58 research outputs found
Modern Aspects of Dynamical Systems: Cetraro, Italy 2021
No full textThis book provides an overview of recent advances in the theory of dynamical systems, with a particular emphasis on their connections to other areas of mathematical research, including number theory, geometry, mathematical physics, complex analysis, and celestial mechanics. Compiling the lecture notes from some of the contributions presented at the C.I.M.E. school "Modern Aspects of Dynamical Systems" held in Cetraro in August 2021, the contributions are the following: “Homogeneous dynamics and Diophantine problems” by Manfred Einsiedler, “Effective ergodic theory for translation flow” by Giovanni Forni, “Integrability and rigidity for convex billiards” by Vadim Kaloshin, “Holomorphic dynamics” by Jasmin Raissy and “Exponentially small phenomena and its role in the dynamics” by Tere Martinez-Seara.
These notes are suitable for graduate students and young researchers interested in an introduction to some of the modern research areas within the field of dynamical systems
Toeplitz operators and Carleson measures in strongly pseudoconvex domains
No full textWe study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in C^n. 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
Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudo convex domains
No full textIn this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudoconvex domains in several complex variables. In particular we prove that a Toeplitz operator built using as kernel a weighted Bergman kernel of weight β and integrating against a measure μ maps continuously (when β is large enough) a weighted Bergman space into another weighted Bergman space if and only if μ is a (λ,γ)-skew Carleson measure, where λ and γ can be computed explicitly. This theorem generalizes results obtained by Pau and Zhao on the unit ball, and extends and makes more precise results obtained by Abate, Raissy and Saracco for a smaller class of Toeplitz operators on bounded strongly pseudoconvex domains
The Julia-Wolff-Carathéodory theorem and its generalizations
No full textInternational audienceThis note is a short introduction to the Julia-Wolff-Carath\'eodory theorem, and its generalizations in several complex variables, up to very recent results for infinitesimal generators of semigroups
Holomorphic linearization of commuting germs of holomorphic maps
No full textAbstract. Let f1,..., fh be h ≥ 2 germs of biholomorphisms of Cn fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if f1,..., fh commute and their linear parts are al-most 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 linerizable. This answers to a multi-dimensional version of a problem raised by Moser. 1
Dynamics of post-critically algebraic endomorphisms
No full textDans cette thèse, j'étudie la dynamique des endomorphismes de l'espace projectif complexe. Je m'intéresse aux endomorphismes post-critiquement algébriques, une notion qui généralise celle de fractions rationnelles post-critiquement finies en dimension 1. En particulier, j'étudie les valeurs propres d'un endomorphisme post-critiquement algébrique le long de l'orbite d'un point périodique. En dimension 1, un résultat bien connu, qui remonte aux travaux de Pierre Fatou, dit que ces valeurs sont soit nulles soit de module strictement plus supérieur à 1. Dans cette thèse, j'étudie une conjecture qui généralise ce résultat en dimension au moins 2. Dans la première partie de cette thèse, j'étudie une famille des endomorphismes post-critiquement algébriques introduite dans la thèse de Sarah Koch. En utilisant la caractérisation topologique des fractions rationnelles de William Thurston, sous certaines conditions, Sarah Koch a associé à une fraction rationnelle post-critiquement finie g un endomorphisme post-critiquement algébrique f. Lorsque g est un polynôme quadratique, je donne une caractérisation détaillée des valeurs propres de l'endomorphisme associé f en ses points fixes. En particulier, je montre que celles-ci sont soit nulles soit de modules strictement supérieurs à 1. Ce résultat suggère la validité de la conjecture. Dans la deuxième partie, je montre que la conjecture est vraie dans le cas de dimension 2 sans hypothèse supplémentaire et en toute dimension lorsque les points périodiques sont en dehors de l'ensemble post-critique et sans autre hypothèse.In this thesis, I study the dynamics of endomorphisms of the complex projective space. I am interested in post-critically algebraic endomorphisms, a notion which generalizes that of post-critically finite rational maps in dimension 1. In particular, I study the eigenvalues of a post-critically algebraic endomorphism along the orbit of a periodic point. In dimension 1, a well-known result, which is due to Pierre Fatou, states that these values are either zero or of modules strictly greater than 1. In this thesis, I study a conjecture which generalizes this result in dimension at least 2. In the first part of this thesis, I study a family of post-critically algebraic endo- morphisms introduced in Sarah Koch's thesis. Using the topological characterization of rational maps of William Thurston, under certain conditions, Sarah Koch associated with a post-critically finite rational map g a post-critically algebraic endomorphism f. When g is a quadratic polynomial, I give a detailed characterization of the eigenvalues of the endomorphism f at its fixed points. In particular, I show that these values are either zero or of modules strictly greater than 1. This result provides evidence of the validity of the conjecture. In the second part, I show that the conjecture is true in the case of dimension 2 without additional hypotheses and in any dimension when the periodic points are outside the post-critical set and without other hypotheses
Skew Carleson measures in strongly pseudoconvex domains
No full textInternational audienceGiven a bounded strongly pseudoconvex domain D in C n with smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of (λ, γ)-skew Carleson measures on D, with λ > 0 and γ > 1 − 1 n+1
A Julia-Wolff-Carathéodory theorem for infinitesimal generators in the unit ball
No full text16 pagesInternational audienceWe prove a Julia-Wolff-Carathédory theorem on angular derivatives of infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit ball , starting from results recently obtained by Bracci and Shoikhet
Introduction to Fatou components in holomorphic dynamics
No full textThis work is part of the C.I.M.E. Lecture Notes ``Modern Aspects of Dynamical Systems'' to be published in the Springer Lecture Notes in Mathematics -- C.I.M.E. subseries, corresponding to the course taught by the second author in Cetraro in August 2021.This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of
Wolff-Denjoy theorems in nonsmooth convex domains
No full textInternational audienceWe give a short proof of Wolff-Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff-Denjoy theorem for weakly convex domains, again without any smoothness assumption on the boundary
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