The supports of higher bifurcation currents

TL;DR

This paper investigates the support of higher bifurcation currents in holomorphic families of rational maps, showing that their support is characterized by parameters where critical points eventually land on repelling cycles, advancing understanding of bifurcation phenomena.

Contribution

It provides a complete characterization of the support of the wedge product of bifurcation currents in rational maps, linking it to critical points falling on repelling cycles.

Findings

Support of higher bifurcation currents is accumulated by parameters with critical points on repelling cycles.

Complete characterization of the support of the wedge product of bifurcation currents.

Results unify and extend previous work by Buff, Epstein, and Gauthier.

Abstract

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).