Extended orbit properties on surfaces

TL;DR

This paper explores the properties of $ $-actions on compact surfaces, establishing relationships between various recurrence and periodicity conditions, and characterizing $R$-closed actions in terms of singularities and recurrence.

Contribution

It provides new characterizations of $R$-closed and pointwise almost periodic actions on surfaces, linking recurrence, non-wandering, and singularity conditions.

Findings

$R$-closed implies p.a.p. implies recurrence implies non-wandering.

Recurrence with finitely many singularities characterizes regular non-wandering actions.

Without locally dense orbits, p.a.p. is equivalent to recurrence without infinitely singular orbits.

Abstract

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 $\R$-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.".