Pseudo-immersions

Joris, Henri; Preissmann, Emmanuel

doi:10.5802/aif.1092

Pseudo-immersions

Joris, Henri ; Preissmann, Emmanuel

Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 195-221.

Résumé
Abstract

Si $f$ est un germe $𝒞^{\infty}$ de $(R^{n}, 0)$ , on dira que $f$ est une pseudo-immersion (on notera $f \in Ψ_{n, m}$ ) si tous les germes continus $g$ de $(R, 0)$ dans $(R^{m}, 0)$ , tels que $f \circ g \in 𝒞^{\infty}$ sont eux-mêmes $𝒞^{\infty}$ . On détermine complètement $Ψ_{n, 1}$ , et on montre que $Ψ_{2, 2} = {Diff}_{2}$ . Par ailleurs, si $K = R$ ou $C$ et si $g$ est une application de $K$ dans $K$ telle que $g^{2}$ et $g^{3}$ sont $𝒞^{\infty}$ , alors $g$ est aussi $𝒞^{\infty}$ . Si $K = H$ (corps des hamiloniens) alors cette implication n’est plus vraie.

Let $f : (R^{m}, 0) \to (R^{n}, 0)$ be a $𝒞^{\infty}$ -germ. $f$ is said to be a pseudo-immersion (noted $f \in Ψ_{n, m}$ ) if for continuous germ $g : (R, 0) \to (R^{m}, 0)$ , $f \circ g \in 𝒞^{\infty}$ implies $g \in 𝒞^{\infty}$ . $Ψ_{n, 1}$ , is completely determined, for each natural $n, Ψ_{2, 2}$ is shown to coincide with ${Diff}_{2}$ . If $K = R$ or $C$ and $g : K \to K$ is such that $g^{2}$ and $g^{3}$ are in $𝒞^{\infty}$ . If $K = H$ (field of Hamiltonians), a counter-exemple shows that this implication is no more valid.

MR Zbl

DOI : 10.5802/aif.1092

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     author = {Joris, Henri and Preissmann, Emmanuel},
     title = {Pseudo-immersions},
     journal = {Annales de l'Institut Fourier},
     pages = {195--221},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {2},
     year = {1987},
     doi = {10.5802/aif.1092},
     mrnumber = {88e:57028},
     zbl = {0596.58004},
     language = {fr},
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Joris, Henri; Preissmann, Emmanuel. Pseudo-immersions. Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 195-221. doi : 10.5802/aif.1092. https://www.numdam.org/articles/10.5802/aif.1092/

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