Natural extensions for [formula omitted]-adic [formula omitted]-shifts and other scaling maps

Abstract

The p-adic β-shift can naturally be thought of as an isometry on the p-adic integers (multiplication by a unit), followed by an iterate of the one-sided shift. We show that all (p−n,pn)-locally scaling maps have such a decomposition into an iterate of the one-sided shift composed with an isometry. We then use this decomposition to give a formula for the natural extension.