Non density of stability for holomorphic mappings on P^k

Abstract

A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that J-stable maps are dense in any holomorphic family of rational maps in dimension 1. In this paper we show that the corresponding result fails in higher dimension. More precisely, we construct open subsets in the bifurcation locus in the space of holomorphic mappings of degree d of P^k (C) for every d $\ge$ 2 and k $\ge$ 2.

v2 now includes a discussion on the Newhouse phenomenon

Cited by in corpus (4)

• Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree
• Dynamics of skew-products tangent to the identity
• Geometric methods in holomorphic dynamics
• Strong probabilistic stability in holomorphic families of endomorphisms of ${\mathbb{P}}^k(\mathbb{C})$ and polynomial-like maps