# Entropy and periodic points of algebraic actions of discrete amenable groups

What Mathematisches Kolloquium 2011-06-15 16:15 2011-06-15 17:00 2011-06-15 from 16:15 to 17:00 C209 Rindler, Hauser vCal iCal

Klaus Schmidt

Abstract. If G is a discrete amenable group, every element f in the integral group ring ZG defines a measure-preserving a ction alpha_f of G by automorphisms of a compact abelian group X_f. This action is automatically ergodic whenever G is no t essentially equal to the group of integers.

The entropy of this 'algebraic' action alpha_f is an interesting quantity associated with the element f in ZG. If G=Z^d, then it is the logarithmic Mahler measure of f (viewed as a Laurent polynomial in d variables with integer coefficients).

If G is residually finite and the action alpha_f is expansive, it is the logarithmic growth rate of the number of periodic points of alpha_f, which in turn coincides with the logarithm of the Fuglede-Kadison determinant associated with f (viewed as an element of the group von Neumann algebra of G).

If alpha_f is nonexpansive, the connection between entropy, the logarithmic growth rate of periodic points, and Fuglede-Kadison determinants holds many mysteries, even for G=Z^d. Some of these problems will be discussed in this lecture.

This talk is based on joint work with C. Deninger, D. Lind and E. Verbitskiy.

Von 15:45 bis 16:15 Uhr, Kaffeejause im Common Room.