Some remarks on the topology of hyperbolic actions of $\mathbb{R}^n$ on $n$-manifolds
Résumé
This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the $\mathbb{R}$-action generated by a fixed vector of $\mathbb{R}^n$, that provides some results on the number of hyperbolic domains and the number of fixed points of the action. We study with more details the case of the $2$-sphere, in particular we investigate some combinatorial properties of the associated $4$-valent graph embedded in $\mathrm{S}^2$. We also construct hyperbolic actions in dimension $3$, on the sphere $\mathrm{S}^2$ and on the projective space $\mathbb{R}P^3$ .
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