Tame semiflows for piecewise linear vector fields

[Semi-flots pour des champs de vecteurs linéaires morcelés]

Panazzolo, Daniel

Annales de l'Institut Fourier, Tome 52 (2002), p. 1593-1628 / Harvested from Numdam

Résumé

Soit ${\mathcal {E}}$ une décomposition disjointe de ${\mathbb {R}}^n$ et soit $X$ un champ de vecteurs sur ${\mathbb {R}}^n$, défini comme étant linéaire sur chaque cellule de la décomposition ${\mathcal {E}}$. 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 ${\mathbb {R}}_{{\rm an}, \exp }$. 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 ${\mathbb {R}}_{{\rm an}, \exp }$ quand $\Gamma $ est non-spiralante.

Abstract

Let ${\mathcal {E}}$ be a disjoint decomposition of ${\mathbb {R}}^n$ and let $X$ be a vector field on ${\mathbb {R}}^n$, defined to be linear on each cell of the decomposition ${\mathcal {E}}$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${\mathbb {R}}_{{\rm an},\exp }$. 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 ${\mathbb {R}}_{{\rm an} ,\exp }$.

Détail

Publié le : 2002-01-01

MR 1952525 | Zbl 1009.37008

DOI : https://doi.org/10.5802/aif.1928
Classification:  03C64,  14P10,  34C25,  37G15
Mots clés: champ de vecteurs linéaire par parties, o-minimale, semi-flot

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