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 of all diffeomorphisms of $\mathbb{R}^d$ fixing the origin?