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 = T 2 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 = S 2 and f is not periodic but orientation-preserving (resp. reversing), then the minimal sets of f (resp. f 2 ) are exactly two fixed points and a family of circloids and S 2 / f ˜ ≅ [ 0 , 1 ] .