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oai:docta.ucm.es:20.500.14352/1332932026-02-26T01:25:20Zcom_20.500.14352_14col_20.500.14352_15

00925njm 22002777a 4500 dc Sánchez Gabites, Jaime Jorge author 2017 Suppose that a closed surface S ⊆ R3 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. 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 defined for any compact set K ⊆ R3 and 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 r must 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. 10.1016/j.aim.2017.05.011 https://hdl.handle.net/20.500.14352/133293 https://doi.org/10.1016/j.aim.2017.05.011 On the set of wild points of attracting surfaces in R^3