I would like to know everything you know (group structure, dense or interesting subsets etc) about the group of diffeomorphisms $$\psi: \mathbb{D}^n \to \mathbb{D}^n$$ that such that $\psi|_{\partial \mathbb{D}^n} = id|_{\partial \mathbb{D}^n}$ ($\mathbb{D}^n $ is the unit disk in $\mathbb{R}^n$).
The obvious observation is that there is an injection of $\Gamma_c(TB^n)$, the set of compactly supported vector fields of the disk. Indeed we can send such a vector field to it's flow at time $1$.
Moreover as $\psi $ extends to an automorphism of $\mathbb{R}^n$ it must be is smoothly isotopic to the identity or to a reflection.
This does not seem to imply that if $\psi$ preserves the orientation it is the flow a vector field at time $1$ though as the isotopy can be not a 1-parameter subgroup of diffeomorphisms.