Smale Space Research Articles

On the K-theory of the stable C ⁎-algebras from substitution tilings

We describe a method to compute the K-theory of the C ⁎-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group K 0 is always torsion free and give an example of a two-dimensional tiling such that K 0 has torsion.

The homoclinic and heteroclinic C*-algebras of a generalized one-dimensional solenoid

AbstractD. Ruelle and I. Putnam have constructed three C*-algebras from the homoclinic and heteroclinic structure of a Smale space. This paper gives gives a complete description of these algebras when the Smale space is one of the generalized one-dimensional solenoids studied by R. Williams and I. Yi.

Optimal Lyapunov metrics of expansive homeomorphisms

We sharpen the following results of Reddy, Sakai and Fried: any expansive homeomorphism of a metrizable compactum admits a Lyapunov metric compatible with the topology, and if we also assume the existence of a local product structure (that is, if the homeomorphism is an A-homeomorphism in the terminology of Alekseev and Yakobson, or possesses hyperbolic canonical coordinates in the terminology of Bowen, or together with the metric compactum constitutes a Smale space in the terminology by Ruelle), then we also obtain the validity of Ruelle's technical axiom on the Lipschitz property of the homeomorphism, its inverse, and the local product structure. It is shown that any expansive homeomorphism admits a Lyapunov metric such that the homeomorphism on local stable (resp. unstable) “manifolds” is approximately representable on a small scale as a contraction (resp. expansion) with constant coefficient (resp. ) in this metric. For A-homeomorphisms, we prove that the desired metric can be approximately represented on a small scale as the direct sum of metrics corresponding to the canonical coordinates determined by the local product structure and that local “manifolds” are “flat” in some sense. It is also proved that the lower bounds for the contraction constants and expansion constants of A-homeomorphisms are attained simultaneously for some metric that satisfies all the conditions described.

SELF-SIMILAR INVERSE SEMIGROUPS AND SMALE SPACES

Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.

Lifting factor maps to resolving maps

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 Smale's 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 Bowen's 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.

Large deviations in Axiom A endomorphisms

By applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

A note on the ideals of groupoid C*-algebras from Smale spaces

In this note, we characterise completely the ideals of the groupoid C*-algebra arising from the asymptotic equivalence relation on the points of a Smale space and show that the related Ruelle algebra is simple when the Smale space is topologically transitive.

Functoriality of the C *-Algebras Associated with Hyperbolic Dynamical Systems

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.

The Structure ofC*-Algebras Associated with Hyperbolic Dynamical Systems

We consider the stable, unstableC*-algebras and the Ruelle algebras associated to a mixing Smale space. In the case of a shift of finite type, these are the AF-algebras studied by W. Krieger and the (stabilized) Cuntz–Krieger algebras. In the general case, we show that the stable and unstable algebras are simple and amenable. We also show that the Ruelle algebras are simple, amenable, and purely infinite.

Shadowing, expansiveness and hyperbolic homeomorphisms

AbstractThe purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mañé in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.