Bifurcation currents and equidistribution on parameter space

In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.Comment: Revised version, 46 pages, to appear in the proceedings of the conference "Frontiers in complex dynamics (Celebrating John Milnor's 80th birthday)

The supports of higher bifurcation currents

Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).Comment: 13 page

Fatou directions along the Julia set for endomorphisms of CP^k

Not much is known about the dynamics outside the support of the maximal entropy measure $\mu$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(\mu)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions.Comment: Final, shorter version, to appear in J. Math. Pures App

A note on the rank of positive closed currents

In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.Comment: This is a complement, not intended for publication, to my paper "Fatou directions along the Julia set for endomorphisms of CP^k" [arXiv:1006.0882