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Presented By: Department of Mathematics

Complex Analysis, Dynamics and Geometry Seminar

Preperiodic points in families of rational maps, arithmetic equidistribution and a variation of heights

We discuss dynamical analogs of two seemingly unrelated questions from number theory. The first fits in the theme of `unlikely intersections' and is inspired by a theorem of Masser-Zannier concerning torsion points in families of elliptic curves. In the dynamical setting it concerns generalizations of a result by Baker and DeMarco asserting that there are at most finitely many parameters t for which two complex numbers a and b are preperiodic for f_t(z) = z^d + t unless a^d = b^d. The second question we discuss is a dynamical generalization of theorems of Tate and Silverman concerning the variation of heights in families of elliptic surfaces. We discuss our progress towards these generalizations and emphasize how the two questions are actually related via arithmetic equidistribution statements. This is joint with Laura DeMarco. Speaker(s): Myrto Mavraki (Northwestern/University of Basel)

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