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Lattès maps and the interior of the bifurcation locus

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  • 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.

    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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