Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlev\'e 6

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 $\Gamma_{A,B,C,D} < {\rm Aut}(S_{A,B,C,D})$ whose action not only describes the dynamics of Painlev'e 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 ${\mathrm{\Gamma }}_{A,B,C,D}$ acts locally discretely and a second open set in $S_{A,B,C,D}$ on which ${\mathrm{\Gamma }}_{A,B,C,D}$ acts locally non-discretely. After removing a countable union of real-algebraic hypersurfaces from $\mathcal{N}$ we show that ${\mathrm{\Gamma }}_{A,B,C,D}$ 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.