Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory

TL;DR

This paper extends classical theorems relating hyperbolicity and bounded solutions in linear difference equations and applies these generalizations to establish a link between limit shadowing and structural stability in dynamical systems.

Contribution

The paper introduces new generalizations of Maizel and Pliss theorems and applies them to connect limit shadowing with structural stability in diffeomorphisms.

Findings

Generalized Maizel and Pliss theorems for hyperbolicity and bounded solutions

Established equivalence between limit shadowing and structural stability

Demonstrated applications in shadowing theory of diffeomorphisms

Abstract

We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.