Abstract. The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here non-Archimedean norms in function fields of positive characteristic are used to exhibit an asymptotic shape in non-mixing sequences for algebraic Z 2-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial. Using these methods, we show that an algebraic dynamical system for which any shape of cardinality three is mixing is mixing of order three, and for any k≥1exhibit examples that are k-fold mixing but not (k + 1)-fold mixing. 1. Introduction and

Complete treatments of the results stated in this appendix may be found in any measure theory book; see for example Parthasarathy [281], Royden [321] or Kingman and Taylor [195]. A similar summary of measure theory without proofs may be found in Walters [375, Chap. 0]. Some of this appendix will use terminology from Appendix B. A.1 Measure Spaces Let X be a set, which will usually be infinite, and denote by P(X) the collection of all subsets of X. Definition A.1. A set S ⊆ P(X) is called a semi-algebra if (1) ∅ ∈ S, (2) A, B ∈ S implies that A ∩ B ∈ S, and (3) if A ∈ S then the complement X�A is a finite union of pairwise disjoint elements in S; if in addition (4) A ∈ S implies that X�A ∈ S, then it is called an algebra. If S satisfies the additional property

Abstract not found

Abstract. We study mixing properties of algebraic actions of Q d, showing in particular that prime mixing Q d actions on connected groups are mixing of all orders, as is the case for Z d-actions. This is shown using a uniform result on the solution of S-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group Q ∗ are shown to behave quite differently, with finite order of mixing possible on connected groups. Mixing properties of Z d-actions by automorphisms of a compact metrizable abelian group are quite well understood. Roughly speaking, the picture has three facets. Firstly, the one-to-one correspondence between such actions and countably generated modules over the integral group ring Rd = Z[Z d] of the acting group Z d due to Kitchens and K. Schmidt [6] allows any mixing problem to be reduced to the case corresponding to a cyclic module of the form Rd/P for a prime ideal P ⊂ Rd. Secondly, in the connected case P ∩Z = {0}, K. Schmidt and Ward [13] showed that mixing implies mixing of all orders by relating the mixing property to S-unit equations and exploiting a deep result of Schlickewei on solutions of such equations [11] (see also [4] and [14]). Finally, in the totally disconnected case P ∩ Z = pZ for some rational prime p, Masser [9] has shown that the order of mixing is determined by the mixing behaviour of shapes, reducing the problem – in principle – to an algebraic one. Our purpose here is to show how some of this changes for algebraic actions of infinitely generated abelian groups. The algebra is more involved, so for simplicity we restrict attention to the simplest extreme examples: actions of Q ×>0 (isomorphic to the direct sum of countably many copies of Z) and actions of Q d (which is a torsion extension of Z d). These groups are the simplest non-trivial examples chosen from the ‘dual ’ categories of free abelian and infinitely divisible groups in the sense of MacLane [8]. The algebraic difficulties mean we cannot present the complete picture found for Z d-actions, and the emphasis is partly on revealing or suggestive examples. Some topological properties (expansiveness and closed invariant sets) for actions of infinitely generated abelian groups have been studied by Berend [1] and Miles [10].

2. R. L. Adler, M. Keane, and M. Smorodinsky, ‘A construction of a normal number

Abstract not found

Abstract: Actions of Z d by automorphisms of compact zero-dimensional groups exhibit a range of mixing behaviour. Schmidt introduced the notion of mixing shapes for these systems, and proved that non-mixing shapes can only arise non-trivially for actions on zero-dimensional groups. Masser has shown that the failure of higher-order mixing is always witnessed by nonmixing shapes. Here we show how valuations can be used to understand the (non-)mixing behaviour of a certain family of examples. The sharpest information arises for systems corresponding to tight polyhedra. 1.