Abstract

Let M2 be a compact connected two-dimensional manifold, with or without boundary, and let f: M2 → M2 be a continuous map. We prove that if M ⊆ M2 is a minimal set of the dynamical system (M2, f) then either M = M2 or M is a nowhere dense subset of M2. Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case M2 is a torus or a Klein bottle. © 2008 Cambridge University Press.