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: |
M. Abadi
and A. Galves
, Inequalities for the occurrence times of rare events in mixing processes, Markov Process. Related Fields, 7 (2001)
, 97-112.
![]() ![]() |
|
R. Bowen,
Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
W. Feller,
An Introduction to Probability Theory and its Application 2 2nd edition, Wiley, New york, 1971.
doi: 10.2307/3029053.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Hirata
, Poisson law for Axiom A diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993)
, 533-556.
doi: 10.1017/S0143385700007513.![]() ![]() ![]() |
|
S. V. Nagaev
, Some limit theorems for stationary Markov chains, Theor. Probab. Appl., 2 (1957)
, 378-406.
![]() ![]() |
|
S. V. Nagaev
, More exact statement of limit theores of homogeneous Markov chains, Theor. Probab. Appl., 6 (1961)
, 62-81.
![]() ![]() |
|
F. Péne
and B. Saussol
, Back to balls in billiards, Comm. Math. Phys., 293 (2010)
, 837-866.
doi: 10.1007/s00220-009-0911-4.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
B. Saussol
, An introduction to quantitative poincaré recurrence in dynamical systems, Reviews in Mathematical Physics, 21 (2009)
, 949-979.
doi: 10.1142/S0129055X09003785.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |