The long-time behavior is one of the most fundamental properties of dynamical systems. Poincaré 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 T₁ (resp. T_1/2) separation axiom. Moreover, Poisson stability for such flows is equivalent to distal property. In addition, T₂ 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.

长时间行为是动力系统最基本的属性之一。庞加莱研究了泊松稳定性，以捕捉点是否任意返回到初始位置附近的性质。伯克霍夫研究了循环点的概念。希尔伯特引入远端属性来描述一组刚性运动。我们证明了表面流动的泊松稳定性、递归性和远端特性都是拓扑特性。事实上，当且仅当轨道空间的柯尔莫哥洛夫商满足T ₁（或T _1/2）分离公理时，连通致密表面上的流才是泊松稳定（或循环）的。此外，此类流动的泊松稳定性相当于远端特性。另外，T ₂分离公理对应于等距性质。此外，我们还构建了“Wada 湖连续体”，即表面上循环非泊松稳定流和泊松稳定远端非等连续流的奇点集。