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

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

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.

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. PéneB. 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. PéneB. 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. PéneB. 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. PéneB. 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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