Dynamical Systems/Number Theory
p-adic affine dynamical systems and applications
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 129-134.

We consider p-adic affine dynamical systems on the ring Zp of all p-adic integers, and we find a necessary and sufficient condition for such a system to be minimal. The minimality is equivalent to the transitivity, the ergodicity of the Haar measure, the unique ergodicity, and the strict ergodicity. When the condition is not satisfied, we prove that the system can be decomposed into strict ergodic subsystems. One of our applications is the study of the divisibility, by a power of prime number, of the sequence of integers anb with positive integers a,b and n.

Nous considérons les systèmes dynamiques affines p-adiques sur l'anneau Zp des entiers p-adiques. Nous obtenons une condition nécessaire et suffisante pour qu'un tel système soit minimal. La minimalité est équivalente à la transitivité, à l'ergodicité de la mesure de Haar, à l'unique ergodicité, et à la stricte ergodicité. Quand la condition n'est pas satisfaite, nous donnons tous les sous-systèmes strictement ergodiques du système affine p-adique en question. L'une de nos applications est l'étude de la divisibilité, par une puissance d'un nombre premier, de la suite des entiers de la forme anb (a,b et n étant des entiers positifs).

Published online:
DOI: 10.1016/j.crma.2005.11.017
Fan, Ai-Hua 1, 2; Li, Ming-Tian 2; Yao, Jia-Yan 2; Zhou, Dan 2

1 LAMFA, UMR 6140 CNRS, université de Picardie, 33, rue Saint-Leu, 80039 Amiens, France
2 Department of Mathematics, Wuhan University, 430072 Wuhan, PR China
     author = {Fan, Ai-Hua and Li, Ming-Tian and Yao, Jia-Yan and Zhou, Dan},
     title = {\protect\emph{p}-adic affine dynamical systems and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {129--134},
     publisher = {Elsevier},
     volume = {342},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2005.11.017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.11.017/}
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Fan, Ai-Hua; Li, Ming-Tian; Yao, Jia-Yan; Zhou, Dan. p-adic affine dynamical systems and applications. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 129-134. doi : 10.1016/j.crma.2005.11.017. http://www.numdam.org/articles/10.1016/j.crma.2005.11.017/

[1] Coelho, Z.; Parry, W. Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 51-70

[2] Khrennikov, A.; Lindahl, K.-O.; Gundlach, M. Ergodicity in the p-adic framework, Operator Methods in Ordinary and Partial Differential Equations, Stockholm, 2000, Oper. Theory Adv. Appl., vol. 132, Birkhäuser, 2002, pp. 245-251

[3] Serre, J.-P. A Course in Arithmetic, Grad. Texts in Math., vol. 7, Springer-Verlag, 1973 (See also Cours d'arithmétique, Paris PUF, 1970)

[4] Walters, P. An Introduction to Ergodic Theory, Grad. Texts in Math., vol. 79, Springer-Verlag, 1982

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