Smoothness of class $C^2$ of nonautonomous linearization without spectral conditions

TL;DR

This paper proves that nonautonomous linearizations are of class C^2 without relying on spectral conditions, expanding the understanding of smoothness in dynamical systems.

Contribution

It establishes C^2 smoothness of nonautonomous linearization without spectral assumptions, incorporating stable and unstable manifolds from nonautonomous hyperbolicities.

Findings

Linearization is of class C^2.

Stable and unstable manifolds exist under nonautonomous hyperbolicities.

Spectral conditions are not necessary for smoothness.

Abstract

We prove that smoothness of nonautonomous linearization is of class $C^2.$ Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities. Moreover, our goal is reached without spectral conditions.