Nonuniform $(\mu,\nu)$-dichotomies and local dynamics of difference equations

TL;DR

This paper develops a local stable manifold theorem for nonautonomous linear difference equations with general nonuniform dichotomies, including cases with zero Lyapunov exponents, and analyzes their stability and perturbation behavior.

Contribution

It introduces a stable manifold theorem for difference equations with broad nonuniform dichotomies, extending classical results to more general and possibly degenerate cases.

Findings

Established a local stable manifold theorem for nonuniform dichotomies.

Analyzed decay rates of manifolds along orbits.

Provided examples of equations with the considered dichotomies.

Abstract

We obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy, possibly with different growth rates in the uniform and nonuniform parts. We note that we consider situations were the classical Lyapunov exponents can be zero. Additionally, we study how the manifolds decay along the orbit of a point as well as the behavior under perturbations and give examples of nonautonomous linear difference equations that admit the dichotomies considered.