February  2014, 34(2): 663-676. doi: 10.3934/dcds.2014.34.663

Non-normal numbers with respect to Markov partitions

1. 

Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France

Received  December 2012 Revised  December 2012 Published  August 2013

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.
Citation: Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663
References:
[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. doi: 10.1007/s11253-006-0001-0. Google Scholar

[2]

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

[3]

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. doi: 10.3934/dcds.2010.27.935. Google Scholar

[4]

G. Barat, V. Berthé, P. Liardet and J. Thuswaldner, Dynamical directions in numeration,, Numération, 56 (2006), 1987. doi: 10.5802/aif.2233. Google Scholar

[5]

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

[6]

K. Dajani and C. Kraaikamp, "Ergodic Theory of Numbers,", Carus Mathematical Monographs, (2002). Google Scholar

[7]

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

[8]

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. doi: 10.4064/aa144-3-6. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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. doi: 10.1016/j.bulsci.2004.01.003. Google Scholar

[13]

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

[14]

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. doi: 10.1112/S0024610702003630. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

show all references

References:
[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. doi: 10.1007/s11253-006-0001-0. Google Scholar

[2]

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

[3]

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. doi: 10.3934/dcds.2010.27.935. Google Scholar

[4]

G. Barat, V. Berthé, P. Liardet and J. Thuswaldner, Dynamical directions in numeration,, Numération, 56 (2006), 1987. doi: 10.5802/aif.2233. Google Scholar

[5]

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

[6]

K. Dajani and C. Kraaikamp, "Ergodic Theory of Numbers,", Carus Mathematical Monographs, (2002). Google Scholar

[7]

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

[8]

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. doi: 10.4064/aa144-3-6. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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. doi: 10.1016/j.bulsci.2004.01.003. Google Scholar

[13]

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

[14]

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. doi: 10.1112/S0024610702003630. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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