We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in , n ≥ 1, is a σ-limit set for some continuous map.
We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms...