Tame semiflows for piecewise linear vector fields

Panazzolo, Daniel

Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1593-1628.

Soit $ℰ$ une décomposition disjointe de ${ℝ}^n$ et soit $X$ un champ de vecteurs sur ${ℝ}^n$, défini comme étant linéaire sur chaque cellule de la décomposition $ℰ$. Sous certaines hypothèses naturelles, nous montrons comment associer un semi-flot à $X$ et nous montrons qu’un tel semi-flot appartient à la structure o-minimale ${ℝ}_{\mathrm{an},exp}$. En particulier, si $X$ est un champ de vecteurs continu et $\Gamma$ est un sous-ensemble invariant par $X$, notre résultat implique que l’application de premier retour de Poincaré associée à $\Gamma$ est également dans ${ℝ}_{\mathrm{an},exp}$ quand $\Gamma$ est non-spiralante.

Let $ℰ$ be a disjoint decomposition of ${ℝ}^n$ and let $X$ be a vector field on ${ℝ}^n$, defined to be linear on each cell of the decomposition $ℰ$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${ℝ}_{\mathrm{an},exp}$. In particular, when $X$ is a continuous vector field and $\Gamma$ is an invariant subset of $X$, our result implies that if $\Gamma$ is non-spiralling then the Poincaré first return map associated $\Gamma$ is also in ${ℝ}_{\mathrm{an},exp}$.

Classification : 03C64, 14P10, 34C25, 37G15
Keywords: piecewise linear vector field, o-minimal, semiflow
Mot clés : champ de vecteurs linéaire par parties, o-minimale, semi-flot

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