Fractal substitution tilings and applications tononcommutative geometry

Mike WhittakerUniversity of Wollongong

Kansas State University Colloquium

January 27, 2015

Plan

Part I. History and motivation for studying aperiodic tilings.

Part II. Aperiodic tilings and their properties.

Part III. Fractal realizations of substitution tilings (joint withNatalie Frank and Sam Webster).

Part IV. Spectral triples associated with fractal realizations oftilings and the noncommutative geometry of tilings (joint withMichael Mampusti).

Part I. History and motivation for studying aperiodictilings.

Sir Roger Penrose

Shechtman’s Diffraction Pattern

Dan Shechtman

Winner of the Nobel prize in Chemistry 2011 for the “discovery ofquasicrystals” in 1984.

Kleenex Toilet Paper

Penrose tiling embossed toilet paper

“So often we read of very large companies riding rough-shod oversmall businesses or individuals, but when it comes to thepopulation of Great Britain being invited by a multi-national towipe their bottoms on what appears to be the work of a Knight ofthe Realm without his permission, then a last stand must bemade.” - David Bradley, director of Pentaplex (the company thatcares for Penrose’s copyrights)

Part II. Aperiodic tilings and their properties.

The Federation Square Building in Melbourne, Australia.

Tilings

A prototile is a compact subset of R2 (shape) with a label (colour).

Let P be a finite set of prototiles. A tiling of R2 is a countablecollection of tiles T = ti : i ∈ N such that:

ti = p − x for some p ∈ P and x ∈ R2;⋃i∈N

ti = R2;

int (ti ) ∩ int (tj) = ∅ if i 6= j .

A patch is a connected finite collection of tiles from a tiling T .

Properties of Tilings

A tiling has finite local complexity (FLC) if the number of two tilepatches (up to translation) is finite.

A tiling is aperiodic if T + x = T implies x = 0.

A substitution rule is a map ω on prototiles P and a scaling factorλ > 1 such that for each p ∈ P, ω (p) is a patch of tiles and suchthat the subsets of R2 containing ω(p) and λ(p) are equal.

INFLATE and SUBDIVIDE!

For the purposes of this talk we assume that all tilings areaperiodic substitution tilings with finite local complexity.

The Penrose Substitution

ωω

ω

ω

Substitution Tilings

After 4 substitutions of the red prototile:

Copy of red tile (in the same orientation) in the centre of thispatch, so there is a fixed point in the interior of the central red tile.

Substitution Tilings

After 4 substitutions of the red prototile:

Copy of red tile (in the same orientation) in the centre of thispatch, so there is a fixed point in the interior of the central red tile.

Constructing the Penrose tiling from the substitution

Let p be the red prototile with fixed point on the origin of R2.

Then p ⊂ ω4(p), and iterating this process again, we see thatω4(p) ⊂ ω8(p) and so on.

Thus we obtain the nested sequence of inclusions

p ⊂ ω4(p) ⊂ ω8(p) ⊂ ω12(p) ⊂ · · ·

Then

T :=∞⋃k=1

ω4k(p)

is a tiling of the plane.

A tiling constructed in this way is called a self-similar tiling.

The Penrose Tiling

Part III. Fractal realizations of substitution tilings.

A fractal version of the Penrose tiling

Dual trees in prototiles

DefinitionSuppose T is a substitution tiling with prototile set P. A dual-treein a prototile p ∈ P consists of a vertex vp in the interior of p anda collection of non-overlapping edges connecting vp to the interiorof each edge of p.We say that the set of prototiles P has a consistent dual-tree G ifeach p ∈ P has a dual-tree such that if two translated prototileedges meet in the tiling T , then the associated boundary verticesof G meet in T as well.

Recurrent pairs

DefinitionSuppose T is a substitution tiling with substitution ω and scalingfactor λ, and let RN := λ−NωN . A pair of consistent dual-trees(G ,S) on P is called a recurrent pair if S ⊂ RN(G ) for someN ∈ N.

