The supports of higher bifurcation currents

Abstract

Let (f λ ) λ∈Λ be a holomorphic family of rational mappings of degree d on P 1 (C), with k marked critical points c1, . . . , c k . To this data is associated a closed positive current T1 ∧ • • • ∧ T k of bidegree (k, k) on Λ, 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 c1, . . . , c k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T1 ∧ • • • ∧ T k ).

The bifurcation measure has maximal entropy

“…Notice that the hypothesis µ bif = 0 is satisfied if and only if there exists a parameter in Λ which admits k critical points that are, in a non persistent way, strictly preperiodic to a repelling cycle ( [BE,Ga,Du2]). It is in particular satisfied in any smooth orbifold parametrization of the moduli space of rational maps of degree d with marked critical points.…”

Classification of Special Curves in the Space of Cubic Polynomials

“…This fact was further explored in [GV]. The support of this measure has been characterized in several ways in a series of works [DF,Du1,Du2,G3], and it was shown by the second author [G1] to have maximal Hausdorff dimension 2(d − 1).…”

Higher bifurcation currents, neutral cycles and the Mandelbrot set

“…To end this section, we want to underline the fact that Theorem 3.1, Proposition 3.4 and the work [Du2] of Dujardin directly give Hausdorff dimension estimates for the support of T 1 ∧ · · · ∧ T k .…”

“…Let us mention that the equality supp(T k bif ) = Prerep(k) is known (see [BE, BG, DF] for the case when k is maximal). Dujardin [Du2,Corollary 5.3] proved it in the general case, using a transversality Theorem for laminar currents.…”