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Poincar\'e recurrence theorem implies the density of recurrent points forvolume-preserving dynamical systems on compact domains. The density of closedorbits in the non-wandering set is one of the essential properties of Axiom Aand chaos. The minimal flows are one of the most fundamental objects intopological dynamics. To analyze minimality, recurrence, and density in thispaper, we introduce a strict limit circuit and a circuit with wanderingholonomy. Using these concepts, we classify non-recurrent non-wandering orbitsof flows on compact surfaces with finitely many singular points, characterizethe minimality of flows on compact surfaces, and generalize the Poincar\'erecurrence theorem for flows on surfaces. Moreover, we characterize the densityof closed orbits in the non-wandering set under finiteness of singular points.Furthermore, we demonstrate the necessity of finiteness of singular points andcompactness of surfaces. In addition, the analogous results hold for flows withfinitely many connected components of the singular point set on non-compactsurfaces using the end compactification and collapsing singular points.

Generalization of Poincar\'e recurrence theorem for flows on surfaces and characterization of minimal flows on compact surfaces