Ergodic functions over Zp

Abstract

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Zp, in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on Z2 and Z3, which are known as B-functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p≥5, an ergodicity criterion of B-functions on Zp is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Zp that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo pμ, which are reduced from all ergodic B-functions on Zp, is provided where μ=μ(p)=3 for p∈{2,3} and μ=2 for p≥5.