Tiling Dynamical Systems as an Introduction to Smale Spaces Michael Whittaker (University of Wollongong) University of Otago Dunedin, New Zealand February 15, 2011
A Penrose Tiling
Sir Roger Penrose
Penrose Toilet Paper Kleenex used the Penrose tiling to prevent layers of toilet paper from sticking together. Penrose was not amused. So often we read of very large companies riding rough-shod over small businesses or individuals, but when it comes to the population of Great Britain being invited by a multi-national to wipe their bottoms on what appears to be the work of a Knight of the Realm without his permission, then a last stand must be made. - David Bradley, director of Pentaplex (the company that cares for Penrose s copyrights)
A Pinwheel Tiling (Conway-Radin)
Federation Square, Melbourne
Tilings Definition A tiling T of R d is a countable set {t 1, t 2,... } of closed subsets of R d, called tiles such that t i t j has empty interior whenever i j and i=1t i = R 2. Furthermore, any finite subset of tiles is called a patch. For x in R d and R > 0, let B(x, R) denote the ball of radius R centred at x. We define the patch T B(x, R) = {t T t B(x, R)}. If T is a tiling and x in R d, the tiling T + x is formed by translating every tile in T by x. The space of all translations of a tiling T is given by T + R d = {T + x x R d }.
Tiling Metric There is a metric on T + R d. We say that T 1 and T 2 are close if 1. T 2 is a small translate of T 1, 2. T 2 agrees with T 1 on a large ball around the origin and can disagree elsewhere. The Tiling Metric The distance between tilings T 1 and T 2 is less than ε if there are vectors x 1 and x 2 with norm less than ε such that T 1 x 1 B(0, ε 1 ) = T 2 x 2 B(0, ε 1 ). The distance d(t 1, T 2 ) is defined as the infimum over the set consisting of each ε satisfying the above and 1 is no such ε exists.
Continuous Hull Definition Given a tiling T, we let Ω T denote the completion of T + R d in the tiling metric. We call Ω T the continuous hull. Remarks Elements of Ω T are tilings. R d acts continuously on Ω T. Definition A tiling T is said to have Finite Local Complexity (FLC) if for every R > 0, the set {T B(x, R) x R d }/R d is finite.
Properties of tilings Theorem (Radin-Wolff) T has finite local complexity if and only if Ω T is a compact metric space. Definition A tiling T is aperiodic if T + x T for any non-zero vector x in R d. Moreover, T is said to be strongly aperiodic if Ω T contains no periodic tilings.
Examples of Hulls Periodic 1-d Tiling Ω T T t t t t t t t t t Aperiodic 1-d Tiling Ω T T t t t t s t t t t
Examples of Hulls Periodic, 2-d Tiling Placement of the origin in any square determines the tiling. T = T x. Thus, Ω T = T 2.
Substitution Rules Definition A set P = {p 1, p 2,..., p N } is called a set of prototiles for T if every tile t in T is a translate of some p in P. Definition A substitution rule on a set of prototiles P consists of A scaling constant λ > 1 A rule ω such that, for each p P, ω(p) is a patch of tiles {t 1,, t k } whose support is λp, each tile in ω(p) is a translate of a prototile p in P, and t i t j has empty interior whenever i j. ω can be applied to tilings by applying it to each tile. ω can be iterated.
Example: Penrose Tiling ω + all rotations by π/5 of the above prototiles.
Example: Pinwheel Tiling (Conway-Radin)
Example: Fractal Pinwheel Tiling (Priebe Frank-W)
A tiling dynamical system The continuous hull Ω T is a compact metric space if and only if T has finite local complexity (Radin-Wolff). The substitution ω is a homeomorphism on Ω T if T is strongly aperiodic (Solomyak). Therefore, (Ω T, ω) is a dynamical system. Proposition (Anderson-Putnam) The dynamical system (Ω T, ω) is a Smale space if T is strongly aperiodic and has finite local complexity. We will sketch the proof of this once we have defined Smale spaces.
Introduction to Smale spaces A Smale space is a dynamical system (X, ϕ), where X is a compact metric space and ϕ is a homeomorphism, such that each point in X is locally a product space with canonical expanding and contracting directions. Smale spaces were introduced by David Ruelle and include: Shifts of finite type, Hyperbolic toral automorphisms, Solenoids (Bob Williams), Dynamical systems associated with certain substitution tiling spaces (Anderson and Putnam), the basic sets of Smale s Axiom A systems (Ruelle)
Example: Hyperbolic Toral Automorphisms Let A be the matrix A = ( 1 1 1 0 ). Consider A : T 2 T 2 as a map on the quotient space T 2 = R 2 /Z 2. (T 2, A) is a Smale space: The eigenvalues of A are γ and γ 1, where γ is the golden ratio. The associated eigenvectors define the local hyperbolic structure on (T 2, A) as per the picture on the following page.
