The Trouv\'e group for spaces of test functions

The Trouv\'e group $\mathcal G_{\mathcal A}$ from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity $\mathcal A(\mathbb R^d,\mathbb R^d)$. For a multitude of regularity classes $\mathcal A$, we prove that the Trouv\'e group $\mathcal G_{\mathcal A}$ coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of $\mathbb R^d$ which differ from the identity by a mapping of class $\mathcal A$. We thus conclude that $\mathcal G_{\mathcal A}$ has a natural regular Lie group structure. In many cases we show that the mapping which takes a time-dependent vector field to its flow is continuous. As a consequence we obtain that the scale of Bergman spaces on the polystrip with variable width is stable under solving ordinary differential equations.