Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms: $$\phi : (D^2,S^1) \to (\Delta,\gamma)$$ and $$\psi : (\mathbb{CP}^1 - \text{Int}\,\Delta,\gamma) \to (D^2,S^1)\,,$$ where $\text{Int}$ means interior and $D^2 \subset \mathbb C$ is the closed unit disk. We have the induced maps $$\alpha:= \phi|_{S^1}\,,\quad \beta:=\psi|_\gamma\,.$$ Let $$\Gamma(\gamma):=\overline \beta \circ \alpha : S^1 \to S^1$$ where the bar denotes complex conjugation. This $\Gamma(\gamma)$ is an orientation-preserving analytic diffeomorphism of $S^1$. Moreover, $\alpha$ and $\beta$ are well-defined up to precomposition and postcomposition with elements of the biholomorphism group $G$ of $(D^2,S^1)$, restricted to $S^1$, respectively, which implies that $\Gamma(\gamma)$ is also well-defined up to pre- and postcomposition with elements of $G$, since conjugation can be moved inside an automorphism of $D^2$ by changing the automorphism. Therefore the double coset of $\Gamma(\gamma)$ in the analytic diffeomorphism group of $S^1$ by $G$ is well-defined.

For instance if $\gamma = S^1$, this is the double coset of the identity. Probably this can also be computed for ellipses in terms of elliptic functions.

**Question:** *What can be said about this double coset in general? Can the curve $\gamma$ be reconstructed from it? What are its dynamical properties?*

I have to admit that I'm asking this out of sheer curiosity, although this does tie with a line of research I tried not so long ago.