Shift spaces are to symbolic dynamics what shapes like polygons and curves are to geometry. We begin by introducing these spaces, and describing a variety of examples to guide the reader's intuition. Later chapters will concentrate on special classes of shift spaces, much as geometry concentrates on triangles and circles. As the name might suggest, on each shift space there is a shift map from the space to itself. Together these form a "shift dynamical system." Our main focus will be on such dynamical systems, their interactions, and their applications.In addition to discussing shift spaces, this chapter also connects them with formal languages, gives several methods to construct new shift spaces from old, and introduces a type of mapping from one shift space to another called a sliding block code. In the last section, we introduce a special class of shift spaces and sliding block codes which are of interest in coding theory. §1.1. Full Shifts Information is often represented as a sequence of discrete symbols drawn from a fixed finite set. This book, for example, is really a very long sequence of letters, punctuation, and other symbols from the typographer's usual stock. A real number is described by the infinite sequence of symbols in its decimal expansion. Computers store data as sequences of 0's and 1's. Compact audio disks use blocks of 0's and 1's, representing signal samples, to digitally record Beethoven symphonies.In each of these examples, there is a finite set A of symbols which we will call the alphabet. Elements of A are also called letters, and they will typically be denoted by a, b, c, . . . , or sometimes by digits like 0, 1, 2, . . . , when this is more meaningful. Decimal expansions, for example, use the alphabet A = {0, 1, . . . , 9}.Although in real life sequences of symbols are finite, it is often extremely useful to treat long sequences as infinite in both directions (or bi-infinite).

Abstract. We study the non-archimedean 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 non-archimedean 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 local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.

SummaryWe compute the joint entropy of d commuting automorphisms of a compact metrizable group. Let R d = Z[u( 1 ..... uf 1] be the ring of Laurent polynomials in d commuting variables, and M be an Ra-module. Then the dual group X M of M is compact, and multiplication on M by each of the d variables corresponds to an action a M of 7/d by automorphisms of Xu. Every action of 7/d by automorphisms of a compact abelian group arises this way. Iffe R d, our main formula shows that the topological entropy of ~R,/ is given bywhere M(f) is the Mahler measure off. This reduces to the classical result for toral automorphisms via Jensen's formula. While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7((3)/4n 2 occur. If p is a non-principal prime ideal, we show h(CtR,ip ) = 0. Using an analogue of the Yuzvinskii-Thomas addition formula, we compute h(~u) for arbitrary Rd-modules M, and then the joint entropy for an action of 7/a on a (not necessarily abelian) compact group. Using a result of Boyd, we characterize those ~t M which have completely positive entropy in terms of the prime ideals associated to M, and show this condition implies that ct M is mixing of all orders. We also establish an analogue of Berg's theorem, proving that if ct M has finite entropy then Haar measure is the unique measure of maximal entropy if and only if a M has completely positive entropy. Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy.

ABSTRACT. Let (Xt,ot) be a shift of finite type, and G = aut(or) denote the group of homeomorphisms of Xt commuting with ctWe investigate the algebraic properties of the countable group G and the dynamics of its action on Xt and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of ot whose exact value is not known. We prove that, modulo a few points of low period, G acts transitively on the set of points with least or-period n. Using p-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of G on the dimension group of o~t is investigated. We show there are no proper infinite compact G-invariant sets. We give a complete characterization of the G-orbit closure of a continuous probability measure, and deduce that the only continuous G-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.