Linearization of germs: regular dependence on the multiplier

TL;DR

This paper demonstrates that the linearization of certain holomorphic germs depends smoothly on the multiplier, with analytic behavior off the unit circle and controlled regularity near it, especially at Brjuno and Diophantine points.

Contribution

It establishes a $C^1$--holomorphic dependence of the linearization on the multiplier, extending the understanding of regularity in holomorphic dynamics near resonances.

Findings

Linearization is analytic for $|ta| eq 1$

Linearization has $C^1$--holomorphic dependence on the multiplier

Asymptotic expansion of the linearization at Brjuno points

Abstract

We prove that the linearization of a germ of holomorphic map of the type $F_\lambda(z)=\lambda(z+O(z^2))$ has a $ C^1$--holomorphic dependence on the multiplier $\lambda$. $C^1$--holomorphic functions are $ C^1$--Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar{\partial}$ operator. The linearization is analytic for $|\lambda|\not= 1$ and the unit circle $S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of $S^1$, namely those points which lie ``far enough from resonances'', i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated spaces of ${\cal C}^1$--holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions, and the corresponding function space is arcwise-quasianalytic. Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points.