Ian F. Putnam

Topological invariants for substitution tilings and their associated

A color organ device for producing a visual display of lights responsive to electrical signals produced on respective channels of a four-channel stereo. The device includes a housing having a plurality of banks of lights therein which is rotated by an electric motor. A interpreter is provided for selectively separating the frequencies produced by the four-channel stereo for selectively energizing the banks of lights. Slip-rings are provided for making electrical contact between respective frequency discriminating circuits and the banks of lights.

C^\ast

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC*-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

-algebras

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤd-actions.

Full groups of Cantor minimal systems

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.

Orbit equivalence for Cantor minimal ℤd-systems

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤ d -actions.

On the topological stable rank of certain transformation group C *-algebras

Abstract:We will study the stable and unstable Ruelle algebras associated to a hyperbolic homeomorphism of a compact space. To do this, we will describe a notion of K-theoretic duality for -algebras which generalizes Spanier-Whitehead duality in topology. A criterion for checking that it holds is presented. As an application it is shown that the Ruelle algebras which are associated to subshifts of finite type, (and agree with Cuntz-Krieger algebras in this case) satisfy this criterion and hence are dual.

Orbit equivalence for Cantor minimal

The author develops a homology theory for Smale spaces, which include the basics sets for an Axiom A diffeomorphism. It is based on two ingredients. The first is an improved version of Bowens result that every such system is the image of a shift of finite type under a finite-to-one factor map. The second is Kriegers dimension group invariant for shifts of finite type. He proves a Lefschetz formula which relates the number of periodic points of the system for a given period to trace data from the action of the dynamics on the homology groups. The existence of such a theory was proposed by Bowen in the 1970s.

\mathbb{Z}^{2}

We consider Smale spaces, that is, homeomorphisms of a compact metric spaces possessing canonical coordinates of contracting (stable) and expanding (unstable) directions. Examples of such dynamical systems include the basic sets for Smales Axiom A systems. We also assume that each point of the space is non-wandering and that there is a dense orbit. We show that any almost one-to-one factor map between two such systems may be lifted in a certain sense to a factor map which is injective on the local stable sets (i.e., s-resolving). We derive several corollaries. One is a refinement of Bowens result that every irreducible Smale space is a factor of an irreducible shift of finite type by an almost one-to-one factor map. We are able to show that there exists such a factor which is the composition of an s-resolving map and a u-resolving map.

-systems

Abstract We study the C ∗ -algebras GJ (X, R ) are I (X, R ) of singular integral operators and Toeplitz operators (respectively) associated with a strictly ergodic flow (X, R ). We show that the commutator ideals of these algebras, CGJ (X, R ) and CL (X, R ), are simple and are closely related to the transformation group C ∗ -algebra, C ∗ (X, R ). We calculate the K -theory of GJ (X, R ), L (X, R ) and their commutator ideals. The main results of this calculation, Corollary 3.8.4 and Theorem 4.1.1, assert that C ∗ (X, R ) is contained in CGJ (X, R ) and if j denotes the inclusion map, then j ∗ : K 0 (C ∗ (X,R)) → K 0 (CGJ(X,R)) is an order isomorphism and there is a short exact sequence 0 → K 1 (C ∗ (R)) → i i K 1 C ∗ (R)) → j i K 1 (CGJ(X,R)) → 0 where i is the canonical imbedding of C ∗ ( R ) into C ∗ (X R ). We show also that, up to a change of scale, there is a unique trace on each of the commutator ideals. The key ingredient of our analysis is Theorem 3.1.1 which asserts a bijective correspondence between Silov representations of the algebra of analytic functions on the flow and C ∗ -representations of GJ (X, R ) and L (X, R ). This simultaneously generalizes Coburns theorem on the uniqueness of the C ∗ -algebra generated by an isometry and Douglass theorem on the uniqueness of the C ∗ -algebra generated by an isometric representation of a dense subsemigroup of R + .

K-Theoretic Duality for Shifts of Finite Type

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.