BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Douglas Lind (University of Washington) DTSTART:20201016T161500Z DTEND:20201016T173000Z DTSTAMP:20260423T021427Z UID:NEDNT/5 DESCRIPTION:Title: De cimation limits of algebraic actions\nby Douglas Lind (University of W ashington) as part of New England Dynamics and Number Theory Seminar\n\nLe cture held in Online.\n\nAbstract\nThis is intended to be an expository ta lk using simple examples to illustrate what’s going on\, and so will (ho pefully) be a gentle introduction to these topics. Given a polynomial in d commuting variables we can define an algebraic action of ℤ^d by commuti ng automorphisms of a compact subgroup of 𝕋^(ℤ^d). Restricting the co ordinates of points in this group to finite-index subgroups of ℤ^d gives other algebraic actions\, defined by polynomials whose support grows poly nomially and whose coefficients grow exponentially. But by “renormalizin g” we can obtain a limiting object that is a concave function on ℝ^d w ith interesting properties\, e.g. its maximum value is the entropy of the action. For some polynomials this function also arises in statistical mech anics models as the “surface tension” of a random surface via a variat ional principle. In joint work with Arzhakova\, Schmidt\, and Verbitskiy\, we establish this limiting behavior\, and identify the limit in terms of the Legendre transform of the Ronkin function of the polynomial. The proof is based on Mahler’s estimates on polynomial coefficients using Mahler measure\, and an idea used by Boyd to prove that Mahler measure is continu ous in the coefficients of the polynomial. Refinements of convergence ques tions involve diophantine issues that I will discuss\, together with some open problems.\n LOCATION:https://researchseminars.org/talk/NEDNT/5/ END:VEVENT END:VCALENDAR