Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows

TL;DR

This work extends the connection between modulo-$p$ transitivity and $p$-adically minimal dynamics for Markoff surfaces by developing a Bell–Poonen flow framework to include modulo $p^2$ reductions, enabling a localized, $p$-adic analysis of stabilizers. The authors decompose $X_D^*(\\mathbb{Z}_p)$ into reduction fibers, identify a level-1 polydisk, and construct group actions via stabilized powers of Vieta involutions using Chebyshev polynomials and companion matrices to achieve minimality on a subdisk, under conditions $D\\equiv 0\\pmod{p^2}$ or $(D-4)/p$ a quadratic residue. They provide explicit local parametrizations and expansions, establish residual transitivity and minimality on the chosen fiber, and discuss generalizations and the role of periodic orbits, including a $p=5$ special case. The results offer a new dynamical toolkit for Markoff-like surfaces over $p$-adic rings, linking modular transitivity to global $p$-adic minimality with potential broad applications in arithmetic dynamics.

Abstract

Let $X_D^\ast$ be the non-singuar locus of the Markoff surface $X_D\colon x^2+y^2+z^2=xyz+D$ and consider the set of its $p$-adic integer points $X_D^\ast(\mathbb{Z}_p)$. It is known to Bourgain, Gamburd, and Sarnak that the modulo $p$ transitivity by algebraic automorphisms of $X_0^\ast$ implies minimality of $X_0^\ast(\mathbb{Z}_p)$ by algebraic automorphisms. In this paper, we provide an alternative proof of this fact, by some techniques to study $p$-adic analytic flows. This establish a slight generalization to those parameters $D$ congruent to $0$ modulo $p^2$ or $(D-4)$ being a nonzero quadratic residue.

Theorems & Definitions (61)

• Theorem : Bourgain--Gamburd--Sarnak
• Theorem 1
• Theorem 2: Bell--Poonen
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof