October  2016, 36(10): 5477-5492. doi: 10.3934/dcds.2016041

Zero-one law of Hausdorff dimensions of the recurrent sets

1. 

Department of Mathematics Education, Dongguk University - Seoul, Seoul 04620, South Korea

2. 

Department of Mathematics, South China University of Technology, Guangzhou, 510641

Received  October 2015 Revised  March 2016 Published  July 2016

Let $(\Sigma, \sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma: \liminf_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\beta\right\},$$ where $\varphi: \mathbb{N}\to \mathbb{R}^+$ is a monotonically increasing function and $0\leq\alpha\leq\beta\leq +\infty$. We show that the Hausdorff dimension of the set $E^\varphi_{\alpha,\beta}$ admits a dichotomy: it is either zero or one depending on $\varphi, \alpha$ and $\beta$.
Citation: Dong Han Kim, Bing Li. Zero-one law of Hausdorff dimensions of the recurrent sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5477-5492. doi: 10.3934/dcds.2016041
References:
[1]

J.-C. Ban and B. Li, The multifractal spectra for the recurrence rates of beta-transformations,, J. Math. Anal. Appl., 420 (2014), 1662.  doi: 10.1016/j.jmaa.2014.06.051.  Google Scholar

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L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincare recurrence,, Comm. Math. Phys., 219 (2001), 443.  doi: 10.1007/s002200100427.  Google Scholar

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M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617.  doi: 10.1007/BF01244320.  Google Scholar

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H.-B. Chen, Z.-X. Wen and M. Yu, The multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system,, J. Math. Anal. Appl., 422 (2015), 1264.  doi: 10.1016/j.jmaa.2014.09.023.  Google Scholar

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K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications,, John Wiley & Sons, (1990).   Google Scholar

[6]

D. J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces,, Nonlinearity, 14 (2001), 81.  doi: 10.1088/0951-7715/14/1/304.  Google Scholar

[7]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).   Google Scholar

[8]

S. Galatolo, Dimension and hitting time in rapidly mixing systems,, Math. Res. Lett., 14 (2007), 797.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[9]

S. Galatolo, D. H. Kim and K. K. Park, The recurrence time for ergodic systems with infinite invariant measures,, Nonlinearity, 19 (2006), 2567.  doi: 10.1088/0951-7715/19/11/004.  Google Scholar

[10]

S. Galatolo and M. J. Pacifico, Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence,, Ergodic Theory Dynam. Systems, 30 (2010), 1703.  doi: 10.1017/S0143385709000856.  Google Scholar

[11]

R. Hill and S. Velani, The ergodic theory of shrinking targets,, Invent. Math., 119 (1995), 175.  doi: 10.1007/BF01245179.  Google Scholar

[12]

K. S. Lau and L. Shu, The spectrum of Poincare recurrence,, Ergod. Th. Dynam. Sys., 28 (2008), 1917.  doi: 10.1017/S0143385707001095.  Google Scholar

[13]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[14]

L. Olsen, First return times: Multifractal spectra and divergence points,, Discrete Contin. Dyn. Syst., 10 (2004), 635.  doi: 10.3934/dcds.2004.10.635.  Google Scholar

[15]

L. Olsen, Applications of multifractal divergence points to sets of numbers defined by their $N$-adic expansion,, Math. Proc. Cambridge Philos. Soc., 136 (2004), 139.  doi: 10.1017/S0305004103007047.  Google Scholar

[16]

D. Ornstein and B. Weiss, Entropy and data compression schemes,, IEEE Trans. Inform. Theory, 39 (1993), 78.  doi: 10.1109/18.179344.  Google Scholar

[17]

L. Peng, Dimension of sets of sequences defined in terms of recurrence of their prefixes,, C. R. Math. Acad. Sci. Paris, 343 (2006), 129.  doi: 10.1016/j.crma.2006.05.005.  Google Scholar

[18]

L. Peng, B. Tan and B.-W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system,, Sci. China Math., 55 (2012), 131.  doi: 10.1007/s11425-011-4303-9.  Google Scholar

[19]

J. Rousseau, Recurrence rates for observations of flows,, Ergodic Theory Dynam. Systems, 32 (2012), 1727.  doi: 10.1017/S014338571100037X.  Google Scholar

[20]

J. Rousseau and B. Saussol, Poincaré recurrence for observations,, Trans. Amer. Math. Soc., 362 (2010), 5845.  doi: 10.1090/S0002-9947-2010-05078-0.  Google Scholar

