Non-normal numbers in dynamical systems fulfilling the specification property

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

Non-normal numbers in dynamical systems fulfilling the specification property

Manfred G. Madritsch ^1, and Izabela Petrykiewicz ^2,

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

Abstract
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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.

Keywords: Numeration systems, dynamical systems, Baire category, non-normal numbers., specification property.

Mathematics Subject Classification: Primary: 11K16, 37B10; Secondary: 11A63, 54H2.

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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