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

We give a characterization of measure-preservation of 1-Lipschitz functions on Fq[[T]]Fq[[T]] in terms of the van der Put expansion and use this result to give sufficient conditions for measure-preserving 1-Lipschitz functions on Fq[[T]]Fq[[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 F2[[T]]F2[[T]]. Moreover, we give an alternate, unified proof of ergodicity criteria of F2[[T]]F2[[T]] in terms of the aforementioned three bases.