doi: 10.3934/dcds.2020339

Dynamical Borel–Cantelli lemmas

Lund University, Sweden

Received  May 2020 Revised  July 2020 Published  October 2020

This paper is a study of Borel–Cantelli lemmas in dynamical systems. D. Kleinbock and G. Margulis [7] have given a very useful sufficient condition for strongly Borel–Cantelli sequences, which is based on the work of W. M. Schmidt [10], [11]. We will obtain a weaker sufficient condition for the strongly Borel–Cantelli sequences. Two versions of the dynamical Borel–Cantelli lemmas will be deduced by extending a theorem by W. M. Schmidt [11], W. J. LeVeque [8], and W. Philipp [9]. Some applications of our theorems will also be discussed. Firstly, a characterization of the strongly Borel–Cantelli sequences in one-dimensional Gibbs–Markov systems will be established. This will improve the theorem of C. Gupta, M. Nicol, and W. Ott in [4]. Secondly, N. Haydn, M. Nicol, T. Persson, and S. Vaienti [5] proved the strong Borel–Cantelli property in sequences of balls in terms of a polynomial decay of correlations for Lipschitz observables. Our theorems will then be applied to relax their inequality assumption.

Citation: Viktoria Xing. Dynamical Borel–Cantelli lemmas. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020339
References:
[1]

E. Borel, Les probabilit$\acute{e}$s d$\acute{e}$nombrables et leurs applications arithmetiques, Rend. Circ. Mat. Palermo, 27 (1909), 247-271.  doi: 10.1007/BF03019651.  Google Scholar

[2]

F. P. Cantelli, Sulla probabilit$\grave{a}$ come limite della frequenza, Atti delta Reale Accademia Nationale dei Lincei, Serie V, Rendicotti, 26 (1917), 39-45.   Google Scholar

[3]

N. Chernov and D. Kleinbock, Dynamical Borel–Cantelli lemmas for Gibbs measures, Isreal J. Math., 122 (2001), 1-27.  doi: 10.1007/BF02809888.  Google Scholar

[4]

C. GuptaM. Nicol and W. Ott, A Borel–Cantelli lemma for non–uniformly expanding dynamical systems, Nonlinearity, 23 (2010), 1991-2008.  doi: 10.1088/0951-7715/23/8/010.  Google Scholar

[5]

N. HaydnM. NicolT. Persson and S. Vaienti, A note on Borel–Cantelli lemmas for non–uniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[6]

D. Khoshnevisan, Probability, Graduate Studies in Mathematics, 80, AMS, 2007. doi: 10.1090/gsm/080.  Google Scholar

[7]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Inv. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.  Google Scholar

[8]

W. J. LeVeque, On the frequency of small fractional parts in certain real sequences III, Journal Reine Angew. Math., 202 (1959), 215-220.  doi: 10.1515/crll.1959.202.215.  Google Scholar

[9]

W. Philipp, Some metrical theorems in number theory, Pacific J. Math, 20 (1967), 109-127.  doi: 10.2140/pjm.1967.20.109.  Google Scholar

[10]

W. M. Schmidt, A metrical theorem in of Diophantine approximation, Canad. J. Math, 12 (1960), 619-631.  doi: 10.4153/CJM-1960-056-0.  Google Scholar

[11]

W. M. Schmidt, Metrical theorems on fractional parts of sequences, Transactions AMS, 110 (1964), 493-518.  doi: 10.1090/S0002-9947-1964-0159802-4.  Google Scholar

[12]

C. E. Silva, Invitation to Ergodic Theory, American Mathematical Soc., 2008. doi: 10.1090/stml/042.  Google Scholar

[13]

V.Sprindžuk, Metric Theory of Diophantine Approximations, J. Wiley & Sons, New York–Toronto–London, 1979. Google Scholar

show all references

References:
[1]

