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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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On Poincaré extensions of rational maps
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by Carlos Cabrera, Peter Makienko and Guillermo Sienra PDF
Conform. Geom. Dyn. 19 (2015), 197-220 Request permission

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
  • MR Author ID: 829036
  • 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
  • MR Author ID: 223466
  • Email: gsl@dinamical.fciencias.unam.mx
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 19 (2015), 197-220
  • MSC (2010): Primary 37F10, 37F30; Secondary 30F99
  • DOI: https://doi.org/10.1090/ecgd/281
  • MathSciNet review: 3373954