On measure-preserving ${\mathcal C}^1$ transformations of compact-open subsets of non-archimedean local fields

TL;DR

This paper introduces locally scaling transformations on non-archimedean local fields, characterizes measure-preserving ${ m C}^1$ maps, and establishes their ergodic and mixing properties, including conditions for Bernoulli behavior.

Contribution

It defines a new class of transformations called locally scaling, proves a structure theorem, and demonstrates ergodic and Bernoulli properties for polynomial maps on non-archimedean fields.

Findings

Locally scaling transformations include all ${ m C}^1$ measure-preserving maps.

Polynomial maps can be ergodic, mixing, or Bernoulli under certain conditions.

Existence of ergodic and mixing Markov transformations on non-archimedean fields.

Abstract

We introduce the notion of a \emph{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 ${\mathcal C}^1$ (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 $\mathbb Z_p \to \mathbb Z_p$ for it to define a Bernoulli transformation.