Ergodicity criteria for non-expanding transformations of 2-adic spheres

TL;DR

This paper establishes precise conditions for ergodicity of non-expanding, measure-preserving transformations on 2-adic spheres, contributing to the understanding of dynamical systems in ultrametric spaces.

Contribution

It provides necessary and sufficient criteria for ergodicity of non-expanding, measure-preserving maps on 2-adic spheres, advancing the theory of ultrametric dynamical systems.

Findings

Derived conditions for ergodicity on 2-adic spheres

Characterized non-expanding measure-preserving transformations

Enhanced understanding of ultrametric dynamical behavior

Abstract

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $<f;\mathbf S_{2^{-r}}(a)>$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.