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

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 $\mathcal{C}^1$ (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.