Homogenization of codimension 1 actions of ${ℝ}^{n}$ near a compact orbit
Annales de l'Institut Fourier, Tome 44 (1994) no. 5, p. 1435-1448
Soit $\Phi$ une ${ℝ}^{n}$-action ${C}^{\infty }$ sur une variété orientable de dimension $n+1$. Supposons que $\Phi$ possède une orbite compacte isolée $T$ et soit $W$ un petit voisinage tubulaire de $T$. À l’aide d’un changement de variables ${C}^{\infty }$, nous pouvons écrire $W={ℝ}^{n}/{ℤ}^{n}×I$ et $T={ℝ}^{n}/{ℤ}^{n}×\left[0\right]$, où $I$ est un intervalle réel contenant 0.Dans ce travail nous montrons que par un changement de variables ${C}^{0}$, qui est ${C}^{\infty }$ au-dehors de $T$, nous pouvons rendre ${\Phi }_{|W}$ invariante par les transformations du type $\left(x,z\right)\to \left(x+a,z\right),\phantom{\rule{0.166667em}{0ex}}a\in {ℝ}^{n}$, où $x\in {ℝ}^{n}/{ℤ}^{n}$ et $z\in I$. Comme corollaire nous pouvons décrire complètement la dynamique de $\Phi$ sur $W$.
Let $\Phi$ be a ${C}^{\infty }\phantom{\rule{0.166667em}{0ex}}{ℝ}^{n}$-action on an orientable $\left(n+1\right)$-dimensional manifold. Assume $\Phi$ has an isolated compact orbit $T$ and let $W$ be a small tubular neighborhood of it. By a ${C}^{\infty }$ change of variables, we can write $W={ℝ}^{n}/{ℤ}^{n}×I$ and $T={𝕋}^{n}×\left[0\right]$, where $I$ is some interval containing 0.In this work, we show that by a ${C}^{0}$ change of variables, ${C}^{\infty }$ outside $T$, we can make ${\Phi }_{|W}$ invariant by transformations of the type $\left(x,z\right)\to \left(x+a,z\right),\phantom{\rule{0.166667em}{0ex}}a\in {ℝ}^{n}$, where $x\in {ℝ}^{n}/{ℤ}^{n}$ and $z\in I$. As a corollary one cas describe completely the dynamics of $\Phi$ in $W$.
@article{AIF_1994__44_5_1435_0,
author = {Craizer, Marcos},
title = {Homogenization of codimension 1 actions of ${\mathbb {R}}^n$ near a compact orbit},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {44},
number = {5},
year = {1994},
pages = {1435-1448},
doi = {10.5802/aif.1440},
zbl = {0820.34021},
mrnumber = {95m:58100},
language = {en},
url = {http://www.numdam.org/item/AIF_1994__44_5_1435_0}
}

Craizer, Marcos. Homogenization of codimension 1 actions of ${\mathbb {R}}^n$ near a compact orbit. Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1435-1448. doi : 10.5802/aif.1440. http://www.numdam.org/item/AIF_1994__44_5_1435_0/

 J.L. Arraut and M. Craizer, A characterization of 2-dimensional foliations of rank 2 on compact orientable 3-manifolds, preprint.

 G. Chatelet, H. Rosenberg and D. Weil, A classification of the topological types of ℝ2-actions on closed orientable 3-manifolds, Publ. Math. IHES, 43 (1973), 261-272. | Numdam | MR 49 #11533 | Zbl 0278.57015

 N. Koppel, Commuting diffeomorphisms. Global Analysis, Proc. of Symp. in Pure Math., AMS, XIV (1970). | Zbl 0225.57020

 R. Mañé, Ergodic theory and differentiable dynamics, Springer-Verlag, 1987. | MR 88c:58040 | Zbl 0616.28007

 F. Sergeraert, Feuilletages et difféomorphismes infiniment tangents à l'identité, Inv. Math., 39 (1977), 253-275. | MR 57 #13973 | Zbl 0327.58004

 G. Szekeres, Regular iteration of real and complex functions, Acta Math., 100 (1958), 163-195. | MR 21 #5744 | Zbl 0145.07903