November  2014, 34(11): 4751-4764. doi: 10.3934/dcds.2014.34.4751

Non-normal numbers in dynamical systems fulfilling the specification property

1. 

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

2. 

Université Joseph Fourier, Institut Fourier, 100 rue des maths, 38402 St Martin d'Hères, France

Received  November 2013 Revised  February 2014 Published  May 2014

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.
Citation: Manfred G. Madritsch, Izabela Petrykiewicz. Non-normal numbers in dynamical systems fulfilling the specification property. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4751-4764. doi: 10.3934/dcds.2014.34.4751
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), 1163-1170. doi: 10.1007/s11253-006-0001-0.

[2]

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

[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), 935-943. doi: 10.3934/dcds.2010.27.935.

[4]

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.

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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.

[10]

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

[11]

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.

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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.

[25]

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

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), 1163-1170. doi: 10.1007/s11253-006-0001-0.

[2]

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

[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), 935-943. doi: 10.3934/dcds.2010.27.935.

[4]

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.

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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.

[10]

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

[11]

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.

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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.

[25]

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

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