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

Appears in collection : 2024 - T2 - WS2 - Group actions with hyperbolicity and measure rigidity

I will describe the dynamics by the group of holomorphic automorphisms of the affine cubic surfaces $$S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 : \textrm{ } x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D},$$ where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlevè 6 Equation as well as the natural dynamics of the mapping class group on the $\mathrm{SL}(2,\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere. For these reasons it has been studied from many perspectives by many people including Bowditch, Goldman, Cantat-Loray, Cantat, Tan-Wong-Zhang, Maloni-Palesi-Tan, and many others.

In this talk I will describe my recent joint with Julio Rebelo and I will focus on several interesting open questions that arose while preparing our work "Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6" and during informal discussions with many people.

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

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