Average shadowing and gluing property

TL;DR

This paper extends shadowing theory to discontinuous, non-invertible systems with more general perturbations, establishing a unified framework based on gluing constructions that handle single perturbation cases.

Contribution

It introduces a general shadowing theory for complex systems using gluing methods, accommodating broader classes of perturbations and non-invertible dynamics.

Findings

Extended shadowing theory to discontinuous systems

Developed a unifying approach based on gluing constructions

Applicable to systems with average-based perturbations

Abstract

The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The "gluing" construction used in the analysis of the last property turns out to be the key point of this theory.