Group Structure of the $$p$$ -Adic Ball and Dynamical System of Isometry on a Sphere

IF 0.5 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-05-06 DOI:10.1134/s2070046624020031

I. A. Sattarov

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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 these 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 is not 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.

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球面上 $$p$$ 阿迪克球的群结构和等值动态系统

摘要本文研究了 $p$-adic 球和球面的群结构。研究了定义在不变球上的等距动态系统。我们在球和球面上分别定义了二元操作（$oplus$和$\odot$），并证明这些集合是关于操作的紧凑拓扑阿贝尔群。然后，我们证明任意两个半径为正的球（球面）作为群是同构的。我们证明在 $\mathbb Z_p$ 中引入的哈量也是任意球和球上的哈量。我们研究了在球面上定义的等距所产生的动力系统，并证明了任何不是固定点的初始点的轨迹都是不收敛的。我们研究了这个 $p$-adic 动力系统关于球面上归一化哈量的遍历性。对于 $p\geq 3$ 我们证明了动力系统不是遍历的。但是对于（p=2），在某些条件下动力学系统可能是遍历的。

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来源期刊

P-Adic Numbers Ultrametric Analysis and Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-

CiteScore

1.10

自引率

20.00%

发文量

期刊介绍： This is a new international interdisciplinary journal which contains original articles, short communications, and reviews on progress in various areas of pure and applied mathematics related with p-adic, adelic and ultrametric methods, including: mathematical physics, quantum theory, string theory, cosmology, nanoscience, life sciences; mathematical analysis, number theory, algebraic geometry, non-Archimedean and non-commutative geometry, theory of finite fields and rings, representation theory, functional analysis and graph theory; classical and quantum information, computer science, cryptography, image analysis, cognitive models, neural networks and bioinformatics; complex systems, dynamical systems, stochastic processes, hierarchy structures, modeling, control theory, economics and sociology; mesoscopic and nano systems, disordered and chaotic systems, spin glasses, macromolecules, molecular dynamics, biopolymers, genomics and biology; and other related fields.