Additivity of local function and dynamical system
Sk. Selim, Chhapikul Miah, Monoj Kumar Das, Shyamapada Modak
TL;DR
This work extends the local-function framework on topological spaces by incorporating generalized open sets and star-type closures, identifying when standard coincidences with closure operators hold and when they fail. It introduces several variants, including $A^{\star s}$, $A^{\star p}$, $A^{\star b}$, $A^{\star β}$ and $A^{\diamond k}$, and analyzes their relations, including strict inclusions and non-additivity. A central contribution is the $\star sp$-local function and its use in defining $\star sp$-transitivity for dynamical systems, illustrated by examples that reveal density, transitivity, and non-wandering phenomena in ideal-topological spaces. Together these results provide a framework linking generalized local-function theory with topological dynamics and offer directions for studying dense sets and transitive behavior in complex topological contexts.
Abstract
The study of local function in topological spaces is remarkable. Various branches have been developed through this study. In this paper, we further consider the local function and exploring the various properties of the same by considering some generalized open sets. In this situation some of the properties of local function fails to hold due to the finite intersection property of the topology. Due to this outcome, we are investigating the situation of dynamical system and Topological Transitivity.