The realization problem of non-connected compacta as attractors

Let K subset of Rn be a compactum which is a disjoint union of compacta K1, . . . , Kr. We consider the question of whether the realizability of K as an attractor is equivalent to that of the Ki individually. We show that in the case of discrete dynamical systems it may happen that all the Ki can be realized as attractors for suitable homeomorphisms but K cannot. The linking of the Ki to each other plays an essential role in this phenomenon, and using this geometric condition we answer the above question completely for a certain class of compacta. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).