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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 |
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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