(0, 1) (1, 1) X s (x, ε) X u (x, ε) x (0, 0) (1, 0)
Heuristic definition of a Smale space Suppose X is a compact metric space with a homeomorphism ϕ : X X. A dynamical system (X, ϕ) is a Smale space if there are constants ε X > 0, λ > 1 such that for x X and 0 < ε < ε X : 1. There are two open sets X s (x, ε) and X u (x, ε) such that X s (x, ε) X u (x, ε) = {x}, 2. The product X s (x, ε) X u (x, ε) is homeomorphic to a neighbourhood of x. 3. X s (x, ε) is called a local stable set and satisfies ϕ(x s (x, ε)) X s (ϕ(x), λ 1 ε), 4. X u (x, ε) is called a local unstable set and satisfies ϕ 1 (X u (x, ε)) X u (ϕ 1 (x), λ 1 ε).
Tilings as Smale spaces Consider the dynamical system (Ω T, ω) along with the expansive constant λ > 1. Set ε > 0 and T 0 in Ω T, we claim that the local stable and unstable sets are defined as follows: X s (T 0, ε) = {T Ω T T B(0, ε 1 ) = T 0 B(0, ε 1 )} X u (T 0, ε) = {T Ω T T = T 0 + x such that x < ε}. Let us show these sets satisfy conditions 1-4: 1. X s (T 0, ε) X u (T 0, ε) = T 0 since any T in the intersection must agree with T 0 on a large ball about the origin as well as be a small translation of T 0. 2. The product X s (T 0, ε) X u (T 0, ε) agrees precisely with an ε-ball in the tiling metric.
Tilings as Smale spaces 3. ω(x s (T 0, ε)) = {T T B(0, λε 1 ) = T 0 B(0, λε 1 )}. 4. ω 1 (X u (T 0, ε)) = {T T = ω 1 (T 0 ) x, x < λ 1 ε}. This shows that (Ω T, ω) is a Smale space. Notice that: X s (T 0, ε) is homeomorphic to a Cantor set. In fact, for 0 < δ < ε, a cylinder set is given by T 0 B(0, δ). X u (T 0, ε) is homeomorphic to a disk, since the sets consists of all translations less than ε.
Another Example: shifts of finite type Suppose G is a directed graph and define X G = { e 2 e 1.e 0 e 1 e 2 i e i to e i+1 is allowable}; that is, X G consists of all possible bi-infinite paths in the graph G. Metric on X G : d(e, f ) = inf{2 n e i = f i for all i < n}.
Homeomorphism on X G : σ( e 2 e 1.e 0 e 1 e 2 ) = e 2 e 1 e 0.e 1 e 2 The dynamical system (X G, σ) is a Smale space: Fix a point e X G and let 0 < ε < 1 2, the local stable and unstable sets are X s (e, ε XG ) = {f X G e i = f i for all i 0}, X u (e, ε XG ) = {f X G e i = f i for all i 0}.
C -algebras associated with a Smale space David Ruelle defined two C -algebras associated with a Smale space, called the stable and unstable algebras, denoted S(X, ϕ) and U(X, ϕ). The homeomorphism ϕ gives rise to integer actions on S(X, ϕ) and U(X, ϕ) and the crossed products are known as the Ruelle algebras: R s (X, ϕ) := S(X, ϕ) Z and R u (X, ϕ) := U(X, ϕ) Z.
Where do I fit in? Theorem (Kaminker, Putnam, W) The Ruelle algebras R s (X, ϕ) and R u (X, ϕ) are Poincaré dual. Corollary The K-theory of R s (X, ϕ) is isomorphic to the K-homology of R u (X, ϕ) and the K-theory of R u (X, ϕ) is isomorphic to the K-homology of R s (X, ϕ). Theorem (W) There are summable spectral triples on both S(X, ϕ) and U(X, ϕ). Moreover, the spectral dimension of each of these spectral triples recovers the topological entropy of the original Smale space.
Open problems Problem 1 Describe R s (X, ϕ) and R u (X, ϕ) for the Smale space associated with a substitution tiling. Problem 2 Give an explicit description of the K-theory of S(X, ϕ) and U(X, ϕ) for the Smale space associated with a substitution tiling. Note: Anderson and Putnam described the K-theory using cohomology. Problem 3 Use Poincaré duality to describe the K-homology of R s (X, ϕ) and R u (X, ϕ) geometrically. Problem 4 Poincaré duality of R s (X, ϕ) and R u (X, ϕ) is circle equivariant. Using Takai duality, does this give Poincaré duality on the stable and unstable algebras S(X, ϕ) and U(X, ϕ)? Moreover, what is the circle equivariant K-theory of these algebras?
References: J. Anderson and I.F. Putnam, Topological invariants for substitution tilings and their C*-algebras, Ergodic Th. and Dynam. Sys. 18 (1998), 509-537. J. Kaminker, I.F. Putnam, and M.F. Whittaker K-Theoretic Duality for Hyperbolic Dynamical Systems, Math. ArXiv: 1009.4999 (2010), 1-36. I.F. Putnam, C -Algebras from Smale Spaces, Canad. J. Math. 48 (1996), 175-195. D. Ruelle, Noncommutative Algebras for Hyperbolic Diffeomorphisms, Invent. Math. 93 (1988), 1-13. S. Smale, Differentiable Dynamical Systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. M.F. Whittaker, Spectral Triples for Hyperbolic Dynamical Systems, Pre-print: arxiv 1011.3292, 2010.