# On the Siegel-Sternberg linearization theorem

## TL;DR

This paper generalizes the Siegel-Sternberg linearization theorem to ultradifferentiable maps, covering analytic, smooth, and Gevrey cases, and explores properties of these classes with implications for dynamical systems.

## Contribution

It provides a unified linearization theorem for ultradifferentiable maps without small divisor conditions, and characterizes classes closed under composition.

## Key findings

- Unified linearization theorem for ultradifferentiable maps
- Characterization of ultradifferentiable classes closed under composition
- Reproves regularity results for solutions of ODEs and PDEs

## Abstract

We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode's and pde's. This will open up new directions in \textsc{kam}-theory and other applications of ultradifferentiable functions.