Extended orbit properties on surfaces

Abstract

In this paper, we study "demi-caractéristique" and (Poisson) stability in the sense of Poincaré. Using the definitions á la Poincaré for $\mathbb{R}$-actions $v$ on compact connected surfaces, we show that "$R$-closed" $\Rightarrow$ "pointwise almost periodicity (p.a.p.)" $\Rightarrow$ "recurrence" $\Rightarrow$ non-wandering. Moreover, we show that the action $v$ is "recurrence" with $|\mathrm{Sing}(v)|<\infty$ iff $v$ is regular non-wandering. If there are no locally dense orbits, then $v$ is "p.a.p." iff $v$ is "recurrence" without "orbits" containing infinitely singularities. If $|\mathrm{Sing}(v)|<\infty$, then $v$ is "$R$-closed" iff $v$ is "p.a.p.".