Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal

Abstract

In this paper, we introduce and analyze several key dynamical properties-namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal-within the framework of uniform transformation semigroups. Given an ideal $\mathcal{I}$ on semigroup $T$, we investigate the interplay between these properties in compact Hausdorff transformation semigroup $(T,X,\mathfrak{X})$. Our main result establishes that if a compact Hausdorff transformation semigroup exhibits the shadowing property modulo $\mathcal{I}$ and is expansive modulo $\mathcal{I}$ then it is also topologically stable modulo $\mathcal{I}$. This extends known stability theorems in classical topological dynamics to the setting of ideal-constrained dynamics. Additionally, we explore the relationship between shadowing modulo $\mathcal{I}$ and the conventional shadowing property.

15 pages