G R1(G ) S

A recurrent pair (G , S) defines a graph iterated function systemand the attractor (or fractal) of (G , S) is a graph whose boundaryvertices meet anytime two translated prototiles meet.

Example: a (G , S) recurrent pair for the Penrose tiling

G R2(G ) S G1

G R2(G ) S G1

G R2(G ) S G1

G R2(G ) S G1

Example: the first iteration for the Penrose tiling

G1 R2(G1) S1 G2

G1 R2(G1) S1 G2

G1 R2(G1) S1 G2

G1 R2(G1) S1 G2

Example: the second iteration for the Penrose tiling

G2 R2(G2) S2 G∞

G2 R2(G2) S2 G∞

G2 R2(G2) S2 G∞

G2 R2(G2) S2 G∞

Example: a fractal Penrose tiling

Example: a fractal Penrose tiling

Example: a fractal Penrose tiling

Example: 2-dimensional Thue-Morse Tiling

G R2(G ) S G∞

G R2(G ) S G∞

Example: 2-dimensional Thue-Morse Tiling

Example: Fractal Thue-Morse Tiling Substitution

Example: The Penrose Tiling

G R1(G ) S Fractal G∞

G R1(G ) S Fractal G∞

Example: A Fractal PenroseTiling

Example: A Fractal Penrose Substitution

Main result

Theorem (Frank, Webster, W)

Suppose T is a substitution tiling with finite local complexity.Then T has an infinite number of distinct dual-tree fractalsubstitution tilings.

Part IV: Spectral triples and the noncommutativegeometry of tilings.

A fractal tree overlaid on the Penrose tiling

Spectral Triples

Theorem (Gelfand-Naimark)

Every C ∗-algebra A is isomorphic to a closed ∗-subalgebra of thebounded operators on a Hilbert space

Theorem (Gelfand-Naimark)

Suppose A is a commutative C ∗-algebra. Then A ∼= C0(X ) for alocally compact and Hausdorff topological space X .

The Gelfand-Naimark Theorem implies that we should think ofC ∗-algebras as noncommutative topological spaces.

Connes’ noncommutative geometry program says that we shouldthink of the existence of a spectral triple on a C ∗-algebra as anoncommutative geometric space.

Example: A spectral triple on the circle

Let S1 denote the circle, S1 := e2πix | x ∈ [0, 1].

Let (A,H,D) be a triple consisting of the C ∗-algebra A = C (S1),the Hilbert space H = L2[0, 1], and the unbounded self-adjointoperator D = −i d

dx with domain of definition C∞(S1) ⊂ C (S1).

Then we have that

For f ∈ C∞(S1), the commutator [D, f ] is a boundedoperator in H.

[D, f ]g = D(fg)− fD(g) = D(f )g + fD(g)− fD(g) = D(f )g .

(1 + D2)−1 is a compact operator on H.

The following formula defines the geodesic metric on S1:

d(x , y) := sup|f (x)− f (y)| | f ∈ C∞ and ‖[D, f ]‖ ≤ 1.

Spectral Triples

A spectral triple is a triple (A,H,D) consisting of:

a separable Hilbert space H;

A C ∗-algebra A ⊂ B(H);

an unbounded, self-adjoint operator D on H satisfying:

a ∈ A | [D, a] ∈ B(H)

is dense in A, and

(1 + D2)−1 is a compact operator.

Our goal is to define a spectral triple on a C ∗-algebra associatedwith a substitution tiling.

Constructing the C ∗-algebra of a tiling

Given a substitution tiling T , consider the orbit of T :

OT :=

T ′ : T ′ = T − x for some x ∈ R2.

The completion of OT in the tiling metric is called the continuoushull ΩT .

Theorem (Radin and Wolff)

A tiling T has finite local complexity ⇐⇒ the continuous hull ΩT

is a compact metric space.

For the remainder of this talk let T be a fixed self-similarsubstitution tiling; that is, let T be a tiling constructed usinga fixed point of the substitution system.

