SEMINARIO DI SISTEMI DINAMICI OLOMORFI (e dintorni) Centro di Ricerca Matematica De Giorgi Sala Conferenze

Lunedi' 19 marzo, ---> ore 14.30 <---

Carlo Carminati (Pisa)

"Linearization of germs: regular dependence on the multiplier"

(abstarct in coda)

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SEMINARIO DI SISTEMI DINAMICI OLOMORFI (e dintorni) Centro di Ricerca Matematica De Giorgi Sala Conferenze

Lunedi' 26 marzo, ---> ore 14.30 <---

Jasmin Raissy (Pisa)

"Linearization of holomorphic germs with quasi-elliptic fixed points"

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ABSTRACT.1 - 19 marzo. [joint work with S. Marmi]

We prove that the linearization of germs of holomorphic (or formal) maps with a fixed point in one complex dimension have a ${\cal C}^1$--holomorphic dependence on the multiplier $\lambda$. % The linearization is analytic for $\lambda\in{\mathbb P}^1\C\setminus{\mathbb S}^1$, the unit circle~$\Su$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of~${\mathbb S}^1$ 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 space of ${\cal C}^1$--holomorphic functions. These are a special case of Borel's uniform monogenic functions, and their space is arcwise-quasianalytic \cite{MS2}. Among the consequences of these results, we can prove that the linearizations are defined and admit asymptotic expansions of Gevrey type at the points of~${\mathbb S}^1$ which satisfy a uniform version of the Brjuno condition first introduced in \cite{Ris}. The regular dependence on the multiplier holds also in the formal ultradifferentiable case considered in \cite{CM}.}

ABSTRACT.2 - 26 marzo

"Linearization of holomorphic germs with quasi-elliptic fixed points"

Abstract: Let f be a germ of holomorphic diffeomorphism of \C^n with the origin O as quasi-elliptic fixed point, i.e., so that the spectrum of df_O contains 0 < s eigenvalues satisfying a Brjuno condition. We shall give sufficient conditions for the holomorphic linearization of f valid even when s< n.

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