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Conformal Geometry and Dynamics

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On Poincaré extensions of rational maps


Authors: Carlos Cabrera, Peter Makienko and Guillermo Sienra
Journal: Conform. Geom. Dyn. 19 (2015), 197-220
MSC (2010): Primary 37F10, 37F30; Secondary 30F99
DOI: https://doi.org/10.1090/ecgd/281
Published electronically: July 29, 2015
MathSciNet review: 3373954
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Abstract: There is a classical extension of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space $ \mathbb{H}^3$ which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $ \mathbb{H}^3$ exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in $ \mathbb{H}^3$ that allows us to construct an extension of any given rational map which is right equivariant with respect to the action of $ PSL(2,\mathbb{C})$.


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Additional Information

Carlos Cabrera
Affiliation: Universidad Nacional Autonoma de Mexico, Unidad Cuernavaca del Instituto de Matematicas, Universidad s/n, Col Lomas de Chamilpa, 62210 Cuernavaca, Mexico
Email: carloscabrerao@im.unam.mx

Peter Makienko
Affiliation: Universidad Nacional Autonoma de Mexico, Unidad Cuernavaca del Instituto de Matematicas, Universidad s/n, Col Lomas de Chamilpa, 62210 Cuernavaca, Mexico
Email: makienko@matcuer.unam.mx

Guillermo Sienra
Affiliation: Universidad Nacional Autonoma de Mexico, Facultad de Ciencias, Av. Universidad 3000, 04510 Mexico, Mexico
Email: gsl@dinamical.fciencias.unam.mx

DOI: https://doi.org/10.1090/ecgd/281
Received by editor(s): June 13, 2013
Received by editor(s) in revised form: November 10, 2014, and April 24, 2015
Published electronically: July 29, 2015
Additional Notes: This work was partially supported by PAPIIT IN-105912 and CONACYT CB2010/153850.
Article copyright: © Copyright 2015 American Mathematical Society