Ergodic functions over Z

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Z p , 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 Z 2 and Z 3 , 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 Z p is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Z p that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo p μ , which are reduced from all ergodic B -functions on Z p , is provided where μ = μ ( p ) = 3 for p ∈ { 2 , 3 } and μ = 2 for p ≥ 5 .