Ergodicity for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p$

We provide an ergodicity criterion for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p$ 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$\acute{c}$'s result regarding Mahler's coefficients estimation for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p.$