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

TL;DR

This paper extends the Poincaré recurrence theorem to flows on surfaces, providing classifications of minimal flows, characterizations of orbit density, and emphasizing the importance of finiteness conditions on singular points.

Contribution

It introduces new concepts like strict limit circuits and circuits with wandering holonomy to classify and analyze flows on surfaces, generalizing classical recurrence results.

Findings

Classified non-recurrent non-wandering orbits on compact surfaces.

Characterized minimality of flows on compact surfaces.

Established the necessity of finiteness of singular points for certain properties.

Abstract

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