Transitive map in bitopological dynamical systems

Abstract

Topological dynamical system is a tool to investigate dynamical
properties in terms of a topological space. Although it has been stud-
ied by many researchers from a long time ago, there may be sev-
eral drawbacks in topological dynamical system as there is only one
topology which is involved in the entire mathematical process. But
in nature, for example, in the development of an organism from zy-
gote, the brain together with central nervous system and the other
body parts grow parallel as dierent stem cell layers generate them
so that they can be represented by two topologies. In this paper we
disprove the conjecture 1 of Nada and Zohny [Nada, S. I., and Zohny,
H., An application of relative topology in biology. Chaos, Solitons and Fractals, 42 (2009), 202{204.] on applying bitopological dynamical system. Also, we introduce bitopological transitivity, point transitivity, pairwise iterated compactness and establish relations between them in bitopological dynamical system. We show that bitopological dynamical system is a more accurate tool to understand the development of an organism from zygote than dynamical topological space. Here, dynamical topological space is not distinguished from topological dynamical system. We procure the term \dynamical topological space" in this paper from Nada and Zohny [Nada, S. I., and Zohny, H., An application of relative topology in biology. Chaos, Solitons and Fractals, 42 (2009), 202{204.]