Differentiable conjugacy for groups of area-preserving circle diffeomorphisms

Abstract:

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb {S}^1\times \mathbb {S}^1\setminus \Delta$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm {PSL}(2,\mathbb {R})$, then discuss the existence of a differentiable conjugacy.

For some groups, we find that this conjugacy is automatically differentiable. These rigidity results can be seen as particular cases of theorems of Herman (for circle diffeomorphisms conjugate to rotations) and Ghys (for actions of surface groups), with much simpler proofs.

For other groups (typically deformations in $\mathrm {Diff}(\mathbb {S}^1)$ of Schottky groups in $\rm {PSL}(2,\mathbb {R})$), we show that there is much more flexibility and that a differentiable conjugacy does not always exist.