Ergodic Theory and Dynamical Systems

Hyperbolic maps in p-adic dynamics

a1 Department of Mathematics, University of Rochester, Rochester, NY 14627, USA (e-mail:


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.

(Received November 6 1998)
(Revised February 19 1999)

p1 Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: