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Arcs, balls and spheres that cannot be attractors in $ \mathbb{R}^3$


Author: J. J. Sánchez-Gabites
Journal: Trans. Amer. Math. Soc. 368 (2016), 3591-3627
MSC (2010): Primary 54H20, 37B25, 37E99
DOI: https://doi.org/10.1090/tran/6570
Published electronically: June 24, 2015
MathSciNet review: 3451887
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Abstract: For any compact set $ K \subseteq \mathbb{R}^3$ we define a number $ r(K)$ that is either a nonnegative integer or $ \infty $. Intuitively, $ r(K)$ provides some information on how wildly $ K$ sits in $ \mathbb{R}^3$. We show that attractors for discrete or continuous dynamical systems have finite $ r$ and then prove that certain arcs, balls and spheres cannot be attractors by showing that their $ r$ is infinite.


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Additional Information

J. J. Sánchez-Gabites
Affiliation: Facultad de Ciencias Económicas y Empresariales, Universidad Autónoma de Madrid, Campus Universitario de Cantoblanco, 28049 Madrid, España
Email: JaimeJ.Sanchez@uam.es

DOI: https://doi.org/10.1090/tran/6570
Received by editor(s): June 20, 2013
Received by editor(s) in revised form: May 14, 2014
Published electronically: June 24, 2015
Additional Notes: The author was partially supported by MICINN (grant MTM 2009-07030).
The author wishes to express his deepest gratitude to Professor Rafael Ortega (Universidad de Granada) for his generous support and encouragement during the writing of this paper
Article copyright: © Copyright 2015 American Mathematical Society