Shadowing in CR-Dynamical Systems

TL;DR

This paper introduces generalized shadowing properties in CR-dynamical systems, extending classical shadowing concepts from topological and set-valued dynamical systems to a broader framework.

Contribution

It defines and explores the $(i, j)$-shadowing properties in CR-dynamical systems, broadening the understanding of shadowing in more general dynamical contexts.

Findings

Introduces $(i, j)$-shadowing properties for CR-dynamical systems.

Extends shadowing theory from classical to set-valued and CR contexts.

Provides foundational results for future research in CR-dynamical systems.

Abstract

A CR-dynamical system is a pair $(X, G)$, where $X$ is a non-empty compact Hausdorff space with uniformity $\mathscr{U}$ and $G$ is a closed relation on $X$. In this paper we introduce the $(i, j)$-shadowing properties in CR-dynamical systems, which generalises the shadowing property from topological dynamical systems $(X, f)$. This extends previous work on shadowing in set-valued dynamical systems.