# Smooth Linearization of Nonautonomous Differential Equations with a Nonuniform Dichotomy

## TL;DR

This paper establishes a smooth linearization theorem for nonautonomous differential equations with nonuniform strong exponential dichotomy, providing conditions for differentiability and Hölder continuity of the linearization near hyperbolic fixed points.

## Contribution

It offers the first rigorous proof of simultaneous differentiable and Hölder linearization for hyperbolic systems without non-resonant conditions in the autonomous case.

## Key findings

- Spectral bounds for linearization are formulated via discretized evolution operators.
- Provides conditions under which the linearization is both differentiable and Hölder continuous.
- First proof of such linearization without non-resonant conditions in the autonomous setting.

## Abstract

In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and H\"older continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and H\"older linearization of hyperbolic systems without any non-resonant conditions.