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Hyperbolic maps in p-adic dynamics

Published online by Cambridge University Press:  07 March 2001

ROBERT L. BENEDETTO
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA (e-mail: benny@math.rochester.edu) Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bene@bu.edu)

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.

Type
Research Article
Copyright
© 2001 Cambridge University Press

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