A note on holomorphic extensions

A note on random holomorphic iteration in convex domains

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore we discuss the relations of such a Bloch type condition with the analogous hyperbolic Lipschitz condition.

关于全纯迭代函数系统的边界动力学的注记

A note on the boundary dynamics of holomorphic iterated function systems

本文研究全纯自映射的单位圆盘迭代函数系统的边界动力学，提出一个充分条件，将内部动力行为扩展到边界，推广了Bourdon等人对经典Denjoy-Wolff定理的扩展。

Extension of germs of holomorphic foliations

We consider the problem of extending germs of plane holomorphic foliations to foliations of compact surfaces. We show that the germs that become regular after a single blow up and admit meromorphic first integrals can be extended, after local changes of coordinates, to foliations of compact surfaces. We also show that the simplest elements in this class can be defined by polynomial equations. On the other hand we prove that, in the absence of meromorphic first integrals there are uncountably many elements without polynomial representations.

A brief survey on local holomorphic dynamics in higher dimensions

We give a brief survey on local holomorphic dynamics in higher dimensions. The main novelty of this note is that we will organize the material by the "level" of local invariants rather than the type of maps.

Geometric methods in holomorphic dynamics

In this note we review a selection of contemporary research themes in holomorphic dynamics. The main topics that will be discussed are: geometric (laminar and woven) currents and their applications, bifurcation theory in one and several variables, and the problem of wandering Fatou components.

The Kobayashi distance in holomorphic dynamics and operator theory

These are the notes of a short course I gave in the school "Aspects métriques et dynamiques en analyse complete", Lille, May 2015. The aim of this notes is to describe how to use a geometric structure (namely, the Kobayashi distance) to explore and encode analytic properties of holomorphic functions and maps defined on complex manifolds. We shall first describe the main properties of the Kobayashi distance, and then we shall present applications to holomorphic dynamics in taut manifolds, strongly pseudo convex domains and convex domains, and to operator theory in Bergman spaces (Carleson measu…