Criteria of measure-preservation for 1-Lipschitz functions on F-q[[T]] in terms of the van der Put basis and its applications

摘要

We give a characterization of measure-preservation of 1-Lipschitz functions on F-q[[T]] in terms of the van der Put expansion and use this result to give sufficient conditions for measure-preserving 1-Lipschitz functions on F-q[[T]] in terms of the three well known bases, Carlitz polynomials, digit derivatives and digit shifts. We show that these conditions are also necessary for F-2[[T]]. Moreover, we give an alternate, unified proof of ergodicity criteria of F-2[[T]] in terms of the aforementioned three bases. (C) 2015 Elsevier Inc. All rights reserved.