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?
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$\begingroup$ Is it just me or why is it cleat that Y is even a subset of the diffeomorphisms? On one hand the matrix exponential is non-injective and having thought a while about this I do not see why the resulting mapping will be injective in general. $\endgroup$– Alexander SchmedingCommented Apr 21, 2020 at 16:18
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$\begingroup$ I made some edits. Basically I want to approximate all diffeomorphisms using a set of smooth modified matrix functions. $\endgroup$– ABIMCommented Apr 22, 2020 at 9:41
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$\begingroup$ @New_Topologist_On_The_Block If $f : \mathbb{R}^d \to \mathbb{R}^d$ is such a map, is it obvious that the derivative $Df : \mathbb{R}^d \to \mathfrak{gl}_d(\mathbb{R})$ is pointwise invertible, so that $f$ is at least everywhere a local diffeomorphism? $\endgroup$– Branimir ĆaćićCommented Apr 22, 2020 at 15:43
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