E. Borel, Les probabilit$\acute{e}$s d$\acute{e}$nombrables et leurs applications arithmetiques, Rend. Circ. Mat. Palermo, 27 (1909), 247-271.  doi: 10.1007/BF03019651.  Google Scholar

[2]

F. P. Cantelli, Sulla probabilit$\grave{a}$ come limite della frequenza, Atti delta Reale Accademia Nationale dei Lincei, Serie V, Rendicotti, 26 (1917), 39-45.   Google Scholar

[3]

N. Chernov and D. Kleinbock, Dynamical Borel–Cantelli lemmas for Gibbs measures, Isreal J. Math., 122 (2001), 1-27.  doi: 10.1007/BF02809888.  Google Scholar

[4]

C. GuptaM. Nicol and W. Ott, A Borel–Cantelli lemma for non–uniformly expanding dynamical systems, Nonlinearity, 23 (2010), 1991-2008.  doi: 10.1088/0951-7715/23/8/010.  Google Scholar

[5]

N. HaydnM. NicolT. Persson and S. Vaienti, A note on Borel–Cantelli lemmas for non–uniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[6]

D. Khoshnevisan, Probability, Graduate Studies in Mathematics, 80, AMS, 2007. doi: 10.1090/gsm/080.  Google Scholar

[7]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Inv. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.  Google Scholar

[8]

W. J. LeVeque, On the frequency of small fractional parts in certain real sequences III, Journal Reine Angew. Math., 202 (1959), 215-220.  doi: 10.1515/crll.1959.202.215.  Google Scholar

[9]

W. Philipp, Some metrical theorems in number theory, Pacific J. Math, 20 (1967), 109-127.  doi: 10.2140/pjm.1967.20.109.  Google Scholar

[10]

W. M. Schmidt, A metrical theorem in of Diophantine approximation, Canad. J. Math, 12 (1960), 619-631.  doi: 10.4153/CJM-1960-056-0.  Google Scholar

[11]

W. M. Schmidt, Metrical theorems on fractional parts of sequences, Transactions AMS, 110 (1964), 493-518.  doi: 10.1090/S0002-9947-1964-0159802-4.  Google Scholar

[12]

C. E. Silva, Invitation to Ergodic Theory, American Mathematical Soc., 2008. doi: 10.1090/stml/042.  Google Scholar

[13]

V.Sprindžuk, Metric Theory of Diophantine Approximations, J. Wiley & Sons, New York–Toronto–London, 1979. Google Scholar

[1]

Dong Han Kim. The dynamical Borel-Cantelli lemma for interval maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 891-900. doi: 10.3934/dcds.2007.17.891

[2]

L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274

[3]

Dominique Lecomte. Hurewicz-like tests for Borel subsets of the plane. Electronic Research Announcements, 2005, 11: 95-102.

[4]

Óscar Vega-Amaya, Joaquín López-Borbón. A perturbation approach to a class of discounted approximate value iteration algorithms with borel spaces. Journal of Dynamics & Games, 2016, 3 (3) : 261-278. doi: 10.3934/jdg.2016014

[5]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1

[6]

Mike Boyle. The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics. Journal of Modern Dynamics, 2019, 15: 427-435. doi: 10.3934/jmd.2019026

[7]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

[8]

Ahmad Deeb, A. Hamdouni, Dina Razafindralandy. Comparison between Borel-Padé summation and factorial series, as time integration methods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 393-408. doi: 10.3934/dcdss.2016003

[9]

Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834

[10]

Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004

[11]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[12]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249

[13]

Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669

[14]

Maxime Herda, Luis Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic & Related Models, 2019, 12 (3) : 593-636. doi: 10.3934/krm.2019024

[15]

Julian Newman. Synchronisation of almost all trajectories of a random dynamical system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4163-4177. doi: 10.3934/dcds.2020176

[16]

Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531

[17]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091

[18]

Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507

[19]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533

[20]

Mourad Bellassoued, Chaima Moufid. Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020394

2019 Impact Factor: 1.338

Article outline

[Back to Top]