A classification of minimal sets for surface homeomorphisms.

We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by T. Jäger et al. [Math. Z. 274, No. 1–2, 405–426 (2013; Zbl 1322.37005)] in the torus. The classification being sensitive to global topology, striking differences with the toral case arise. We also show that the given classification can be strengthened when considered for non-wandering (conservative) systems, and in the homotopy class of pseudo-Anosov maps.