Mahler’s work and algebraic dynamical systems

Abstract

After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact Abelian group, further key examples motivated the development of an extensive theory of such actions.

Two of Mahler's achievements, the recognition of the significance of Mahler measure of multivariate polynomials in relating the lengths and heights of products of polynomials in terms of the corresponding quantities for the constituent factors, and his work on additive relations in fields, have unexpectedly played important roles in the study of entropy and higher order mixing for these actions.

This article briefly surveys these connections between Mahler's work and dynamics. It also sketches some of the dynamical outgrowths of his work that are very active today, including the investigation of the Fuglede-Kadison determinant of a convolution operator in a group von Neumann algebra as a noncommutative generalisation of Mahler measure, as well as Diophantine questions related to the growth rates of periodic points and their relation to entropy.