Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AGT] Artur Avila, Sébastien Gouëzel, and Masato Tsujii, Smoothness of solenoidal attractors, Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 21-35. MR 2191383 (2008m:37053), https://doi.org/10.3934/dcds.2006.15.21
  • [AMS] Nobuo Aoki, Kazumine Moriyasu, and Naoya Sumi, $ C^1$-maps having hyperbolic periodic points, Fund. Math. 169 (2001), no. 1, 1-49. MR 1852352 (2002h:37028), https://doi.org/10.4064/fm169-1-1
  • [BKRU] Rodrigo Bamón, Jan Kiwi, Juan Rivera-Letelier, and Richard Urzúa, On the topology of solenoidal attractors of the cylinder, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 2, 209-236 (English, with English and French summaries). MR 2201152 (2007b:37042), https://doi.org/10.1016/j.anihpc.2005.03.002
  • [Bue] Jorge Buescu, Exotic attractors, Progress in Mathematics, vol. 153, Birkhäuser Verlag, Basel, 1997. From Liapunov stability to riddled basins. MR 1488418 (99f:58135)
  • [Con] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133 (80c:58009)
  • [IPR] J. Iglesias, A. Portela, and A. Rovella, $ C^1$ stable maps: examples without saddles, Fund. Math. 208 (2010), no. 1, 23-33. MR 2609218 (2011f:37035), https://doi.org/10.4064/fm208-1-2
  • [IPR1] J. Iglesias, A. Portela, and A. Rovella, $ C^1$ stability of endomorphisms on two-dimensional manifolds, Fund. Math. 219 (2012), no. 1, 37-58. MR 2990545, https://doi.org/10.4064/fm219-1-3
  • [Prz] Feliks Przytycki, On $ U$-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math. 60 (1977), no. 1, 61-77. MR 0445553 (56 #3891)
  • [Tho] Carsten Thomassen, The Jordan-Schönflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), no. 2, 116-130. MR 1144352 (92k:57026), https://doi.org/10.2307/2324180
  • [Tsu] Masato Tsujii, Fat solenoidal attractors, Nonlinearity 14 (2001), no. 5, 1011-1027. MR 1862809 (2002j:37035), https://doi.org/10.1088/0951-7715/14/5/306

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37C70, 37B25

Retrieve articles in all journals with MSC (2010): 37C70, 37B25


Additional Information

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.

American Mathematical Society