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The cycle structure of a Markoff automorphism over finite fields. (English) Zbl 1459.37096

Summary: We begin an investigation of the action of pseudo-Anosov elements of \(\text{Out}(\mathbf{F}_2)\) on the Markoff-type varieties \[\mathbb{X}_\kappa : x^2 + y^2 + z^2 = x y z + 2 + \kappa\] over finite fields \(\mathbb{F}_p\) with \(p\) prime. We first make a precise conjecture about the permutation group generated by \(\text{Out}( \mathbf{F}_2)\) 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 \(g \in \text{Out}(\mathbf{F}_2)\), 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 \(\text{GL}_2(\mathbf{Z})\). This improves on a result of J. H. Silverman from [New York J. Math. 14, 601–616 (2008; Zbl 1153.11028)] 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 P. Sarnak [Clay Math. Proc. 7, 217–237 (2007; Zbl 1198.11039)]. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture in Conjecture 1.10.

MSC:

37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P35 Arithmetic properties of periodic points
11D25 Cubic and quartic Diophantine equations
11G35 Varieties over global fields
11F06 Structure of modular groups and generalizations; arithmetic groups
20B27 Infinite automorphism groups
11G20 Curves over finite and local fields
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