# 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 & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
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##### References:
 [1] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems, 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, 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, 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, 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, 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] Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete & 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 & 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 & 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 & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167 [12] 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 [13] 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 [14] 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 [15] Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $\omega$-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096 [16] Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic & Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031 [17] 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 & Management Optimization, 2021  doi: 10.3934/jimo.2021032 [18] Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639 [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

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