Non density of stability for holomorphic mappings on P^k

TL;DR

This paper demonstrates that the density of J-stable maps, a property in one-dimensional complex dynamics, does not extend to higher dimensions, by constructing specific open subsets in the bifurcation locus.

Contribution

It shows the failure of J-stability density in higher-dimensional holomorphic mappings, contrasting with known results in one dimension.

Findings

J-stable maps are not dense in higher dimensions

Constructed open subsets in the bifurcation locus for P^k

Demonstrated failure of classical stability results in higher dimensions

Abstract

A well-known theorem due to Ma\~n\'e-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.