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Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form $$ f(x) = \exp\left( \sum_{i=1}^n f_i(x) A_i \right)x, \qquad f_i \in C_c(\mathbb{R}^d,[0,1]), A_i \in \mathfrak{gl}_d(\mathbb{R}) . $$ Edit: What additional constrains do I need on $f_i$ so that $f$ is a diffeomorphism and $Y$ is dense in the subset of $C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ of diffeomorphisms fixing $0$?

Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form $$ f(x) = \exp\left( \sum_{i=1}^n f_i(x) A_i \right)x, \qquad f_i \in C_c(\mathbb{R}^d,[0,1]), A_i \in \mathfrak{gl}_d(\mathbb{R}) . $$ Edit: What additional constrains do I need on $f_i$ so that $f$ is a diffeomorphism and $Y$ is dense in the subset of $C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ of diffeomorphisms fixing $0$? Or is it just dense in some space of embeddings?

Let $\exp$ denote the matrix exponential map and let $X \in C^{\infty}(\mathbb{R}^d,\mathrm{Mat}_{d\times d}(\mathbb{R}))$ denote the set of all continuous injective maps. Let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ denote the set of all maps of the form $$ f(x)= \exp(g(x))x ,\quad g \in X. $$ Is $Y$ a studied object? Is it the collection dense in the set of all diffeomorphisms of $\mathbb{R}^d$ fixing the origin? For the compact-convergence topology?

Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form $$ f(x) = \exp\left( \sum_{i=1}^n f_i(x) A_i \right)x, \qquad f_i \in C_c(\mathbb{R}^d,[0,1]), A_i \in \mathfrak{gl}_d(\mathbb{R}) . $$ Edit: What additional constrains do I need on $f_i$ so that $f$ is a diffeomorphism and $Y$ is dense in the subset of $C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ of diffeomorphisms fixing $0$?

Let $\exp$ denote the matrix exponential map and let $X \in C(\mathbb{R}^d,\mathrm{Mat}_{d\times d}(\mathbb{R}))$ denote the set of all continuous injective maps. Let $Y\subset C(\mathbb{R}^d,\mathbb{R}^d)$ denote the set of all maps of the form $$ f(x)= \exp(g(x))x ,\quad g \in X. $$ Is $Y$ a studied object? Is it the collection dense in the set of all diffeomorphisms of $\mathbb{R}^d$ fixing the origin? For the compact-convergence topology?

Let $\exp$ denote the matrix exponential map and let $X \in C^{\infty}(\mathbb{R}^d,\mathrm{Mat}_{d\times d}(\mathbb{R}))$ denote the set of all continuous injective maps. Let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ denote the set of all maps of the form $$ f(x)= \exp(g(x))x ,\quad g \in X. $$ Is $Y$ a studied object? Is it the collection dense in the set of all diffeomorphisms of $\mathbb{R}^d$ fixing the origin? For the compact-convergence topology?