# Group of automorphisms of the disk fixing the boundary

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.

• The three dimensional case is interesting. en.wikipedia.org/wiki/Smale_conjecture – Cheerful Parsnip Feb 25 at 0:38
• Here are some notes from a course on diffeomorphisms of disks. – JHF Feb 25 at 3:21
• Aren't the continuous functions $\overline{\mathbb{D}^n} \to \mathbb{R}^n$ vanishing on $\partial \mathbb{D}^n$ the Lie algebra of the group $G$ of diffeomorphisms $\overline{\mathbb{D}^n} \to \overline{\mathbb{D}^n}$ fixing $\partial \mathbb{D}^n$ ? (the exponential $Lie(G) \to G$ is $v \mapsto f(1)$ where $f(0)(x)=x, f'(t)(x)=v(x)$) – reuns Feb 25 at 5:26