The pentagram zoo and From positive current to positive measure

For the next few meetings of the Algebraic Dynamics seminar, we will have two talks instead of one.  These are mostly informal, with several of them describing work in progress.

Max Weinreich (Harvard)

The pentagram zoo

Abstract: Schwartz’s pentagram map is an algebraic dynamical system defined on moduli spaces of polygons by intersecting diagonals. It is an integrable system, meaning that in appropriate coordinates, the map becomes a family of translations on complex tori. Some natural generalizations of the pentagram map produce integrable systems, but numerical experiments by Khesin-Soloviev suggest that others do not. In this talk, we use tools from algebraic dynamics to prove that the skew pentagram map is non-integrable.

Laura DeMarco (Harvard)

From positive current to positive measure

Abstract: This is work in progress, and I’ll focus on a technical point. In studying holomorphic families of maps on P^1 or P^N, bifurcation currents detect and quantify stability properties. For a particular class of maps on P^2, Astorg and Bianchi showed a few years ago that the support of a bifurcation current can coincide with the support of its higher wedge powers. I will dig into their proof and show how it can be applied in another setting. Consequences of interest include control of the geometry of periodic points in families.