Linearization of germs: regular dependence on the multiplier
| dc.creator | Carminati, Carlo | |
| dc.creator | Marmi, Stefano | |
| dc.date | 2008-01-18 | |
| dc.date | 2008-02-27 | |
| dc.date.accessioned | 2026-07-07T09:23:12Z | |
| dc.date.available | 2026-07-07T09:23:12Z | |
| dc.description | We prove that the linearization of a germ of holomorphic map of the type $F_λ(z)=λ(z+O(z^2))$ has a $ C^1$--holomorphic dependence on the multiplier $λ$. $C^1$--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 $|λ|\not= 1$ 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. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/0801.2844 | |
| dc.identifier | http://arxiv.org/abs/0801.2844 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155651 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.title | Linearization of germs: regular dependence on the multiplier | |
| dc.type | text |