Decimation limits of algebraic actions

Douglas Lind (University of Washington)

16-Oct-2020, 16:15-17:30 (3 years ago)

Abstract: This is intended to be an expository talk using simple examples to illustrate what’s going on, and so will (hopefully) be a gentle introduction to these topics. Given a polynomial in d commuting variables we can define an algebraic action of ℤ^d by commuting automorphisms of a compact subgroup of 𝕋^(ℤ^d). Restricting the coordinates of points in this group to finite-index subgroups of ℤ^d gives other algebraic actions, defined by polynomials whose support grows polynomially and whose coefficients grow exponentially. But by “renormalizing” we can obtain a limiting object that is a concave function on ℝ^d with interesting properties, e.g. its maximum value is the entropy of the action. For some polynomials this function also arises in statistical mechanics models as the “surface tension” of a random surface via a variational 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 continuous in the coefficients of the polynomial. Refinements of convergence questions involve diophantine issues that I will discuss, together with some open problems.

dynamical systemsnumber theory

Audience: general audience


New England Dynamics and Number Theory Seminar

Organizers: Dmitry Kleinbock, Han Li*, Lam Pham, Felipe Ramirez
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