Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6

Abstract

We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces \begin{align*} S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\}, \end{align*} where $A,B,C,$ and $D$ are complex parameters. We focus on a finite index subgroup ${\Gamma }_{A,B,C,D}<\mathrm{Aut}(S_{A,B,C,D})$ whose action not only describes the dynamics of Painlevé 6 differential equations but also arises naturally in the context of character varieties. We define the Julia and Fatou sets of this group action and prove that there is a dense orbit in the Julia set. In order to show that the Julia set is ``large'' we consider a second dichotomy, between locally discrete and locally non-discrete dynamics. For an open set in parameter space, $\mathcal{N}\subset {\mathbb{C}}^4$, we show that there simultaneously exists an open set in $S_{A,B,C,D}$ on which ${\Gamma }_{A,B,C,D}$ acts locally discretely and a second open set in $S_{A,B,C,D}$ on which ${\Gamma }_{A,B,C,D}$ acts locally non-discretely. After removing a countable union of real-algebraic hypersurfaces from $\mathcal{N}$ we show that ${\Gamma }_{A,B,C,D}$ simultaneously exhibits a non-empty Fatou set and also a Julia set having non-trivial interior. The open set $\mathcal{N}$ contains a natural family of parameters previously studied by Dubrovin-Mazzocco. The interplay between the Fatou/Julia dichotomy and the locally discrete/non-discrete dichotomy plays a major theme in this paper and seems bound to play an important role in further dynamical studies of holomorphic automorphism groups.

52 pages. Several further changes have been made to improve the exposition following comments from the referee. To appear in the Indiana University Math Journal

Cited by in corpus (1)

• Birational Kleinian groups