The cycle structure of a Markoff automorphism over finite fields

Abstract

We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Markoff-type varietiesXκ:x2+y2+z2=xyz+2+κ over finite fields Fp with p prime. We first make a precise conjecture about the permutation group generated by Out(F2) on X−2(Fp) 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∈Out(F2), there is always an orbit of g of length ≥Clog⁡p+O(1) on Xκ(Fp) where C>0 is given in terms of the eigenvalues of g viewed as an element of GL2(Z). This improves on a result of Silverman from [25] that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic (p→∞) behavior of the longest orbit of a fixed pseudo-Anosov g acting on X−2(Fp) 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 [23]. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture in Conjecture 1.10.