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Orbits in Dynamical Systems Defined by Groups

V. V. S. Ramachandram*

Department of Humanities and Basic Sciences, Dadi Institute of Engineering and Technology, Anakapalle, Visakhapatnam District, Andhra Pradesh, India.

Periodicity:January - June'2021
DOI : https://doi.org/10.26634/jmat.10.1.18197

Abstract

The aim of this paper is to study some basic properties of the orbits of elements in dynamical systems defined by groups. The relation between orbit of an element and the orbit of its inverse element has been established. The nature of the orbit of the identity element in the dynamical system has been studied and a singleton set containing the identity element itself has been obtained. It is observed that the set of all fixed points in the dynamical system is strongly invariant (Sinvariant) and is a closed subgroup of the group. By using the topological conjugacy between dynamical systems defined by groups, a relation between the orbit of an element and the orbit of its image element in the other dynamical system has been obtained. Finally, we obtained that if an element is in the kernel of the homomorphism, then it must be eventually fixed at the identity element of the group.

Keywords

Discrete Dynamical System, Homomorphism, Periodic Point, Fixed Point, Trajectory, Orbit.

How to Cite this Article?

Ramachandram, V. V. S. (2021). Orbits in Dynamical Systems Defined by Groups. i-manager's Journal on Mathematics, 10(1), 32-36. https://doi.org/10.26634/jmat.10.1.18197

References

[1]. Block, L. S., & Coppel, W. A. (1992). Dynamics in one dimension - Lecture notes in mathematics book series. Berlin: Springer.

[2]. Block, L., & Ledis, D. (2014). Topological conjugacy, transitivity, and patterns. Topology and its Applications, 167, 53-61.

[3]. Brin, M., & Stuck, G. (2002). Introduction to dynamical systems. Cambridge University Press.

[4]. Devaney, R. L. (1989). Chaotic dynamical systems (2nd ed.). Addison-Wesley.

[5]. Ramachandram, V. V. S. (2012). Dynamics in groups. International Review of Pure and Applied Mathematics, 8(1).

[6]. Walter, L. (1953). Introduction to the theory of finite groups. London: Oliver and Boyd.

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