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Conformal Geometry and Dynamics

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Uniformly quasiregular maps with toroidal Julia sets

Authors: Riikka Kangaslampi, Kirsi Peltonen and Jang-Mei Wu
Journal: Conform. Geom. Dyn. 16 (2012), 81-88
MSC (2010): Primary 53A30, 53C20; Secondary 30C65
Published electronically: March 21, 2012
MathSciNet review: 2899679
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Abstract: The iterates of a uniformly quasiregular map acting on a Riemannian manifold are quasiregular with a uniform bound on the dilatation. There is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We construct the first examples of uniformly quasiregular mappings that have a 2-torus as the Julia set. The spaces supporting this type of mappings include the Hopf link complement and its lens space quotients.

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Additional Information

Riikka Kangaslampi
Affiliation: Aalto University, P.O. Box 11100, 00076 Aalto, Finland

Kirsi Peltonen
Affiliation: Aalto University, P.O. Box 11100, 00076 Aalto, Finland

Jang-Mei Wu
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801-2975

Keywords: Uniformly quasiregular mapping, Lattès-type mapping, Julia set, conformal structure, lens space
Received by editor(s): October 10, 2011
Published electronically: March 21, 2012
Additional Notes: The first author was supported by the Emil Aaltonen Foundation
The second author was supported by the Väisälä Foundation of the Finnish Academy of Science and Letters
The third author was supported by the National Science Foundation Grant DMS-1001669
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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