On Dynamical System and Topological Transitivity via Ideals
Chhapikul Miah, Shyamapada Modak
TL;DR
The paper develops and analyzes the notion of ideal-based dynamical transitivity, introducing I-transitivity and ideal-non-wandering points to generalize classical topological transitivity and non-wandering concepts. It establishes several equivalent definitions for both standard and ideal transitivity, and corrects a known remark by showing that the equivalence between I-dense, star-dense, and dense requires a completely codense ideal. It proves that I-transitivity implies topological transitivity but not vice versa, and examines the interplay between I-dense and star-dense sets, invariant open sets, and generalized K-open transitivity classes, including a discussion of Hayashi-Samuel spaces and compatibility conditions. The work offers a framework for extending transitivity analyses to ideal-structured topologies, provides illustrative examples and counterexamples, and sets the stage for further exploration of generalized transitivity notions such as beta-, b-, and semi-I-transitivity.
Abstract
This paper will discuss the problem of defining the new topological transitivity. To do this several equivalent topological transitive and non-wandering point has been discussed through this paper. This paper also consider the ideal version of transitivity with the help of the amendment of the result Remark $6.9(2)$ of \cite{LL2013}. Corrected version of the Remark: ``If $\mathcal{\bf I}$ is codense, then $\mathcal{\bf I}$-denseness, $*$-denseness and denseness are equivalent" will be ``If $\mathcal{\bf I}$ is completely codense, then $\mathcal{\bf I}$-denseness, $*$-denseness and denseness are equivalent".