Proper minimal sets on compact connected 2-manifolds are nowhere dense

Abstract

AbstractLet $\mathcal {M}^2$ be a compact connected two-dimensional manifold, with or without boundary, and let $f:{\mathcal {M}}^2\to \mathcal {M}^2$ be a continuous map. We prove that if $M \subseteq \mathcal {M}^2$ is a minimal set of the dynamical system $(\mathcal {M}^2,f)$ then either $M = \mathcal {M}^2$ or M is a nowhere dense subset of $\mathcal {M}^2$. Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case $\mathcal {M}^2$ is a torus or a Klein bottle.