Full paper in PDF:
$%S. De Smedt and A. Khrennikov, A p-adic behavior of dynamical systems, Rev. Mat. Complut. 12 (1999), no. 2, 301–323.%$

A p -adic Behavior of Dynamical Systems
Stany DE SMEDT and Andrew KHRENNIKOV
Faculty of Applied Sciences
Vrije Universiteit Brussel
Pleinaan 2
1050 Brussel — Belgium
stany.desmedt@kb.be
Department of Mathematics
Rikkyo University
Ikebukuro, Toshima-ku
Tokyo 171 — Japan

Received: September 20, 1997
Revised: February 1, 1999

ABSTRACT

We study dynamical systems in the non-Archimedean number fields (i.e., fields with non-Archimedean valuation). The main results are obtained for the fields of p -adic numbers and complex p -adic numbers. Already the simplest p -adic dynamical systems have a very rich structure. There exist attractors, Siegel disks and cycles. There also appear new structures such as “fuzzy cycles.” A prime number p plays the role of parameter of a dynamical system. The behavior of the iterations depends on this parameter very much. In fact, by changing p we can change crucially the behavior: attractors may become centers of Siegel disks and vice versa, cycles of different length may appear and disappear…

1991 Mathematics Subject Classification: 46S10, 58F12.