saussol, Quantitative recurrence in two-dimensioinal extended processes, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 1065-1084.  doi: 10.1214/08-AIHP195.

[12]

F. Péne, B. Saussol and R. Zweimüller, Recurrence rates and hitting-time distributions for random walks on the line, The Annals of Probability, 41 (2013), 619-635.  doi: 10.1214/11-AOP698.

[13]

F. Péne, B. Saussol and R. Zweimüller, Return and hitting time limits for rare events of null-recurrent Markov maps, Ergod. Th. Dynam. Sys., 37 (2017), 244-276.  doi: 10.1017/etds.2015.38.

[14]

B. Saussol, An introduction to quantitative poincaré recurrence in dynamical systems, Reviews in Mathematical Physics, 21 (2009), 949-979.  doi: 10.1142/S0129055X09003785.

[15]

B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete and Continuous Dynamical Systems, 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259.

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