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April  2021, 41(4): 1737-1754. doi: 10.3934/dcds.2020339

Dynamical Borel–Cantelli lemmas

Viktoria Xing

Lund University, Sweden

Received  May 2020 Revised  July 2020 Published  April 2021 Early access  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.

Keywords: Borel–Cantelli lemma, strongly Borel–Cantelli sequence, dynamical system, Gibbs–Markov system, Lipschitz observables.
Mathematics Subject Classification: Primary: 37A05; Secondary: 60F15.
Citation: Viktoria Xing. Dynamical Borel–Cantelli lemmas. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1737-1754. 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. Gupta, M. 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. Haydn, M. Nicol, T. 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. Gupta, M. 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. Haydn, M. Nicol, T. 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

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