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On the set of wild points of attracting surfaces in R3

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URI: https://hdl.handle.net/10016/25276
ISSN: 0001-8708
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set_sanchez_AM_2017_ps.pdf (1.2 MB)
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2017-07
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Elsevier
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Suppose that a closed surface S⊆R3is an attractor, notne-cessarily global, for a discrete dynamical system. Assuming that its set of wild points Wis totally disconnected, we prove that (up to an ambient homeomorphism) it has to be con-tained in a straight line. As a corollary we show that there exist uncountably many different 2-spheres in R3 none of which can be realized as an attractor for a homeomorphism. Our techniques hinge on a quantity r(K)that can be de-fined for any compact set K⊆R3and is related to “how wildly” it sits in R3. We establish the topological results that (i)r(W) ≤r(S)and (ii) any totally disconnected set having a finite rmust be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r.
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Sánchez-Gabites, J.J. (2017). On the set of wild points of attracting surfaces in R3. Advances in Mathematics, v. 315, pp. 246-284
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