Dynamics of the $p$-adic shift and applications

James Kingsbery; Alex Levin; Anatoly Preygel; Cesar E. Silva

doi:10.3934/dcds.2011.30.209

Article Contents

2011, Volume 30, Issue 1: 209-218. Doi: 10.3934/dcds.2011.30.209

This issue Previous Article Dynamics and abstract computability: Computing invariant measures Next Article Existence and non-existence of global solutions for a discrete semilinear heat equation

Dynamics of the $p$-adic shift and applications

1.
400 E 71 St. Apt. 5B, New York, NY 10021
2.
Massachusetts Institute of Technology, Cambridge, MA 02139
3.
Williams College, Williamstown, MA 01267

Received: February 28, 2010

Revised: June 30, 2010

Published: December 2010

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Abstract

There is a natural continuous realization of the one-sided Bernoulli shift on the $p$-adic integers as the map that shifts the coefficients of the $p$-adic expansion to the left. We study this map's Mahler power series expansion. We prove strong results on $p$-adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift. As a result, these polynomials share the shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.

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Mathematics Subject Classification: Primary 37A05; Secondary 37F10.

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