Let $\exp$ denote the matrix exponential map and let $X \in C(\mathbb{R}^d,mat_{d\times d})$ 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 $$ f(x)= \exp(g(x))x, g \in X. $$ Is $Y$ a studied object? Is it the collection of all diffeomorphisms of $\mathbb{R}^d$ fixing the origin?