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Unlikely intersections problem for automorphisms of Markov surfaces
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Unlikely intersections problem for automorphisms of Markov surfaces
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We study the problem of unlikely intersections for automorphisms of Markov surfaces of positive entropy. We show for certain parameters that two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate. Our proof uses arithmetic equidistribution for adelic line bundles over quasiprojective varieties, the theory of laminar currents and quasi-Fuchsian representation theory.
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Open surfaces with a triangle at infinity
Triangle surfaces are exactly the Markov-type cubics xyz=x^2+y^2+z^2+ax+by+cz+d, and their automorphism groups are Gσ⋊Γ with Γ one of five finite groups from Table 1.
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