Symbolic dynamics and tilings of

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Symbolic dynamics and tilings of
E. Arthur Robinson, Jr. Department of
Mathematics George Washington UniversityWashingto
n, DC 20052 robinson_at_gwu.edu
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1. Introduction

• Tilings tiling spaces
• Tiling dynamical systems
• Local matching rules
• Decidability issues
• Substitution tiling spaces
• Topological dynamics
• Repetitive tilings and minimality
• Ergodic theory quasicrystals
• Patch densities
• Weak mixing

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2. Tilings

• is a tile if homeomorphic to B1
• A tiling of is a collection of tiles that
• pack (have disjoint interiors)
• cover (union or support ).
• if they are translates.
• Prototile equivalence class.
• T finite set of inequivalent prototiles.

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Definition 2.1. The set XT of all tilings by
translates of tiles in T is called the full
tiling space.
Definition 2.2. We denote the action of
translation on XT by T.

• A patch y is a finite set of tiles with supp(y)
  homeomorphic to B1.
• The set of all patches is denoted T
• The ntile patches are denoted T(n)

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Definition 2.3. XT is locally finite if T(2) is
finite.
We always assume XT is locally finite.
(a) Shows edge-to edge squares. (b) shows a
continuous fault. (c) shows key cuts that force
local finiteness
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Examples
(b) Rn The set of all tilings by n-rhombs.
(a) Sn The set of all tilings by squares with n
colors.
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The tiling topology.
For a tiling x and compact let xK
be the smallest patch y in x with K ? supp(x).

Theorem 2.8. (Rudolph) If XT is locally finite
then XT is compact.
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• Sketch of proof
• First, show that d is complete Given a Cauchy
  sequence xn,, let yn ? yn1be a sequence of
  patches slightly translated from xn. Then x?
  yn.
• Next, observe that local finiteness for XT is
  equivalent to XT being a totally bounded metric
  space.

The translation action T on XT is continuous.
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3. Tiling dynamical systems

• Let XT be a full tiling space. A tiling space is
  a closed and T-invariant subset X ? XT.
• The pair (X,T) is called a tiling dynamical
  system.

• Tiling dynamical systems as symbolic
  dynamicalsystems
• TilesT ? the alphabet (i.e.,symbols)
• Metric d ? the product topology
• Translation T ? The shift

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Finite type
Any tiling space X ? XT can be defined by
excluding a set F ? T of forbidden patches.
Definition 3.3. X ? XT is finite type if F is
finite.

• When F? T(2) we usually think in terms of the
  allowed 2-patches Q(2)T(2)\F.
• We call Q(2) a local matching rule.

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Example 3.4. Penrose tiles

• Here T is all rotations of the above tiles by
  multiples of 2?/10.
• Q(2) is the local matching rule arrows on
  adjacent tiles must match.
• Penrose tilings are x? X satisfying Q(2)

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A patch of Penrose tiling
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The tiling problem
The preceding picture shows a patch of Penrose
tiling, but how do we know Penrose tilings exist?
Tiling problem. Given forbidden patches F,is the
corresponding tiling space X nonempty?
Extension Theorem. (Wang) Let T be locally
finite. Let F?T. Let T?T be the patches with
no forbidden sub-patches. Then the tiling space
X is nonempty iff ? r ? y?T s.t. Br? supp(y).
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Now assume F is finite.

• Wangs conjecture
• There is an algorithm to decide the tiling
  problem.
• If a finite type X ? ?, then ? a periodic x ? X.

Note Both of these are true for 1-dimensional
shifts of finite type.

• Bergers Answer
• The tiling problem is undecidable.
• There exists X?? with no periodic tilings x.

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• A prototile set T with a local matching rule Q(2)
  such that X contains no periodic tilings is
  called an aperiodic prototile set.
• We will show below that the Penrose tiles are
  aperiodic (but we still must show X ? ?!!)
• Aperiodicity is strictly a higher dimensional
  phenomenon such examples do not exist in
  1-dimension!

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4. Substitution tiling spaces

• Let be expansive (an
  affinity).
• , where M is an isometry is called
  a similarity.
• Given T we define

We call C self-affine or self-similar according
to L
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Some decompositions

• A few self-similar polyomino decompositions

A mapping S LC is called a tiling substitution.
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The chair tiling space
Show
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Substitution tiling spaces

• Let S be a tiling substitution on T and D?T
• Put x1D put xk Sxk-1.
• Define F such that y in F if y in not a
  sub-patch of any xk.
• Define a substitution tiling space X using F.

