Jasmin Raissy

Simultaneous linearization of holomorphic germs in presence of resonances

Let f1, . . . , fm be m ≥ 2 germs of biholomorphisms of C , fixing the origin, with (df1)O diagonalizable and such that f1 commutes with fh for any h = 2, . . . ,m. We prove that, under certain arithmetic conditions on the eigenvalues of (df1)O and some restrictions on their resonances, f1, . . . , fm are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under f1, . . . , fm.

Brjuno conditions for linearization in presence of resonances

We present a new proof, under a slightly different (and more natural) arithmetic hypothesis, and using direct computations via power series expansions, of a holomorphic linearization result in presence of resonances originally proved by Russmann.

Holomorphic Linearization of Commuting Germs of Holomorphic Maps

Let f1,…,fh be h≥2 germs of biholomorphisms of ℂn fixing the origin. We investigate the shape that a (formal) simultaneous linearization of the given germs can have, and we prove that if f1,…,fh commute and their linear parts are almost simultaneously Jordanizable, then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commute and our Brjuno-type condition holds, then they are holomorphically simultaneously linearizable. This answers a multi-dimensional version of a problem raised by Moser.

The Julia-Wolff-Carathéodory Theorem and Its Generalizations

This note is a short introduction to the Julia-Wolff-Caratheodory theorem, and its generalizations in several complex variables, up to very recent results for infinitesimal generators of semigroups.