The discrete hull of a tiling

For each prototile p ∈ P, we assign a point in the interior of pcalled a puncture. We then extend the punctures to all tiles in ΩT

by translation. We denote the puncture of a tile t by x(t).

The discrete hull Ωpunc of a tiling consists of all tilings T ′ in ΩT

such that the tile in T ′ at the origin of R2 has its puncture on theorigin.

The Hilbert space and groupoid of a tiling

For tilings T1,T2 ∈ Ωpunc , define an equivalence relation

T1 ∼ T2 ⇐⇒ T2 = T1 − x for some x ∈ R2.

Let [T ] denote the equivalence class of T , then [T ] is countableand dense in Ωpunc .

Let H be the Hilbert space H := `2([T ]) with standard basisδT ′ | T ′ ∈ [T ].

Let Rpunc be the equivalence relation groupoid

Rpunc := (T1,T2) | T1 ∼ T2.

with inverse (T1,T2)−1 = (T2,T1) and partially defined product(T1,T2)(T2,T3) = (T1,T3).

Kellendonk’s C ∗-algebra of a tiling

Let Cc(Rpunc) denote the continuous functions of compact supporton the equivalence relation groupoid.

Then for f ∈ Cc(Rpunc) and T0 ∈ [T ], the formula

f δT ′ :=∑

(T ′′,T ′)∈ supp(f )

f (T ′′,T ′)δT ′′

makes f a bounded operator on H = `2([T ]).

The completion of Cc(Rpunc) in operator norm of H defines aC ∗-algebra A = C ∗(Rpunc).

So all that is left to define is an unbounded operator D on H.

Recall the (G , S) recurrent pair for the Penrose tiling

G S G∞

G S G∞

G S G∞

G S G∞

Constructing a fractal tree for the Penrose tiling

F1

λ(F1)

ω(F1)

F2 = λ(F1) ∪ ω(F1)

Constructing a fractal tree for the Penrose tiling

F1

λ(F1)

ω(F1)

F2 = λ(F1) ∪ ω(F1)

Constructing a fractal tree for the Penrose tiling

F3 = λ(F2) ∪ ω(F2)

This process can be used to construct a fractal tree F∞ on a tilingof the plane.

Fractal Trees

We would like to use the fractal trees to define a distance functionbetween any two tiles in T . Since fractals typically have infinitearc length, we use Perron-Frobenius theory to define the length ofa fractal edge.

Each fractal tree defines an associated irreducible substitutionmatrix. Perron-Frobenius Theory then implies that there is

a unique eigenvalue κ > 1 (this defines a scaling factor for thefractal tree), and

a normalized eigenvector associated to κ, which is used toassign length to the fractal edge between each pair of tiles inT .

For T1,T2 ∈ [T ], define

dF (T1,T2) = sum of the fractal edges connecting

the origins of T1 and T2.

The Spectral Triple

To define an unbounded operator D on H, for T ′ ∈ [T ], let

DδT ′ := ln(dF

(T ′,T

))δT ′ ,

and extend by linearity to a dense subset of H.

We have constructed:

separable Hilbert space H := `2 ([T ]);

a C ∗-algebra C ∗(Rpunc) ⊂ B(H);

unbounded self-adjoint operator D on H.

Theorem (Mampusti-W)

The triple, (A,H,D) is a spectral triple.

Thus every fractal realisation of a tiling T can be thought of asdefining a metric on Kellendonk’s C ∗(Rpunc).

References

N.P. Frank, S.B.G. Webster, and M.F. Whittaker, Fractal dualsubstitution tilings, preprint, arXiv:1410.4708.

N.P. Frank and M.F. Whittaker, A fractal version of thePinwheel tiling, Math. Intel. 33 (2011), 7–17.

J. Kellendonk, Noncommutative geometry of tilings and gaplabelling, Rev. Math. Phy. 7 (1995), 1133–1180.

M. Mampusti, Spectral triples on substitution tilings viafractal trees, Honours Thesis, University of Wollongong, 2014.

L. Sadun, Topology of Tiling Spaces, University Lecture Series46, AMS, Providence, 2008.