The cycle structure of a Markoff automorphism over finite fields

We begin an investigation of the action of pseudo-Anosov elements of $\mathrm{Out}(\mathbf{F}_{2})$ on the Markoff-type varieties \[ \mathbb{X}_{\kappa}:\:x^{2}+y^{2}+z^{2}=xyz+2+\kappa \] over finite fields $\mathbb{F}_{p}$ with $p$ prime. We first make a precise conjecture about the permutation group generated by $\mathrm{Out}(\mathbf{F}_{2})$ on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov $g\in\mathrm{Out}(\mathbf{F}_{2})$, there is always an orbit of $g$ of length $\geq C\log p+O(1)$ on $\mathbb{X}_{\kappa}(\mathbb{F}_{p})$ where $C>0$ is given in terms of the eigenvalues of $g$ viewed as an element of $\mathrm{GL}_{2}(\mathbf{Z})$. This improves on a result of Silverman (2007) that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic $(p\to\infty)$ behavior of the longest orbit of a fixed pseudo-Anosov $g$ acting on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture.