Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal

TL;DR

This paper extends classical topological dynamics by analyzing stability, shadowing, and expansivity properties within uniform transformation semigroups constrained by an ideal, establishing new stability results under these conditions.

Contribution

It introduces and studies shadowing, expansivity, and stability modulo an ideal in transformation semigroups, extending known dynamical stability theorems to ideal-constrained settings.

Findings

Shadowing plus expansivity modulo an ideal implies topological stability modulo the same ideal.

The work generalizes classical stability theorems to the context of ideal-based dynamics.

Explores the relationship between shadowing modulo an ideal and the classical shadowing property.

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.