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A Survey of Algebraic Actions of the Discrete Heisenberg Group (with Klaus Schmidt)
Abstract: The study of actions of countable groups by automorphisms of compact abelian groups has recently undergone intensive development, revealing deep connections with operator algebras and other areas. The discrete Heisenberg group is the simplest noncommutative example, where dynamical phenomena related to its noncommutativity already illustrate many of these connections. The explicit structure of this group means that these phenomena have concrete descriptions, which are not only instances of the general theory but are also testing grounds for further work. We survey here what is known about such actions of the discrete Heisenberg group, providing numerous examples and emphasizing many of the open problems that remain.

Homoclinic Points, Atoral Polynomials, and Periodic Points of Algebraic \(\mathbb{Z}^d\)actions (with Klaus Schmidt and Evgeny Verbitskiy)
Abstract: Cyclic algebraic \(\mathbb{Z}^d\)actions are defined by ideals of Laurent polynomials in \(d\) commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative \(d\)torus. For such expansive actions it is known that the limit for the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the \(d\)torus in a finite set. Here we further extend it to the case when the dimension of intersection of the variety with the \(d\)torus is at most \(d2\). The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

Entropy and Growth Rate of Periodic Points of Algebraic \(\mathbb{Z}^d\)actions (with Klaus Schmidt and Evgeny Verbitskiy)
Abstract: Expansive algebraic \(\mathbb{Z}^d\)actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.

Nonarchimedean Amoebas and Tropical Varieties (with Mikhail Kapranov and Manfred Einsiedler)
Abstract: We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the localglobal principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.

Lehmer’s Problem for Compact Abelian Groups
Abstract: We formulate Lehmer’s Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.