Lacunary series, resonances, and automorphisms of $\mathbb{C}^2$ with a round Siegel domain
We construct transcendental automorphims of $\mathbb{C}^{2}$ having an unbounded and regular Siegel domain.
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arXiv subjects
Francois Berteloot
Publications and source records attributed to Francois Berteloot.
The Mandelbrot set is the shadow of a Julia set
Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.
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A distortion theorem for the iterated inverse branches of a holomorphic endomorphism of CP(k)
We linearize the inverse branches of the iterates of holomorphic endomorphisms of CP(k) and thus overcome the lack of Koebe distortion theorem in this setting when k $\ge$ 2. We review several applications of this result in holomorphic dynamics.
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Distribution of polynomials with cycles of given multiplier
In the family of degree $d$ polynomials the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier $e^{iθ}$ are shown to equiditribute the bifurcation current.
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Une caracterisation des endomorphismes de Lattes par leur mesure de Green
We show that the Lattes endomorphisms are the only holomorphic endomorphisms of the complex k-dimensional projective space whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a consequence, Lattes endomorphisms are also characterized by other extremal properties as the maximality of the Hausdorff dimension of their measure of maximal entropy or the minimality of their Liapounov exponents. Our proof uses a linearization method which is of independant interest and a previous characterization by the regularity of the Green current.
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Linearisation d'endomorphismes holomorphes de CP(k) et caracterisation des exemples de Lattes par leur mesure de Green
Let f an holomorphic endomorphism of CP(k) with degree larger than 2. We show that if the Green measure of f is not singular, then f is rigid : it is a Lattes example. The proof relies on a linearization property of the iterates of f, along typical orbits. This property allow us to regularize the Green current, and to prove the rigidity. -- -- -- Soit f un endomorphisme holomorphe de CP(k) de degre plus grand que 2. Nous montrons que si la mesure de Green de f n'est pas singuliere, alors f est tres rigide : c'est un exemple de Lattes. La demonstration repose sur une propriete de linearisation des iteres de f le long d'orbites typiques. Cette propriete nous permet de "regulariser" le courant de Green, et d'en deduire la rigidite.
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