Lattès-type mappings on compact manifolds

Astola, Laura; Kangaslampi, Riikka; Peltonen, Kirsi

doi:10.1090/S1088-4173-2010-00220-1

Lattès-type mappings on compact manifolds
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by Laura Astola, Riikka Kangaslampi and Kirsi Peltonen

Conform. Geom. Dyn. 14 (2010), 337-367

DOI: https://doi.org/10.1090/S1088-4173-2010-00220-1

Published electronically: December 29, 2010

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Abstract:

A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. Such maps are rational with respect to some measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Lattès’ construction of chaotic rational functions acting on the extended plane $\bar {\mathbb {C}}$. We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.

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