EVENT DETAILS AND ABSTRACT

Title: Equidistribution and preperiodic points for families of rational maps
Speaker: Myrto Mavraki
Speaker Info: Northwestern University
Brief Description:
Special Note:
Abstract:

Motivated by a question of Zannier, it was shown by Baker and DeMarco that for any fixed complex numbers a and b and integer d ≥ 2, there are infinitely many t ∈ C such that both a and b are preperiodic under iteration by f_t(z) = z^d + t if and only if a^d = b^d. Their result fits into the so-called theme of `unlikely intersections' and has been generalized to other 1-parameter families f_t of rational maps by various authors. A key ingredient in the proofs is an arithmetic equidistribution theorem for small points with respect to an adelic measure, proved independently by Baker–Rumely and Favre–Rivera-Letelier. In this talk we show that many 1-parameter families of rational maps fail to satisfy a crucial hypothesis in the aforementioned equidistribution theorem. To rectify this, we generalize the notion of an adelic measure to that of a quasi-adelic measure and present an equidistribution theorem for quasi-adelic measures. We then connect our work back to questions arising in the theme of unlikely intersections and to an old question concerning the variation of the canonical height in families of rational maps. This is joint work with Hexi Ye.