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.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

Formal Poincare'-Dulac renormalization for holomorphic germs

We shall describe an alternative approach to a general renormalization procedure for formal self-maps, originally suggested by Chen-Della Dora and Wang-Zheng-Peng, giving formal normal forms simpler than the classical Poincare-Dulac normal form. As example of application we shall compute 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-Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena

Automorphisms of C-k with an invariant non-recurrent attracting Fatou component biholomorphic to C x (C*)(k-1)

We prove the existence of automorphisms of C-k , k >= 2, having an invariant, non-recurrent Fatou component biholomorphic to C x (C*)(k-1) which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. As a corollary, we obtain a Runge copy of C x (C*)(k-1) in C-k. The constructed Fatou component also avoids k analytic discs intersecting transversally at the fixed point

Fatou flowers and parabolic curves

In this survey we collect the main results known up to now (July 2015) regarding possible generalizations to several complex variables of the classical Leau-Fatou flower theorem about holomorphic parabolic dynamics

Severe Asthma Standard-of-Care Background Medication Reduction With Benralizumab: ANDHI in Practice Substudy

peer reviewedBackground: The phase IIIb, randomized, parallel-group, placebo-controlled ANDHI double-blind (DB) study extended understanding of the efficacy of benralizumab for patients with severe eosinophilic asthma. Patients from ANDHI DB could join the 56-week ANDHI in Practice (IP) single-arm, open-label extension substudy. Objective: Assess potential for standard-of-care background medication reductions while maintaining asthma control with benralizumab. Methods: Following ANDHI DB completion, eligible adults were enrolled in ANDHI IP. After an 8-week run-in with benralizumab, there were 5 visits to potentially reduce background asthma medications for patients achieving and maintaining protocol-defined asthma control with benralizumab. Main outcome measures for non–oral corticosteroid (OCS)-dependent patients were the proportions with at least 1 background medication reduction (ie, lower inhaled corticosteroid dose, background medication discontinuation) and the number of adapted Global Initiative for Asthma (GINA) step reductions at end of treatment (EOT). Main outcomes for OCS-dependent patients were reductions in daily OCS dosage and proportion achieving OCS dosage of 5 mg or lower at EOT. Results: For non–OCS-dependent patients, 53.3% (n = 208 of 390) achieved at least 1 background medication reduction, increasing to 72.6% (n = 130 of 179) for patients who maintained protocol-defined asthma control at EOT. A total of 41.9% (n = 163 of 389) achieved at least 1 adapted GINA step reduction, increasing to 61.8% (n = 110 of 178) for patients with protocol-defined EOT asthma control. At ANDHI IP baseline, OCS dosages were 5 mg or lower for 40.4% (n = 40 of 99) of OCS-dependent patients. Of OCS-dependent patients, 50.5% (n = 50 of 99) eliminated OCS and 74.7% (n = 74 of 99) achieved dosages of 5 mg or lower at EOT. Conclusions: These findings demonstrate benralizumab's ability to improve asthma control, thereby allowing background medication reduction. © 202

Toeplitz operators and skew carleson measures for weighted bergman spaces on strongly pseudoconvex domains

In this paper we study properties of Toeplitz operators on weighted Bergman spaces of bounded strongly pseudoconvex domains. We prove that a Toeplitz operator built using a weighted Bergman kernel of weight β and integrating against a measure μ maps continuously a weighted Bergman space Ap1α1 (D) into Ap2α2 (D) if and only if μ is a (λ, γ)-skew Carleson measure, where λ = 1 + 1/p1 - 1/p2 and γ = 1 l (β + α1/p1 - α2/p2). This generalizes results obtained by Pau and Zhao on the unit ball, and by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on strongly pseudoconvex domains

Corrigendum to “Backward iteration in strongly convex domains” [Adv. Math. 228 (5) (2011) 2837–2854] (*** (2011) 228(5) (2837–2854), (S000187081100274X), (10.1016/j.aim.2011.06.044))

We correct a gap in two lemmas in [2], providing a new proof of the main results of that paper for hyperbolic and strongly elliptic self-maps of a bounded strongly convex domain with C2 boundary

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C^2 domain in C^d necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of C^d

Dynamics of multi-resonant biholomorphisms

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in such that the resonances among the first 1≤r≤n eigenvalues of the differential are generated over by a finite number of-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 Poincaré-Dulac normal form