A Directional Interpretation of Shadowing and Semicontinuity of Omega-Limit Sets

Abstract

In this paper, we study the relationship between shadowing properties and the semi-continuity of ω-limit sets in dynamical systems. Building on known equivalence results, we provide a directional and topological interpretation of the asymmetry between upper and lower semi-continuity. Our approach clarifies how forward shadowing naturally supports upper semi-continuity, while the absence of backward control obstructs lower semi-continuity in non-invertible and semi-continuous systems. Several illustrative discussions and examples are provided to support this viewpoint.