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On \((\epsilon,k)\)-normal words in connecting dynamical systems. (English) Zbl 0648.10032

It was shown by A. Copeland and P. Erdős [Bull. Am. Math. Soc. 52, 857-860 (1946)] that a real number \(\alpha\) of the form \(\alpha =.a_ 1a_ 2...,\) obtained by catenating the base \(g\) (\(g\geq 2\); integer) digit blocks of a sequence \(a_ 1<a_ 2<..\). of integers, is normal if the \(a_ n\) satisfy a certain growth condition (example: primes).
The authors generalize this result to infinite words obtained by catenating words from a language over a finite alphabet, satisfying a certain “connecting condition” (For any two words a, b there exists a word u of bounded length such that aub is a word). In particular, this leads to a generalization of the Copeland-Erdős Theorem to digit expansion with a Pisot number as base.
Reviewer: B.Volkmann

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11K36 Well-distributed sequences and other variations
94A17 Measures of information, entropy
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References:

[1] Bertrand-Mathis, A.: Développements en base ?, répartion modulo un de la suitex ? n n?0,, langages codés et ?-shifts. Bull. Soc. Math. France114, 271-323 (1986). · Zbl 0628.58024
[2] Bertrand-Mathis, A.: Points génériques de Champernowne sur certains systèmes codés; applications aux ?-shifts. Ergodic Theory and Dynamical Systems8, 35-51 (1988). · Zbl 0657.28014
[3] Bertrand-Mathis, A.: Spécification et synchronisation. University of Bordeaux I: Preprint.
[4] Bertrand-Mathis, A.: Comment écrire des nombres entiers dans une base qui n’est pas entière. Acta Math. Hung. To appear. · Zbl 1383.11009
[5] Besicovitch, A. S.: The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers. Math. Z.39, 146-156 (1935). · Zbl 0009.20002
[6] Billingsley, P.: Ergodic Theory and Information. New York: Wiley. 1965. · Zbl 0141.16702
[7] Blanchard, F., Hansel, G.: Systèmes codés. Theoretical Computer Science44, 17-49 (1986). · Zbl 0601.68056
[8] Champernowne, D.: The construction of decimals normal in the scale of ten. J. London Math. Soc.8, 254-260 (1933). · Zbl 0007.33701
[9] Copeland, A., Erdös, P.: Note on normal numbers. Bull. Amer. Math. Soc.52, 857-860 (1946). · Zbl 0063.00962
[10] Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lect. Notes Math.527. Berlin-Heidelberg-New York: Springer 1976. · Zbl 0328.28008
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