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Non-normal numbers with respect to Markov partitions

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  • We call a real number normal if for any block of digits the asymptotic frequency of this block in the $N$-adic expansion equals the expected one. In the present paper we consider non-normal numbers and, in particular, essentially and extremely non-normal numbers. We call a real number essentially non-normal if for each single digit there exists no asymptotic frequency of its occurrence. Furthermore we call a real number extremely non-normal if all possible probability vectors are accumulation points of the sequence of frequency vectors. Our aim now is to extend and generalize these results to Markov partitions.
    Mathematics Subject Classification: Primary: 11K16, 37B10; Secondary: 11A63, 54H20.


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  • [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.


    G. Barat, V. Berthé, P. Liardet and J. Thuswaldner, Dynamical directions in numeration, Numération, pavages, substitutions, Ann. Inst. Fourier (Grenoble), 56 (2006), 1987-2092.doi: 10.5802/aif.2233.


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


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


    K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl., 1 (1994), 131-170.doi: 10.1007/s00041-001-4007-6.


    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.


    D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511626302.


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


    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.


    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.


    B. Volkmann, Über Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI, Math. Z., 68 (1958), 439-449.


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

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