arXiv:2112.11012v1 [math.NT] 21 Dec 2021 Ergodicity for uniformly differentiable modulo p functions on Z p Sangtae Jeong Department of Mathematics, Inha University, Incheon, Korea 22212 December 22, 2021 Abstract We provide an ergodicity criterion for uniformly differentiable modulo p functions on Zp in regard to the minimal level of the reduced functions by showing that ergodic conditions are explicitly found in terms of the coefficients of Mahler or van der Put for each prime p. To this end, it is essential to give an alternative, natural proof of Memi´ c’s result regarding Mahler’s coefficients estimation for uniformly differentiable modulo p functions on Zp. Contents 1 Introduction 2 2 Preliminaries 2 2.1 Two bases of Mahler and van der Put .......................... 2 2.2 Basics on p-adic dynamical systems ............................ 4 2.3 Uniformly differentiable modulo p functions ....................... 6 3 Estimation of the Mahler coefficients 7 3.1 Coefficients estimation for p ≥ 3 ............................. 7 3.2 Coefficients estimation for p =2 ............................. 16 4 Ergodicity criterion for uniformly differentiable modulo p functions 19 4.1 Measure-preservation criterion ............................... 19 4.2 Memi´ c et al.’s criterion for egrodicity revisited ..................... 21 4.3 Characterization of p =2 ................................. 23 4.4 Characterization of p =3 ................................. 24 4.5 Characterization of p ≥ 5 ................................. 27 Keywords:1-Lipschitz, Uniformly differentiable modulo p, Ergodic,Van der Put basis, Mahler basis. The author is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education, Science, and Technology( 2020R1A2C1A01003498). Mathematics Subject Classification 2000 Primary: 37P05; Secondary: 11S82, 37B05. 1 E-mail: stj@inha.ac.kr 1