A general framework for investigating topological actions of Z d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lowerdimensional subspaces of R d . Here we completely describe this expansive behavior for the class of algebraic Z d -actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables.We introduce two notions of rank for topological Z d -actions, and for algebraic Z d -actions describe how they are related to each other and to Krull dimension. For a linear subspace of R d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component. Expansive subdynamics for algebraic actions 1697Definition 2.2. Let β be a Z d -action on (X, ρ) and F be a subset of R d . Then F is expansive for β, or β is expansive along F , if there are ε > 0 and t > 0 such thatIf F fails to meet this condition it is non-expansive for β, or β is non-expansive along F . Remark 2.3. Every subset of a non-expansive set for β is clearly also non-expansive for β. Every translate of an expansive set is expansive [BL, p. 57]. In the above definition we can take for ε a fixed expansive constant for β [BL, Lemma 2.3].Next we examine subsets F that are linear subspaces of R d . Let G k = G d,k denote the Grassmann manifold of k-dimensional subspaces (or simply k-planes) of R d . Recall that G k is a compact manifold of dimension k(d − k) whose topology is given by declaring two subspaces to be close if their intersections with the unit sphere are close in the Hausdorff metric. A k-plane and its (d − k)-dimensional orthogonal complement determine each other, giving a natural homeomorphism between G k and G d−k .An expansive component of k-planes for β is a connected component of E k (β).Example 2.5. (Ledrappier's example) Take d = 2,and let β be the Z 2 -action generated by the horizontal and vertical shifts. If L is a line that is not parallel to one of the sides of the unit simplex in R 2 and t ≥ 2, then for each x ∈ X the coordinates of x within L t determine all of x, so that L ∈ E 1 (β). On the other hand, the three lines parallel to the sides of the simplex do not have this property, and they comprise N 1 (β) (see [BL, Example 2.7] for details).

We present results and background rationale in support of a Pólya-Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of dynamical zeta functions of compact group automorphisms.

This paper investigates the problem of finding the possible sequences of periodic point counts for endomorphisms of solenoids. For an ergodic epimorphism of a solenoid, a closed formula is given that expresses the number of points of any given period in terms of sets of places of finitely many algebraic number fields and distinguished elements of those fields. The result extends to more general epimorphisms of compact abelian groups.

An algebraic Z d -action of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.

Abstract. There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbitcounting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.