We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

We realize the Hecke C*-algebra $\mathcal{C}_{\mathbb{Q}}$ of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by $\mathbb{N}^F$, for F a finite set of primes. For each approximating Hecke C*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C*-algebras. We can describe the simple summands as ordinary crossed products by actions of $\mathbb{Z}^S$ for S a finite set of primes. When $\vert S\vert =1$, these actions are odometers and the crossed products are Bunce–Deddens algebras; when $\vert S\vert >1$, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras.

The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*-algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of s-resolving, finite-to-one factor maps, then the construction of the stable C*-algebra is contravariant, while that of the unstable C*-algebra is covariant. The paper also discusses the constructions of these C*-algebras for Smale spaces which are not mixing.

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

We consider the C*-algebras constructed from certain hyperbolic dynamical systems. The construction, due to Ruelle, generalizes the C*-algebras of Cuntz and Krieger. We discuss relations between the C*-algebras, show the existence of natural asymptotically abelian systems and investigate the K-theory and E-theory of these C*-algebras.

We consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.

The C*-algebras associated with irrational rotations of the circle were classified up to strong Morita equivalence by M. A. Rieffel. As a corollary, he gave a complete classification of the C*-algebras arising from irrational or Kronecker flows on the 2-torus up to *-isomorphism. Here, we extend the result to the socalled Denjoy homeomorphisms. Specifically, we give a necessary and sufficient condition for the strong Morita equivalence of two C*-algebras arising from homeomorphisms of the circle without periodic points. As a corollary, we show that two C*-algebras arising from flows on the 2-torus obtained from such homeomorphisms by the “flow under constant function” construction are *-isomorphic if and only if the flows themselves are topologically conjugate.