Geometric methods in holomorphic dynamics

Abstract

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.that one-dimensional rational maps are generically structurally stable, using surprisingly elementary arguments. For the quadratic family z 2 + c, c ∈ C, the bifurcation locus is the celebrated Mandelbrot set, whose intricate structure was thoroughly studied since then, using a variety of combinatorial and geometric methods. This research area was profoundly renewed in the 2000's by the systematic investigation of higher dimensional phenomena, and in particular with the introduction of bifurcation currents by DeMarco [33]. The bifurcation theory of holomorphic dynamical systems is nowadays a very active research domain, and a meeting point between the communities of one and several variable dynamicists. We relate this continuing story in Section 2.Finally, one recent breakthrough is the construction of wandering Fatou components in higher dimensional polynomial dynamics, which at the same time solves an old problem and raises many questions. We review these recent developments in Section 3.Let us conclude this introduction with a little notice: some important theorems will be mentioned only in passing, while other are isolated within numbered environments: this is meant to keep the reading flow, not to reflect a hierarchy of importance. Likewise, the list of references is already quite long, but not exhaustive, and we apologize in advance for any serious omission.