zbMATH — the first resource for mathematics

On a \(p\)-adic Julia set. (English. Russian original) Zbl 1054.37027
Russ. Math. Surv. 58, No. 6, 1194-1195 (2003); translation from Usp. Mat. Nauk 58, No. 6, 151-152 (2003).
Introduction: Non-archimedean dynamical systems have recently become the subject of intense study; see, for example [A. Khrennikov, Non-archimedean analysis: quantum paradoxes, dynamical systems and biological modesl, Mathematics and its Applications, 427, Dordrecht: Kluwer Academic Publishers (1997; Zbl 0920.11087)]. In this note, we consider the class of dynamical systems of the form \(x\mapsto x^2-\frac {a^2} {p^2}\) on the set of \(p\)-adic numbers. We study the structure of the Julia set, defined for these systems by analogy with the classical case, and describe the action on it of our map, which reduces to symbolic dynamics. We prove here that all periodic points are contained in the base field.

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
11S85 Other nonanalytic theory
37B10 Symbolic dynamics
Full Text: DOI