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arXiv · 1603.05917

On the set of wild points of attracting surfaces in $\mathbb{R}^3$

J. J. Sánchez-Gabites
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Abstract

Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. Using this result and a modification of the classical construction of a wild sphere due to Antoine we show that there exist uncountably many different $2$--spheres in $\mathbb{R}^3$ none of which can be realized as an attractor for a homeomorphism.

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J. J. Sánchez-Gabites. 2016-03-18. On the set of wild points of attracting surfaces in $\mathbb{R}^3$. https://arxiv.org/abs/1603.05917

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