Hyperbolic maps in p-adic dynamics

Abstract

In this paper we study the dynamics of a rational function \phi\in K(z) defined over some finite extension K of \mathbb{Q}_p. After proving some basic results, we define a notion of ‘components’ of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps.

Measurable Dynamics of Simple p-adic Polynomials

“…While most of this work deals with topological or complex dynamical properties of the p-adics, for which the reader may refer to [1], [4], [13], [17], and the references cited in these articles, measurable dynamics on the p-adics has also received some exposure, particularly in [14], [2], [3] [12], [7], and [8].…”

Onp-adic Gibbs measures of the countable state Potts model on the Cayley tree

“…We remark that first investigations of non-Archimedean dynamical systems have appeared in [28]. We also point out that intensive development of padic (and more general algebraic) dynamical systems has happened few years, see [6,7,11,14,40,57,59,65]. More extensive lists may be found in the p-adic dynamics bibliography maintained by Silverman [57] and the algebraic dynamics bibliography of Vivaldi [60].…”

On the number of equivalence classes of attracting dynamical systems

“…Arrowsmith and Vivaldi [2], Benedetto [3], Khrennikov [5,6], Khrennikov and Nilsson [7,8], Lubin [11], Nyqvist [12], and Svensson [14]. Among the applications one can find for instance cryptography (generating pseudorandom sequences to be used for stream ciphers), see e.g.…”