Topological characterizations of recurrence, Poisson stability, and isometric property of flows on surfaces

Abstract

The long-time behavior is one of the most fundamental properties of dynamical systems. Poincar\'e studied the Poisson stability to capture the property of whether points return arbitrarily near the initial positions. Birkhoff studied the concept of recurrent points. Hilbert introduced distal property to describe a rigid group of motions. We show that Poisson stability, recurrence, and distal property of flows on surfaces are topological properties. In fact, a flow on a connected compact surface is Poisson stable (resp. recurrent) if and only if the Kolmogorov quotient of the orbit space satisfies $T_{1}$ (resp. $T_{1/2}$) separation axiom. Moreover, Poisson stability for such flows is equivalent to distal property. In addition, $T_2$ separation axiom corresponds to the isometric property. In addition, we construct ``Lakes of Wada continua'' which are the singular point sets of recurrent non-Poisson-stable flows and Poisson stable distal non-equicontinuous flows on surfaces.