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.

  • $\begingroup$ The three dimensional case is interesting. en.wikipedia.org/wiki/Smale_conjecture $\endgroup$ – Cheerful Parsnip Feb 25 at 0:38
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    $\begingroup$ Here are some notes from a course on diffeomorphisms of disks. $\endgroup$ – JHF Feb 25 at 3:21
  • $\begingroup$ 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 $f(0)(x)=x, f'(t)(x)=v(x)$) $\endgroup$ – reuns Feb 25 at 5:26

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