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Jasmin Raissy

Publications and source records attributed to Jasmin Raissy.

28 records · Page 2Linked to original sources
arXiv2012

Dynamics of multi-resonant biholomorphisms

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in C^n such that the resonances among the first 1<= r<= n eigenvalues of the differential are generated over N by a finite number of Q-linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. Furthermore, we obtain a generalization of the Leau-Fatou flower theorem, providing a complete description of the dynamics in a full neighborhood of the origin for 1-resonant parabolically attracting holomorphic germs in Poincare'-Dulac normal form.

math.CV↗
arXiv2012

Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of $\C^n$ fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ 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 commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.

math.DS↗
arXiv2012

Toeplitz operators and Carleson measures in strongly pseudoconvex domains

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in n complex variables. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A^p(D) into A^r(D) with r>p, generalizing and making more precise results by Cuckovic and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoglu for the unit ball.

math.CV↗
arXiv2011

Formal Poincare'-Dulac renormalization for holomorphic germs

In this revised version, applying a general renormalization procedure for formal self-maps, producing a formal normal form simpler than the classical Poincaré-Dulac normal form, we shall give a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang, Zheng and Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.

math.CV↗
arXiv2009

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 Rüssmann.

math.DS↗
arXiv2009

Simultaneous linearization of holomorphic germs in presence of resonances

Let $f_1, ..., f_m$ be $m\ge 2$ germs of biholomorphisms of $\C^n$, fixing the origin, with $(\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,..., m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\d f_1)_O$ and some restrictions on their resonances, $f_1, ..., f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,..., f_m$.

math.CV↗
arXiv2009

Torus actions in the normalization problem

Let $f$ be a germ of biholomorphism of $\C^n$, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to $f$, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of $f$. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincaré-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of $df_O$ to the weight matrix of the action. The link and the structure we found are more complicated than what one would expect; a detailed study was needed to completely understand the relations between torus actions, holomorphic Poincaré-Dulac normalizations, and torsion phenomena. We end the article giving an example of techniques that can be used to construct torus actions.

math.DS↗
arXiv2008

Linearization of holomorphic germs with quasi-Brjuno fixed points

Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.

math.DS↗
arXiv2006

About the Trimmed and the Poincare-Dulac normal form of diffeomorphisms

We study two particular continuous prenormal forms as defined by Jean Ecalle and Bruno Vallet for local analytic diffeomorphism: the Trimmed form and the Poincare-Dulac normal form. We first give a self-contain introduction to the mould formalism of Jean Ecalle. We provide a dictionary between moulds and the classical Lie algebraic formalism using non-commutative formal power series. We then give full proofs and details for results announced by J. Ecalle and B. Vallet about the Trimmed form of diffeomorphisms. We then discuss a mould approach to the classical Poincare-Dulac normal form of diffeomorphisms. We discuss the universal character of moulds taking place in normalization problems.

math.DS↗