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Conformal Geometry and Dynamics

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Lattès-type mappings on compact manifolds


Authors: Laura Astola, Riikka Kangaslampi and Kirsi Peltonen
Journal: Conform. Geom. Dyn. 14 (2010), 337-367
MSC (2010): Primary 53A30, 53C20; Secondary 30C65
DOI: https://doi.org/10.1090/S1088-4173-2010-00220-1
Published electronically: December 29, 2010
MathSciNet review: 2746722
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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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Additional Information

Laura Astola
Affiliation: Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Email: l.j.astola@tue.nl

Riikka Kangaslampi
Affiliation: Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Aalto, Finland
Email: riikka.kangaslampi@tkk.fi

Kirsi Peltonen
Affiliation: Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Aalto, Finland
Email: kirsi.peltonen@helsinki.fi

DOI: https://doi.org/10.1090/S1088-4173-2010-00220-1
Keywords: Uniformly quasiregular mapping, Lattès-type mapping, Julia set, conformal structure
Received by editor(s): August 23, 2010
Published electronically: December 29, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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