Advanced Search
Article Contents
Article Contents

Non-normal numbers in dynamical systems fulfilling the specification property

Abstract Related Papers Cited by
  • In the present paper we want to focus on the dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Cesàro variants diverge.
    Mathematics Subject Classification: Primary: 11K16, 37B10; Secondary: 11A63, 54H20.


    \begin{equation} \\ \end{equation}
  • [1]

    S. Albeverio, M. Pratsiovytyi and G. Torbin, Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their $s$-adic digits, Ukraïn. Mat. Zh., 57 (2005), 1163-1170.doi: 10.1007/s11253-006-0001-0.


    S. Albeverio, M. Pratsiovytyi and G. Torbin, Topological and fractal properties of real numbers which are not normal, Bull. Sci. Math., 129 (2005), 615-630.doi: 10.1016/j.bulsci.2004.12.004.


    I.-S. Baek and L. Olsen, Baire category and extremely non-normal points of invariant sets of IFS's, Discrete Contin. Dyn. Syst., 27 (2010), 935-943.doi: 10.3934/dcds.2010.27.935.


    A. Bertrand-Mathis, Points génériques de Champernowne sur certains systèmes codes; application aux $\theta$-shifts, Ergodic Theory Dynam. Systems, 8 (1988), 35-51.doi: 10.1017/S0143385700004302.


    A. Bertrand-Mathis and B. Volkmann, On $(\epsilon,k)$-normal words in connecting dynamical systems, Monatsh. Math., 107 (1989), 267-279.doi: 10.1007/BF01517354.


    E. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Palermo Rend., 27 (1909), 247-271.


    K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, vol. 29 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 2002.


    A. O. Gelfond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814.


    J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesàro averages, frequencies of digits, and Baire category, Acta Arith., 144 (2010), 287-293.doi: 10.4064/aa144-3-6.


    S. Ito and I. Shiokawa, A construction of $\beta $-normal sequences, J. Math. Soc. Japan, 27 (1975), 20-23.doi: 10.2969/jmsj/02710020.


    M. G. Madritsch, Non-normal numbers with respect to markov partitions, Discrete Contin. Dyn. Syst., 34 (2014), 663-676.doi: 10.3934/dcds.2014.34.663.


    L. Olsen, Extremely non-normal continued fractions, Acta Arith., 108 (2003), 191-202.doi: 10.4064/aa108-2-8.


    L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. (9), 82 (2003), 1591-1649.doi: 10.1016/j.matpur.2003.09.007.


    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.


    L. Olsen, Applications of multifractal divergence points to some sets of {$d$}-tuples of numbers defined by their $N$-adic expansion, Bull. Sci. Math., 128 (2004), 265-289.doi: 10.1016/j.bulsci.2004.01.003.


    L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53.doi: 10.1017/S0305004104007601.


    L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc. (2), 67 (2003), 103-122.doi: 10.1112/S0024610702003630.


    L. Olsen and S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II. Non-linearity, divergence points and Banach space valued spectra, Bull. Sci. Math., 131 (2007), 518-558.doi: 10.1016/j.bulsci.2006.05.005.


    W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.doi: 10.1007/BF02020954.


    A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493.


    T. Šalát, Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen, Czechoslovak Math. J., 18 (93) (1968), 489-522.


    T. Šalát, A remark on normal numbers, Rev. Roumaine Math. Pures Appl., 11 (1966), 53-56.


    T. Šalát, Über die Cantorschen Reihen, Czechoslovak Math. J., 18 (93) (1968), 25-56.


    K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299.doi: 10.1090/S0002-9947-1974-0352411-X.


    B. Volkmann, On non-normal numbers, Compositio Math., 16 (1964), 186-190.

  • 加载中

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint