# American Institute of Mathematical Sciences

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. 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.
 [1] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 [2] V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73 [3] 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 [4] Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399 [6] Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete and Continuous Dynamical Systems, 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 and Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639 [8] Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258 [9] Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 [10] Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597 [11] James Nolen. A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167 [12] Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems and Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631 [13] Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near the anti-integrable limit. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 587-598. doi: 10.3934/dcdsb.2005.5.587 [14] Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $\omega$-limit sets of multivalued semiflows. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096 [15] Saloni Rathee, Nilam. Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3115-3129. doi: 10.3934/dcdsb.2015.20.3115 [16] Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic and Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031 [17] Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639 [18] Baoyin Xun, Kam C. Yuen, Kaiyong Wang. The finite-time ruin probability of a risk model with a general counting process and stochastic return. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1541-1556. doi: 10.3934/jimo.2021032 [19] 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 [20] Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2

2020 Impact Factor: 1.392