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Attracting sets on surfaces


Authors: J. Iglesias, A. Portela, A. Rovella and J. Xavier
Journal: Proc. Amer. Math. Soc. 143 (2015), 765-779
MSC (2010): Primary 37C70; Secondary 37B25
DOI: https://doi.org/10.1090/S0002-9939-2014-12274-X
Published electronically: October 16, 2014
MathSciNet review: 3283663
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Abstract: Let $ f$ be a continuous endomorphism of a surface $ M$, and $ A$ an attracting set such that the restriction $ f\vert _A: A \to A$ is a $ d:1$ covering map. We show that if $ f$ is a local homeomorphism, then $ f$ is also a $ d:1$ covering of the immediate basin of $ A$. Moreover, the techniques provide a characterization of invariant $ d:1$ continua on surfaces. These results are no longer true on manifolds of dimensions at least three.


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J. Iglesias
Affiliation: Facultad de Ingenieria, Universidad de La República, IMERL, Julio Herrera y Reissig 565, C. P. 11300, Montevideo, Uruguay
Email: jorgei@fing.edu.uy

A. Portela
Affiliation: Facultad de Ingenieria, Universidad de La República, IMERL, Julio Herrera y Reissig 565, C. P. 11300, Montevideo, Uruguay
Email: aldo@fing.edu.uy

A. Rovella
Affiliation: Facultad de Ciencias, Universidad de La República, Centro de Matemática, Iguá 4225, C. P. 11400, Montevideo, Uruguay
Email: leva@cmat.edu.uy

J. Xavier
Affiliation: Facultad de Ingenieria, Universidad de La República, IMERL, Julio Herrera y Reissig 565, C. P. 11300, Montevideo, Uruguay
Email: jxavier@fing.edu.uy

DOI: https://doi.org/10.1090/S0002-9939-2014-12274-X
Received by editor(s): August 3, 2012
Received by editor(s) in revised form: March 13, 2013, May 3, 2013, May 7, 2013, May 20, 2013, May 22, 2013, and May 27, 2013
Published electronically: October 16, 2014
Communicated by: Nimish Shah
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.