Questions about the dynamics on a natural family of affine cubic surfaces

TL;DR

This paper explores open questions about the dynamics of automorphism groups acting on a family of affine cubic surfaces, linking to Painlevé 6 equations and character varieties in complex dynamics.

Contribution

It introduces several new questions about the dynamics of automorphism groups on affine cubic surfaces related to Painlevé 6 and character varieties, inspired by ongoing research and discussions.

Findings

Formulated open questions on automorphism group dynamics.

Connected dynamics to Painlevé 6 monodromy and character varieties.

Stimulated further research directions in complex dynamics.

Abstract

We present several questions about the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces $$S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\},$$ where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlev\'e 6 Equation as well as the natural dynamics of the mapping class group on the ${\rm SL}(2,\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere. The questions presented here arose while preparing our work ``Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlev\'e~6'' \cite{RR} and during informal discussions with many people. Several of the questions were posed at the Simons Symposium on Algebraic, Complex and Arithmetic Dynamics that was held at Schloss Elmau, Germany, in August 2022 as well as the MINT Summer School ``Facets of Complex Dynamics'' that was held in Toulouse, France, in June 2023.