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

In this paper, the group structure of the -adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations and on a ball and sphere, respectively, and prove that these sets are compact topological abelian group with resp...

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Bibliographic Details
Published in:P-adic numbers, ultrametric analysis, and applications ultrametric analysis, and applications, 2024, Vol.16 (2), p.128-135
Main Author: Sattarov, I. A.
Format: Article
Language:English
Subjects:
14/34
639/766/189
639/766/530
639/766/747
Algebra
Dynamical systems
Ergodic processes
Fixed points (mathematics)
Group theory
Mathematics
Mathematics and Statistics
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Summary:In this paper, the group structure of the -adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations and 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 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 -adic dynamical system with respect to normalized Haar measure reduced on the sphere. For we prove that the dynamical systems are not ergodic. But for under some conditions the dynamical system may be ergodic.
ISSN:2070-0466
2070-0474
DOI:10.1134/S2070046624020031