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 & Continuous Dynamical Systems - A, 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.

[1]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581

[2]

Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001

[3]

V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73

[4]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[5]

Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399

[6]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039

[7]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639

[8]

Saloni Rathee, Nilam. Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3115-3129. doi: 10.3934/dcdsb.2015.20.3115

[9]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597

[10]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167

[11]

Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631

[12]

Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near the anti-integrable limit. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 587-598. doi: 10.3934/dcdsb.2005.5.587

[13]

Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639

[14]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477

[15]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

[16]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143

[17]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677

[18]

Alex Eskin, Gregory Margulis and Shahar Mozes. On a quantitative version of the Oppenheim conjecture. Electronic Research Announcements, 1995, 1: 124-130.

[19]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[20]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (15)
  • HTML views (19)
  • Cited by (0)

Other articles
by authors

[Back to Top]