Bifurcation currents in holomorphic dynamics on

Abstract

Potential theory has been introduced in one dimensional rational dynamics by Brolin and Tortrat ([4], [29]) but does not play a central role there. In higher dimension however, as the classical tools are not any longer efficient, pluri-potential theory has revealed itself to be essential. The fundamental works of Hubbard-Papadopol, Fornaess-Sibony, BedfordSmillie, Briend-Duval (see [26] for precise references) enlight the remarkable efficency of pluri-potential theory in holomorphic dynamics on Pk or Ck. It is therefore tempting to study the parameter spaces in a similar way. More precisely, one would like to relate the bifurcations of an holomorphic family {fλ}λ∈X of endomorphisms of Pk to the powers of a certain current on the parameter space X. Let us recall that in dimension k = 1, a bifurcation is said to occur at some point λ0 ∈ X if the Julia set of fλ does not move continuously around λ0. The famous work of Mane-SadSullivan [16], which is based on the λ-lemma and the Fatou-Cremer-Sullivan classification, relates the bifurcations with the instability of the critical orbits. It also establishes that the bifurcations concentrate on the complement of an open dense subset of X (for the quadratic family {z2 + λ}λ∈X=C the bifurcation locus is precisely the boundary of the Mandelbrot set).