Geometry of periodic points, in families

TIFR International Colloquium 2024

Laura DeMarco (Harvard University)

We will look at the (algebraic) geometry of periodic and preperiodic points in families of maps on P^N and present a conjectural characterization of subvarieties containing many of them. (More precisely, if the family is parameterized by a complex algebraic variety S, we aim to describe subvarieties of S x P^N that arise as the Zariski-closure of some infinite subset of preperiodic points.) The characterization will be formulated in terms of a dynamically-defined current in S x P^N. This is work in progress with Myrto Mavraki, inspired by recent theorems of Gao-Habegger and others on families of abelian varieties. In this dynamical context, there are interesting connections to the theory of J-stability in families and questions about parameter~spaces.