Pseudo-immersions

Joris, Henri; Preissmann, Emmanuel

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

Si $f$ est un germe ${𝒞}^{\infty }$ de $({\mathbf{R}}^n,0)$, on dira que $f$ est une pseudo-immersion (on notera $f\in {\Psi }_{n,m}$) si tous les germes continus $g$ de $(\mathbf{R},0)$ dans $({\mathbf{R}}^m,0)$, tels que $f\circ g\in {𝒞}^{\infty }$ sont eux-mêmes ${𝒞}^{\infty }$. On détermine complètement ${\Psi }_{n,1}$, et on montre que ${\Psi }_{2,2}={\mathrm{Diff}}_2$. Par ailleurs, si $\mathbf{K}=\mathbf{R}$ ou $\mathbf{C}$ et si $g$ est une application de $\mathbf{K}$ dans $\mathbf{K}$ telle que $g^2$ et $g^3$ sont ${𝒞}^{\infty }$, alors $g$ est aussi ${𝒞}^{\infty }$. Si $\mathbf{K}=\mathbf{H}$ (corps des hamiloniens) alors cette implication n’est plus vraie.

Let $f:({\mathbf{R}}^m,0)\to ({\mathbf{R}}^n,0)$ be a ${𝒞}^{\infty }$-germ. $f$ is said to be a pseudo-immersion (noted $f\in {\Psi }_{n,m}$) if for continuous germ $g:(\mathbf{R},0)\to ({\mathbf{R}}^m,0)$, $f\circ g\in {𝒞}^{\infty }$ implies $g\in {𝒞}^{\infty }$. ${\Psi }_{n,1}$, is completely determined, for each natural $n,{\Psi }_{2,2}$ is shown to coincide with ${\mathrm{Diff}}_2$. If $\mathbf{K}=\mathbf{R}$ or $\mathbf{C}$ and $g:\mathbf{K}\to \mathbf{K}$ is such that $g^2$ and $g^3$ are in ${𝒞}^{\infty }$. If $\mathbf{K}=\mathbf{H}$ (field of Hamiltonians), a counter-exemple shows that this implication is no more valid.

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