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Zero-one law of Hausdorff dimensions of the recurrent sets

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  • 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 $\beta$.
    Mathematics Subject Classification: 28A80.

    Citation:

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  • [1]

    J.-C. Ban and B. Li, The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420 (2014), 1662-1679.doi: 10.1016/j.jmaa.2014.06.051.

    [2]

    L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincare recurrence, Comm. Math. Phys., 219 (2001), 443-463.doi: 10.1007/s002200100427.

    [3]

    M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.doi: 10.1007/BF01244320.

    [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), 1264-1276.doi: 10.1016/j.jmaa.2014.09.023.

    [5]

    K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.

    [6]

    D. J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, 14 (2001), 81-85.doi: 10.1088/0951-7715/14/1/304.

    [7]

    H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.

    [8]

    S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.doi: 10.4310/MRL.2007.v14.n5.a8.

    [9]

    S. Galatolo, D. H. Kim and K. K. Park, The recurrence time for ergodic systems with infinite invariant measures, Nonlinearity, 19 (2006), 2567-2580.doi: 10.1088/0951-7715/19/11/004.

    [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), 1703-1737.doi: 10.1017/S0143385709000856.

    [11]

    R. Hill and S. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.doi: 10.1007/BF01245179.

    [12]

    K. S. Lau and L. Shu, The spectrum of Poincare recurrence, Ergod. Th. Dynam. Sys., 28 (2008), 1917-1943.doi: 10.1017/S0143385707001095.

    [13]

    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511623813.

    [14]

    L. Olsen, First return times: Multifractal spectra and divergence points, Discrete Contin. Dyn. Syst., 10 (2004), 635-656.doi: 10.3934/dcds.2004.10.635.

    [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), 139-165.doi: 10.1017/S0305004103007047.

    [16]

    D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory, 39 (1993), 78-83.doi: 10.1109/18.179344.

    [17]

    L. Peng, Dimension of sets of sequences defined in terms of recurrence of their prefixes, C. R. Math. Acad. Sci. Paris, 343 (2006), 129-133.doi: 10.1016/j.crma.2006.05.005.

    [18]

    L. Peng, B. Tan and B.-W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.doi: 10.1007/s11425-011-4303-9.

    [19]

    J. Rousseau, Recurrence rates for observations of flows, Ergodic Theory Dynam. Systems, 32 (2012), 1727-1751.doi: 10.1017/S014338571100037X.

    [20]

    J. Rousseau and B. Saussol, Poincaré recurrence for observations, Trans. Amer. Math. Soc., 362 (2010), 5845-5859.doi: 10.1090/S0002-9947-2010-05078-0.

    [21]

    B. Saussol, Recurrence rate in rapidly mixing dynamical system, Discrete. Contin. Dyn. Sys. ser.A, 15 (2006), 259-267.doi: 10.3934/dcds.2006.15.259.

    [22]

    B. Saussol and J. Wu, Recurrence spectrum in smooth dynamical system, Nonlinearity, 16 (2003), 1991-2001.doi: 10.1088/0951-7715/16/6/306.

    [23]

    B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.doi: 10.1016/j.aim.2011.06.034.

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