Simultaneous linearization of commuting germs of holomorphic diffeomorphisms

TL;DR

This paper extends the theory of linearization for commuting holomorphic germs by establishing new arithmetic conditions under which simultaneous linearizability occurs, generalizing previous results and providing bounds on Siegel disk radii.

Contribution

It introduces a Brjuno-type condition for simultaneous linearization of commuting germs, generalizing Moser's result and adapting Yoccoz's renormalization to this setting.

Findings

A Brjuno-type condition ensures simultaneous linearizability.

Weaker arithmetic conditions suffice when no periodic orbits are present.

Lower bounds for Siegel disk radii are derived based on arithmetic functions.

Abstract

Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f_1,...,f_N. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizability. We also obtain lower bounds for the conformal radii of the Siegel disks in both cases in terms of the arithmetic functions defining the arithmetic conditions.