Topological $(\mathscr{F},\mathscr{G})-$shadowing property

TL;DR

This paper introduces a new topological shadowing property based on filter pairs in uniform spaces and explores its equivalences and limitations within compact chain recurrent dynamical systems.

Contribution

It defines the $( ext{F}, ext{G})$-shadowing property in uniform spaces and establishes equivalences among various forms of topological shadowing, also identifying conditions for its absence.

Findings

Equivalence of several topological shadowing notions in compact chain recurrent systems

Introduction of the $( ext{F}, ext{G})$-shadowing property in uniform spaces

Non-existence of certain shadowing properties when minimal points are not dense

Abstract

We define the concept of $(\mathscr{F},\mathscr{G})-$shadowing property on uniform space and say it as a topological $(\mathscr{F},\mathscr{G})-$shadowing property. We show that topological shadowing, topological $(\mathscr{D},\mathscr{F}_t)-$shadowing, topological $(\mathscr{F}_t,\mathscr{F}_t)-$shadowing and topological $(\mathbb{N},\mathscr{F}_t)-$shadowing are equivalent in compact chain recurrent dynamical system. We also prove that if minimal points of $f$ are not dense in $X$ then the system does not have $(\mathscr{P}(\mathbb{N}),\mathscr{F}_{ps})-$shadowing.