Dynamics of rational maps: A current of the bifucation locus

Abstract

Let fλ : P1 → P1 be a family of rational maps of degree d > 1, parametrized holomorphically by λ in a complex manifold X. We show that there exists a canonical closed, positive (1,1)-current T on X supported exactly on the bifurcation locus B(f) ⊂ X. If X is a Stein manifold, then the stable regime X - B(f) is also Stein. In particular, each stable component in the space Polyd (or Ratd) of all polynomials (or rational maps) of degree d is a domain of holomorphy.