Lattès maps and the interior of the bifurcation locus

Abstract

We study the phenomenon of robust bifurcations in the space of holomorphic maps of \begin{document}$ \mathbb{P}^2(\mathbb{C}) $\end{document} . We prove that any Lattes example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattes map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in \begin{document}$ \mathbb{C}^2 $\end{document} with a well-oriented complex curve. Then we show that any Lattes map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.