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

Julio Rebelo, Roland Roeder

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 Painlevé 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 Painlevé~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.

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

Abstract

We present several questions about the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces where and are complex parameters. This group action describes the monodromy of the famous Painlevé 6 Equation as well as the natural dynamics of the mapping class group on the 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 Painlevé~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.
Paper Structure (18 sections, 44 equations)

This paper contains 18 sections, 44 equations.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Invariant holomorphic two-form
  4. Fatou/Julia Dichotomy
  5. Locally Non-discrete/Locally Discrete Dichotomy
  6. Dynamics on the character variety of the one-holed torus.
  7. Picard Parameter $D=4$
  8. Markoff Parameter $D=0$
  9. Fatou/Julia decomposition of $S_0$
  10. Locally Non-discrete/Locally Discrete decomposition of $S_0$
  11. Arbitrary Parameters $(A,B,C,D)$
  12. Bowditch Set:
  13. Locally Non-Discrete Set
  14. Restating Questions 1-4
  15. Invariant Boundary
  16. ...and 3 more sections

Theorems & Definitions (4)

  • proof : Sketch of the proof
  • proof
  • proof : Proof of Proposition \ref{['PROP:DERIVINGSXSYSZ']}
  • proof