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?