Criteria of ergodicity for $p$-adic dynamical systems in terms of coordinate functions

TL;DR

This paper establishes criteria for ergodicity of $p$-adic dynamical systems using coordinate functions, facilitating analysis of non-smooth transformations and potential cryptographic applications.

Contribution

It introduces new ergodicity criteria based on coordinate functions and van der Put series, enabling analysis of non-smooth $p$-adic transformations.

Findings

Criteria for ergodicity in terms of coordinate functions

Application to non-smooth $p$-adic transformations

Concrete classes of ergodic functions presented

Abstract

This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The coordinate representation can be useful, e.g., for applications to cryptography. Moreover, by using this representation we can consider non-smooth $p$-adic transformations. The basic technical toolsare van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions.