Measurable Dynamics of Maps on Profinite Groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on $\mathbb Z_p$.Comment: 18 page

Dynamics of the p -adic Shift and Applications

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.Williams College (Bronfman Science Center)National Science Foundation (U.S.) (REU Grant DMS – 0353634

On measure-preserving c1 transformations of compactopen subsets of non-Archimedean local fields

Abstract. We introduce the notion of a locally scaling transformation defined on a compact-open subset of a non-archimedean local field. We show that this class encompasses the Haar measure-preserving transformations defined by C1 (in particular, polynomial) maps, and prove a structure theorem for locally scaling transformations. We use the theory of polynomial approximation on compact-open subsets of non-archimedean local fields to demonstrate the existence of ergodic Markov, and mixing Markov transformations defined by such polynomial maps. We also give simple sufficient conditions on the Mahler expansion of a continuous map Zp → Zp for it to define a Bernoulli transformation. 1