Metrical properties of beta-transformations on power series rings

초록

Let A(infinity) be an integer ring of a formal Laurent series field in one variable 1t1t over a finite field of order q, with a maximal ideal M-infinity. Then, we introduce beta-transformations U beta U beta on A infinity A infinity that are topologically isomorphic and measurably isomorphic to the well-known beta-transformation T-beta on M-infinity. We prove various metrical properties of U-beta such as (total) ergodicity and mixing of any order through explicit representations of both U-beta and its nth iterate U-beta(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.

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