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Expansions in Complex Bases

Published online by Cambridge University Press:  20 November 2018

Vilmos Komornik  and
Paola Loreti
Vilmos Komornik
Affiliation:
Département de mathématique, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg Cedex, France e-mail: komornik@math.u-strasbg.fr
Paola Loreti
Affiliation:
Dipartimento di Metodi e Modelli, Matematici per le Scienze Applicate, Sapienza Università di Roma, Via A. Scarpa, 16, 00161 Roma, Italy e-mail: loreti@dmmm.uniroma1.it
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Abstract

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Beginning with a seminal paper of Rényi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daróczy and Kátai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

Keywords

11A6711A63 11B85non-integer basesgreedy expansionsbeta-expansions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 3 , 01 September 2007 , pp. 399 - 408
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Daróczy, Z., Járay, A., and Kátai, I., Intervallfüllende Folgen und volladditive Funktionen. Acta Sci. Math. (Szeged) 50(1986), no. 3–4, 337–350.Google Scholar
[2] Daróczy, Z. and Kátai, I., Generalized number systems in the complex plane. Acta Math. Hungar. 51(1988), no. 3–4, 409–416.Google Scholar
[3] Erdőos, P., Horváth, M., and Joó, I., On the uniqueness of the expansions 1 = Σq–ni . Acta Math. Hungar. 58(1991), no. 3-4333-342.Google Scholar
[4] Erdőos, P., Joó, I. and Komornik, V., Characterization of the unique expansion and related problems. Bull. Soc. Math. France 118(1990), no. 3, 377–390.Google Scholar
[5] Frougny, C. and Solomyak, B., Finite β-expansions. Ergodic Theory Dynam. Systems 12(1992), no. 4, 713–723.Google Scholar
[6] Komornik, V. and Loreti, P., Subexpansions, superexpansions and uniqueness properties in non-integer bases. Period. Math. Hungar. 44(2002), no. 2, 197–218.Google Scholar
[7] Parry, W., On the β-expansion of real numbers. Acta Math. Acad. Sci. Hungar. 11(1960), 401–416.Google Scholar
[8] Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8(1957), 477–493.Google Scholar
[9] Sidorov, N., Arithmetic dynamics. In: Topics in Dynamics and Ergodic Theory. London Math. Soc. Lecture Note Ser. 310, Cambridge University Press, Cambridge, 2003, pp. 145–189.Google Scholar