Group structure of the $p$-adic ball and dynamical system of isometry on a sphere

TL;DR

This paper explores the group structure of p-adic balls and spheres, establishing their properties as topological abelian groups, and analyzes the dynamics and ergodicity of isometries on these spheres.

Contribution

It introduces binary operations making p-adic balls and spheres into topological abelian groups and investigates the ergodic behavior of isometries on spheres, including conditions for ergodicity.

Findings

Balls and spheres with positive radius are isomorphic as groups.

Haar measure on dic integers extends to arbitrary balls and spheres.

For p adgeq 3, the dynamical system is not ergodic; for p=2, it may be ergodic under certain conditions.

Abstract

In this paper the group structure of the $p$-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations $\oplus$ and $\odot$ on a ball and sphere respectively, and prove that this sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in $\mathbb Z_p$ is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere, and show that the trajectory of any initial point that is not a fixed point isn't convergent. We study ergodicity of this $p$-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For $p\geq 3$ we prove that the dynamical systems are not ergodic. But for $p=2$ under some conditions the dynamical system may be ergodic.