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

Abstract

Let $X_D^{\ast }$ be the non-singuar locus of the Markoff surface $X_D:x^2+y^2+z^2=xyz+D$ 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.