Metrical properties of beta-transformations on power series rings

Abstract

Let A ∞ be an integer ring of a formal Laurent series field in one variable 1 t over a finite field of order q, with a maximal ideal M ∞ . Then, we introduce beta-transformations U β on A ∞ that are topologically isomorphic and measurably isomorphic to the well-known beta-transformation T β on M ∞ . We prove various metrical properties of U β such as (total) ergodicity and mixing of any order through explicit representations of both U β and its nth iterate U β n with respect to the shift operators. Furthermore, we examine dynamical connections between the measure-preserving property of 1-Lipschitz functions and the locally scaling property of ( q − k , q k ) -Lipschitz functions, in terms of the coefficients of van der Put and Carlitz–Wagner.