R-closed homeomorphisms on surfaces

Let f be an R-closed homeomorphism on a connected orientable closed surface M. In this paper, we show that if M   has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If M=T2M=T2 and f   is neither minimal nor periodic, then either each minimal set is a finite disjoint union of essential circloids or there is a minimal set which is an extension of a Cantor set. If M=S2M=S2 and f is not periodic but orientation-preserving (resp. reversing), then the minimal sets of f   (resp. f2f2) are exactly two fixed points and a family of circloids and S2/f˜≅[0,1].