Tropical Dynamics of Markov Surfaces

TL;DR

The paper develops a tropical framework for the Markov surface dynamics under Vieta involutions, revealing a sharp dichotomy: meromorphic parameters yield a hyperbolic (∞,∞,∞)-triangle group dynamics on the tropical skeleton, while holomorphic parameters induce a linear action on the plane modulo antipodes. It constructs the invariant skeleton Sk(a,b,c,d), proves tropical equivariance and a robust ping-pong structure, and uses this to connect tropical dynamics to non-archimedean Fatou domains and Diophantine questions about rational points with prime-power denominators. The approach blends tropical geometry with hyperbolic geometry to obtain global dynamical models, and applies these models to both Fatou-domain phenomena and arithmetic of Markov surfaces over non-archimedean fields. The work provides a detailed combinatorial and geometric bridge between tropicalizations, group actions, and arithmetic dynamics, with concrete consequences for p-adic and real points on Markov cubic families.

Abstract

We discuss the algebraic dynamics on Markov cubics generated by Vieta involutions, in the tropicalized setting. It turns out that there is an invariant subset of the tropicalized Markov cubic where the action by Vieta involutions can be modeled by that of $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane. This understanding of the tropicalized algebraic dynamics produces some results on Markov cubics over non-archimedean fields, including the existence of the Fatou domain and finitude of orbits with rational points having prime power denominators.

Figures (13)

• Figure 1: Tropicalization of $S_{000D}$, $\operatorname{val}(D)<0$
• Figure 3: Projection of $Sk(\infty,\infty,\infty,d)$ ($d<0$) to the Plane $\Pi$, with Cells and Fixed sets sketched
• Figure 4: Reflections of the Tripod
• Figure 5: More Reflections of the Tripod
• Figure 6: The $(\infty,\infty,\infty)$-triangle Tessellation of $\mathbb{H}^2$

Theorems & Definitions (128)

• Theorem A
• Theorem B: Fatou Sets over non-archimedean fields; see Theorem \ref{['thm:p-adic-fatou-domain']}
• Theorem C: $\mathbb{Z}[\frac{1}{p}]$-points in the Compact Component; see Theorem \ref{['thm:rational-points-compact-component']}
• Theorem 2.1: Kapranov
• Lemma 2.2: Nonemptiness
• Lemma 2.3: Adjacency
• proof
• Corollary 2.4
• Proposition 3.1
• proof