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

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

2007-10-30

arxiv.org/abs/0710.5562v1

Trans. Amer. Math. Soc. 361 (2009), no. 1, 61--85

Mathematics

Dynamical Systems

Scientific paper

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.

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