Lattès maps and the interior of the bifurcation locus

Sébastien Biebler

doi:10.3934/jmd.2019014

Article Contents

2019, Volume 15: 95-130. Doi: 10.3934/jmd.2019014

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Lattès maps and the interior of the bifurcation locus

Sébastien Biebler

LAMA, UMR8050, Université Paris-Est Marne-La-Vallée, 5 Boulevard Descartes, 77454 Champs-sur-Marne, France

Received: March 21, 2018

Revised: October 14, 2018

Published: May 2019

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Abstract

We study the phenomenon of robust bifurcations in the space of holomorphic maps of $ \mathbb{P}^2(\mathbb{C}) $. We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattès 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 $ \mathbb{C}^2 $ with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.

Keywords:

Lattès maps,
stability,
robust bifurcations,
blender property,
holomorphic endomorphisms of $ \mathbb{P}^{2}(\mathbb{C}) $.

Mathematics Subject Classification: Primary: 37F45; Secondary: 37F10.

Citation:

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Figure 1. The yellow color stands for $\mathscr{U}_{x} \backslash (\mathscr{U}_{x} \cap \mathscr{U}''_{x})$, the red for $\mathscr{U}'_{x}$, the blue for $\mathscr{U}''_{x} \backslash \mathscr{U}'_{x}$. The arrows show a typical sequence of matrices: one multiplies $I_{2}$ by $I_{2}+M_{0}$ (with $M_{0} \in x \cdot V^{0} $) a finite number of times, then by $I_{2}+M_{p}$ (with $M_{p} \in x \cdot V^{p} $)

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