Measure-preservation criteria for a certain class of 1-Lipschitz functions on $Z_p$ in Mahler's expansion

Chunlan Li,Sangtae Jeong

doi:10.3934/dcds.2017160

Abstract

In this paper, we formulate a conjecture for a measure-preservation criterion of 1-Lipschitz functions defined on the ring Zp of p-adic integers, in terms of Mahler's expansion. We then provide evidence for this conjecture in the case that p = 3, and verify that it also holds for a wider class of 1-Lipschitz functions that are everywhere differentiable on Zp, which we call $\mathcal{ B}$-functions, in the sense of Anashin.

Highlights

Introduction and statement of resultsThe ergodic theory of non-Archime dean dynamical systems is a rapidly developing area of research, from which the results are of theoretical value, and have practical applications in various applied sciences, such as computer science, cryptology, and numerical analysis [6, 27]
Anashin [2] first provided a criterion for the ergodicity of 1-Lipschitz functions on the 2-adic integer ring Z2 in Mahler’s expansion
Motivated by the work of Khrennikov and Yurova [19], Jang et al [13, 14] recently provided a measure-preservation criterion for 1-Lipschitz functions on Fq[[T ]] in terms of the four aforementioned bases. Inspired by these results over function fields, we present a conjecture for a measure-preservation criterion of 1-Lipschitz functions on Zp in Mahler’s expansion, excluding p = 2, for which Anashin provided the criterion in Theorem 2.13 (2)

Summary

Introduction

Introduction and statement of resultsThe ergodic theory of non-Archime dean dynamical systems is a rapidly developing area of research, from which the results are of theoretical value, and have practical applications in various applied sciences, such as computer science, cryptology, and numerical analysis [6, 27]. F is measure-preserving if and only if the following conditions are satisfied: (i) {f (0), f (1), · · · , f (p − 1)} is a complete set of distinct residues modulo p; (ii) For all s ≥ 1 and all m = m− + msps with 0 ≤ m− < ps and 2 ≤ ms ≤ p − 1, cm ≡ 0 (mod p); (iii) For all s ≥ 1 and all 0 ≤ k < ps, s pr −1 r=0 i=0 k i ci+pr ≡ 0 (mod p).

Results

Conclusion