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Perturbations of flexible Lattès maps. (Perturbations des exemples de Lattès flexibles.) (English. French summary) Zbl 1326.37035
A rational map of degree \(D\geq 2\) is strictly postcritically finite if the orbit of each critical point intersects a repelling cycle. A flexible Lattès map is a map \(f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}\) which is double-covered by an integral torus endomorphism, that is an endomorphism of the Riemann sphere such that there exists a \(2\)-to-\(1\) cover \(\Theta:T\to \hat{\mathbb{C}}\) from a torus \(T=\mathbb{C} / \Lambda\) such that the pull-back of \(f\) by \(\Theta\) is an endomorpism \(L\) of \(T\) of the form \(L(\tau)=a\tau+b\pmod{\Lambda}\) for an integer \(a\). Such morphisms constitute examples of maps with Julia sets which are the entire Riemann sphere.
The main result of this article is the following:
Theorem: Every flexible Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37D05 Dynamical systems with hyperbolic orbits and sets
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