Latt{\`e}s maps and the interior of the bifurcation locus

TL;DR

This paper demonstrates that high-degree Lattès maps can be perturbed to exhibit bifurcations, revealing open sets of bifurcations near these maps and their relation to the bifurcation locus.

Contribution

The authors develop a method to intersect limit sets of specific IFS with curves and apply it to show perturbations of Lattès maps lead to bifurcations.

Findings

Existence of open bifurcation sets near high-degree Lattès maps

Every Lattès map has an iterate in the closure of the bifurcation locus interior

Method to intersect IFS limit sets with curves for bifurcation analysis

Abstract

We show the existence of open sets of bifurcations near Latt{\`e}s maps of sufficiently high degree. In particular, every Latt{\`e}s map has an iterate which is in the closure of the interior of the bifurcation locus. To show this, we design a method to intersect the limit set of some particular type of IFS with a well-oriented curve. Then, we show that a Latt{\`e}s map of sufficiently high degree can be perturbed to exhibit this geometry.