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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

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, 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.  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), 615-630. 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), 935-943. 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), 35-51. 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), 267-279. doi: 10.1007/BF01517354.  Google Scholar

[6]

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

[7]

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

[8]

A. O. Gelfond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814.  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), 287-293. 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), 20-23. 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), 663-676. doi: 10.3934/dcds.2014.34.663.  Google Scholar

[12]

L. Olsen, Extremely non-normal continued fractions, Acta Arith., 108 (2003), 191-202. 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), 1591-1649. 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), 139-165. 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), 265-289. doi: 10.1016/j.bulsci.2004.01.003.  Google Scholar

[16]

L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. 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), 103-122. 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), 518-558. 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), 401-416. doi: 10.1007/BF02020954.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

B. Volkmann, On non-normal numbers, Compositio Math., 16 (1964), 186-190.  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), 1163-1170. 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), 615-630. 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), 935-943. 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), 35-51. 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), 267-279. doi: 10.1007/BF01517354.  Google Scholar

[6]

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

[7]

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

[8]

A. O. Gelfond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814.  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), 287-293. 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), 20-23. 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), 663-676. doi: 10.3934/dcds.2014.34.663.  Google Scholar

[12]

L. Olsen, Extremely non-normal continued fractions, Acta Arith., 108 (2003), 191-202. 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), 1591-1649. 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), 139-165. 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), 265-289. doi: 10.1016/j.bulsci.2004.01.003.  Google Scholar

[16]

L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. 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), 103-122. 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), 518-558. 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), 401-416. doi: 10.1007/BF02020954.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

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