A connection between iteration theory and the study of sets of commuting holomorphic maps is investigated, in the unit disc of $C$. In particular, given two holomorphic maps $f$ and $g$ of the unit disc into itself, it is proved that if $g$ belongs to the pseudo-iteration semigroup of $f$ (in the sense of Cowen) then - under certain conditions on the behaviour of their iterates - the maps $f$ and $g$ commute.

We give a characterization for two different concepts of quasi-analyticity in Carleman ultraholomorphic classes of functions of several variables in polysectors. Also, working with strongly regular sequences, we establish generalizations of Watson’s Lemma under an additional condition related to the growth index of the sequence.

This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension for hyperbolic points. For such points, Ruelle’s thermodynamical formalism applies, hence computing the dimension amounts to computing the zero point of a pressure function. It is this pressure function that we approximate by a Monte-Carlo process combined with a shift method that considerably decreases the computational...

Dans cet article, nous démontrons deux résultats. L’un concerne les séries $f^{\text{'}}\left(z\right)=\sum a\left(n\right)z^n/n!$ telles que $\sum a\left(n\right)x^n$ est une série algébrique. Soit $AE$ cet ensemble de fonctions. Si $f$ appartient à $AE$, et si $g\left(z\right)$ est un polynôme-exponentiel tel que $h\left(z\right)=f\left(z\right)/g\left(z\right)$ est entière, alors il existe un polynôme $P\left(z\right)$ tel que $P\left(z\right)h\left(z\right)$ appartienne à $AE$.L’autre résultat est parallèle au premier. Soit $\sum u\left(n\right)x^n$ une série algébrique à coefficients dans un corps $𝕂$ (qui est soit $𝕂$, soit un corps quadratique imaginaire). Soit $\sum v\left(n\right)x^n$ une série rationnelle à coefficients dans $𝕂$. Avec...

A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $\Delta =z\in ℂ:\left|z\right|<1$ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{i\theta }$ for which the radial variation $V(f,e^{i\theta })={\int }_0^1\left|f^{\text{'}}\left(r{e}^{i\theta }\right)\right|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{i\theta }$ such that $(1-r)|f^{\text{'}}\left(r{e}^{i\theta }\right)|\ne o\left(1\right)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give...

We study the ramification of the Gauss map of complete minimal surfaces in ${ℝ}^m$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

Let a sequence $\left\{{\lambda }_n\right\}\subset ℝ$ be given such that the exponential system $\left\{\exp \right(i{\lambda }_{n}x\left)\right\}$ forms a Riesz basis in $L^2(-\pi ,\pi )$ and $\left\{{\xi }_n\right\}$ be a sequence of independent real-valued random variables. We study the properties of the system $\{exp(i({\lambda }_n+{\xi }_n)x\left)\right\}$ as well as related problems on estimation of entire functions with random zeroes and also problems on reconstruction of bandlimited signals with bandwidth $2\pi$ via their samples at the random points $\{{\lambda }_n+{\xi }_n\}$.

Holomorphic correspondences are multivalued maps $f=Q̃₊Q̃{₋}^{-1}:Z\to W$ between Riemann surfaces Z and W, where Q̃₋ and Q̃₊ are (single-valued) holomorphic maps from another Riemann surface X onto Z and W respectively. When Z = W one can iterate f forwards, backwards or globally (allowing arbitrarily many changes of direction from forwards to backwards and vice versa). Iterated holomorphic correspondences on the Riemann sphere display many of the features of the dynamics of Kleinian groups and rational maps, of which...

Définitions et propriétés des notions nouvelles de demi-plans, droites et abscisses de régularité et de suprarégularité pour une famille de germes dirichlétiens, par rapport à un support commun de référence. Conditions suffisantes (du type de Landau-Fekete) d’égalité de ces abscisses et expressions algorithmiques de majorants. Relations de dépendance (du type de V. Bernstein) entre les différentes abscisses considérées d’une famille donnée. Extensions de résultats classiques relatifs à la famille...