Extended orbit properties on surfaces

In this paper, we study “demi-caract'eristique” and (Poisson) stability in the sense of Poincar'e. Using the definitions 'a la Poincar'e 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 $\mathrm{\mid }\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(v)\mathrm{\mid }<\mathrm{\infty }$ 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 $\mathrm{\mid }\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(v)\mathrm{\mid }<\mathrm{\infty }$, then $v$ is “$R$-closed” iff $v$ is “p.a.p.”.