[21]

B. Saussol, Recurrence rate in rapidly mixing dynamical system,, Discrete. Contin. Dyn. Sys. ser.A, 15 (2006), 259.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[22]

B. Saussol and J. Wu, Recurrence spectrum in smooth dynamical system,, Nonlinearity, 16 (2003), 1991.  doi: 10.1088/0951-7715/16/6/306.  Google Scholar

[23]

B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system,, Adv. Math., 228 (2011), 2071.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

show all references

References:
[1]

J.-C. Ban and B. Li, The multifractal spectra for the recurrence rates of beta-transformations,, J. Math. Anal. Appl., 420 (2014), 1662.  doi: 10.1016/j.jmaa.2014.06.051.  Google Scholar

[2]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincare recurrence,, Comm. Math. Phys., 219 (2001), 443.  doi: 10.1007/s002200100427.  Google Scholar

[3]

M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617.  doi: 10.1007/BF01244320.  Google Scholar

[4]

H.-B. Chen, Z.-X. Wen and M. Yu, The multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system,, J. Math. Anal. Appl., 422 (2015), 1264.  doi: 10.1016/j.jmaa.2014.09.023.  Google Scholar

[5]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications,, John Wiley & Sons, (1990).   Google Scholar

[6]

D. J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces,, Nonlinearity, 14 (2001), 81.  doi: 10.1088/0951-7715/14/1/304.  Google Scholar

[7]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).   Google Scholar

[8]

S. Galatolo, Dimension and hitting time in rapidly mixing systems,, Math. Res. Lett., 14 (2007), 797.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[9]

S. Galatolo, D. H. Kim and K. K. Park, The recurrence time for ergodic systems with infinite invariant measures,, Nonlinearity, 19 (2006), 2567.  doi: 10.1088/0951-7715/19/11/004.  Google Scholar

[10]

S. Galatolo and M. J. Pacifico, Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence,, Ergodic Theory Dynam. Systems, 30 (2010), 1703.  doi: 10.1017/S0143385709000856.  Google Scholar

[11]

R. Hill and S. Velani, The ergodic theory of shrinking targets,, Invent. Math., 119 (1995), 175.  doi: 10.1007/BF01245179.  Google Scholar

[12]

K. S. Lau and L. Shu, The spectrum of Poincare recurrence,, Ergod. Th. Dynam. Sys., 28 (2008), 1917.  doi: 10.1017/S0143385707001095.  Google Scholar

[13]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[14]

L. Olsen, First return times: Multifractal spectra and divergence points,, Discrete Contin. Dyn. Syst., 10 (2004), 635.  doi: 10.3934/dcds.2004.10.635.  Google Scholar

[15]

L. Olsen, Applications of multifractal divergence points to sets of numbers defined by their $N$-adic expansion,, Math. Proc. Cambridge Philos. Soc., 136 (2004), 139.  doi: 10.1017/S0305004103007047.  Google Scholar

[16]

D. Ornstein and B. Weiss, Entropy and data compression schemes,, IEEE Trans. Inform. Theory, 39 (1993), 78.  doi: 10.1109/18.179344.  Google Scholar

[17]

L. Peng, Dimension of sets of sequences defined in terms of recurrence of their prefixes,, C. R. Math. Acad. Sci. Paris, 343 (2006), 129.  doi: 10.1016/j.crma.2006.05.005.  Google Scholar

[18]

L. Peng, B. Tan and B.-W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system,, Sci. China Math., 55 (2012), 131.  doi: 10.1007/s11425-011-4303-9.  Google Scholar

[19]

J. Rousseau, Recurrence rates for observations of flows,, Ergodic Theory Dynam. Systems, 32 (2012), 1727.  doi: 10.1017/S014338571100037X.  Google Scholar

[20]

J. Rousseau and B. Saussol, Poincaré recurrence for observations,, Trans. Amer. Math. Soc., 362 (2010), 5845.  doi: 10.1090/S0002-9947-2010-05078-0.  Google Scholar

[21]

B. Saussol, Recurrence rate in rapidly mixing dynamical system,, Discrete. Contin. Dyn. Sys. ser.A, 15 (2006), 259.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[22]

B. Saussol and J. Wu, Recurrence spectrum in smooth dynamical system,, Nonlinearity, 16 (2003), 1991.  doi: 10.1088/0951-7715/16/6/306.  Google Scholar

[23]

B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system,, Adv. Math., 228 (2011), 2071.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

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