American Institute of Mathematical Sciences

Advanced
August  2008, 21(3): 729-747. doi: 10.3934/dcds.2008.21.729

Large deviations for short recurrence

Miguel Abadi 1, and  Sandro Vaienti 2,

1. 

Instituto de Matemática, Estatística e Computaçâo Científica, Universidade Estadual de Campinas, Pça Sergio B. de Holanda 651, Cid. Univ. Campinas, SP, Brazil
2. 

PHYMAT, Université de Toulon et du Var, Centre de Physique Théorique, CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France

Received  May 2007 Revised  October 2007 Published  April 2008

Over a $\psi$-mixing dynamical system we consider the function $\tau(C_n)$ $/n$ in the limit of large $n$, where $\tau(C_n)$ is the first return of a cylinder of length $n$ to itself. Saussol et al. ([30]) proved that this function is constant almost everywhere if the $C_n$ are chosen in a descending sequence of cylinders around a given point. We prove upper and lower general bounds for its large deviation function. Under mild assumptions we compute the large deviation function directly and show that the limit corresponds to the Rényi's entropy of the system. We finally compute the free energy function of $\tau(C_n)$ $/n$. We illustrate our results with a few examples.
Keywords: rare event, large deviations., overlapping, recurrence, Mixing, short correlation.
Mathematics Subject Classification: Primary: 60F05; Secondary: 60G10, 60G55, 37A5.
Citation: Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[1]

Idriss Mazari. Trait selection and rare mutations: The case of large diffusivities. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6693-6724. doi: 10.3934/dcdsb.2019163
[2]

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
[3]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[4]

Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168
[5]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[6]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881
[7]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[8]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[9]

Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021021
[10]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[11]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[12]

Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277
[13]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic and Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
[14]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
[15]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[16]

Wei-Wen Hu. Integer-valued Alexis sequences with large zero correlation zone. Advances in Mathematics of Communications, 2017, 11 (3) : 445-452. doi: 10.3934/amc.2017037
[17]

Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040
[18]

Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049
[19]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[20]

Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy