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 & Continuous Dynamical Systems - A, 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. 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]	A. Bertrand-Mathis, Points génériques de Champernowne sur certains systèmes codes; application aux $\theta$-shifts,, Ergodic Theory Dynam. Systems, 8 (1988), 35. doi: 10.1017/S0143385700004302. Google Scholar
[5]	A. Bertrand-Mathis and B. Volkmann, On $(\epsilon,k)$-normal words in connecting dynamical systems,, Monatsh. Math., 107 (1989), 267. doi: 10.1007/BF01517354. Google Scholar
[6]	E. Borel, Les probabilités dénombrables et leurs applications arithmétiques,, Palermo Rend., 27 (1909), 247. Google Scholar
[7]	K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, vol. 29 of Carus Mathematical Monographs,, Mathematical Association of America, (2002). Google Scholar
[8]	A. O. Gelfond, A common property of number systems,, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809. Google Scholar
[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. doi: 10.4064/aa144-3-6. Google Scholar
[10]	S. Ito and I. Shiokawa, A construction of $\beta $-normal sequences,, J. Math. Soc. Japan, 27 (1975), 20. doi: 10.2969/jmsj/02710020. Google Scholar
[11]	M. G. Madritsch, Non-normal numbers with respect to markov partitions,, Discrete Contin. Dyn. Syst., 34 (2014), 663. doi: 10.3934/dcds.2014.34.663. Google Scholar
[12]	L. Olsen, Extremely non-normal continued fractions,, Acta Arith., 108 (2003), 191. doi: 10.4064/aa108-2-8. Google Scholar
[13]	L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl. (9), 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar
[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. doi: 10.1017/S0305004103007047. Google Scholar
[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. doi: 10.1016/j.bulsci.2004.01.003. Google Scholar
[16]	L. Olsen, Extremely non-normal numbers,, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43. doi: 10.1017/S0305004104007601. Google Scholar
[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. doi: 10.1112/S0024610702003630. Google Scholar
[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. doi: 10.1016/j.bulsci.2006.05.005. Google Scholar
[19]	W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. doi: 10.1007/BF02020954. Google Scholar
[20]	A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar, 8 (1957), 477. Google Scholar
[21]	T. Šalát, Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen,, Czechoslovak Math. J., 18 (93) (1968), 489. Google Scholar
[22]	T. Šalát, A remark on normal numbers,, Rev. Roumaine Math. Pures Appl., 11 (1966), 53. Google Scholar
[23]	T. Šalát, Über die Cantorschen Reihen,, Czechoslovak Math. J., 18 (93) (1968), 25. Google Scholar
[24]	K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar
[25]	B. Volkmann, On non-normal numbers,, Compositio Math., 16 (1964), 186. Google Scholar

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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]	A. Bertrand-Mathis, Points génériques de Champernowne sur certains systèmes codes; application aux $\theta$-shifts,, Ergodic Theory Dynam. Systems, 8 (1988), 35. doi: 10.1017/S0143385700004302. Google Scholar
[5]	A. Bertrand-Mathis and B. Volkmann, On $(\epsilon,k)$-normal words in connecting dynamical systems,, Monatsh. Math., 107 (1989), 267. doi: 10.1007/BF01517354. Google Scholar
[6]	E. Borel, Les probabilités dénombrables et leurs applications arithmétiques,, Palermo Rend., 27 (1909), 247. Google Scholar
[7]	K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, vol. 29 of Carus Mathematical Monographs,, Mathematical Association of America, (2002). Google Scholar
[8]	A. O. Gelfond, A common property of number systems,, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809. Google Scholar
[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. doi: 10.4064/aa144-3-6. Google Scholar
[10]	S. Ito and I. Shiokawa, A construction of $\beta $-normal sequences,, J. Math. Soc. Japan, 27 (1975), 20. doi: 10.2969/jmsj/02710020. Google Scholar
[11]	M. G. Madritsch, Non-normal numbers with respect to markov partitions,, Discrete Contin. Dyn. Syst., 34 (2014), 663. doi: 10.3934/dcds.2014.34.663. Google Scholar
[12]	L. Olsen, Extremely non-normal continued fractions,, Acta Arith., 108 (2003), 191. doi: 10.4064/aa108-2-8. Google Scholar
[13]	L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl. (9), 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar
[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. doi: 10.1017/S0305004103007047. Google Scholar
[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. doi: 10.1016/j.bulsci.2004.01.003. Google Scholar
[16]	L. Olsen, Extremely non-normal numbers,, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43. doi: 10.1017/S0305004104007601. Google Scholar
[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. doi: 10.1112/S0024610702003630. Google Scholar
[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. doi: 10.1016/j.bulsci.2006.05.005. Google Scholar
[19]	W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. doi: 10.1007/BF02020954. Google Scholar
[20]	A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar, 8 (1957), 477. Google Scholar
[21]	T. Šalát, Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen,, Czechoslovak Math. J., 18 (93) (1968), 489. Google Scholar
[22]	T. Šalát, A remark on normal numbers,, Rev. Roumaine Math. Pures Appl., 11 (1966), 53. Google Scholar
[23]	T. Šalát, Über die Cantorschen Reihen,, Czechoslovak Math. J., 18 (93) (1968), 25. Google Scholar
[24]	K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar
[25]	B. Volkmann, On non-normal numbers,, Compositio Math., 16 (1964), 186. Google Scholar

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