Lemma 4.6. X is nonempty.
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Penrose decompositions

• The rhombic Penrose decomposition
  (imperfectnote the overlap)

Show

• Can get self-similar decomposition using half
  tiles

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New tiling substitution induced on half-tiles

• Iterate this substitutions (get triangular
  Penrose tilings).
• Then erase all3- and 4-arrow lines.
• The resulting tiles satisfy the Penrose matching
  rule and hence are Penrose tilings.

Corollary 4.7. Penrose tilings exist.
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More imperfect self-similar decompositions

• The octagonal or Ammann-Beenker tiling

Show

• The binary decomposition (note the required 2?/20
  rotation)

Show
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The pinwheel tiling
Federation Square,Melbourne, Australia
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A self-affine polynomial tiling substitution.
In this example (i.e. its non
self-similar)
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5. Topological dynamics

• A decomposition C is called invertible if it has
  a continuous inverse

We say the substitution SMC is invertible.
For let A be the
matrix where
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One consequence of primitivity is that X is
independent of D.
Defintion 5.1. A tiling x is called repetitive if
for any patch y in x there is an rgt0 so that a
patch equivalent to y occurs in every r-ball in
x.
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A tiling is properly repetitive if it is
repetitivebut not periodic.
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A (X,T) is minimal if there are no proper closed
T- invariant subspaces of X.
First a preliminary result
Proposition 5.6. A tiling x in a substitution
tiling space corresponding to an invertible
primitive substitution is non-periodic.
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Proof.

• One first shows the Lemma.
• x?X if and only if S-nx exists for all n.
• Now suppose
• Then
• Choose n so large that
• This contradicts ()

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Now, the fundamental property of substitutions
Theorem 5.8. If X be a tiling substitution space
corresponding to a primitive tiling substitution
S, then any x?X is repetitive. Moreover, every
tile has dense orbit.
Corollary. Substitution tiling dynamical systems
are minimal and non-periodic.
Corollary 5.22. There are uncountably many
incongruent Penrose tilings.
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Proof of Theorem 5.8.

• Assume wolog Agt0.
• Let x?X and y a patch in x.
• Fix D?T. Choose k so large that y is a patch in
  Sk-1D.
• Then y is a patch in SkD for all D?T.
• Now, there exists xk ? X s.t. Sk xkx.
• Let x'kLk xk.

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• Proof (continued). Let s be the max diameter of
  LkD ? LkT r 2s.
• For any D' equivalent to LkD, if t ? D ', then D
  ' is in the ball Brt.
• Thus any r-ball in Rd contains a tile D' ? x'k,
  and
• ...the patch CkD' in x, with support D',
  contains
• a sub-patch y.

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Factors and isomorphisms.

• Definition 5.9.
• A continuous surjective mapping QX? Y such that
  TQQT is called a factor.
• A factor is local Q(x)0 depends only on
  xBr.

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• We say Q(x) is locally derivable from x.
• If Q is invertible, we say x and Q(x) are
  mutually locally derivable.

Example. The rhombic Penrose tilings are
mutually locally derivable from the triangular
Penrose tilings
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Local mappings are analogous to sliding block
codes in symbolic dynamics.
Lemma 5.10. A mutual local equivalence between
tiling spaces is a topological conjugacy.
Caution. (Petersen Radin Sadun) Not every
factor/conjugacy is local (No Curtis-Lyndon-Hedlun
d Theorem).
Definition 5.12. A factor mapping is almost 11
if card(Q-1(y))1 for some y.
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Earlier versions of similar results are due to
Mozes and Radin.
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• Observation. (de Bruijn) The octagonal tiles can
  be marked to achieve a matching rule that forces
  the octagonal tilings.
• However, for a few tilings the markings are
  necessarily not unique.
• The unmarked octagonal tilings are a strictly
  almost 11 factor of the marked octagonal tilings

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F
F-1
Debruijn showed that the marked Penrose
tilingsare mutually locally derivable from the
unmarked ones.
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6. Ergodic Theory
Now we will consider T-invariant measures on
tiling spaces X, and what they tell about the
tilings x.

• Let U?,y be the open set of all x?X so that an ?
  - translation of y occurs in x.

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A dynamical system (X,T) is called uniquely
ergodic if it has a unique T-invariant measure.
Theorem 6.1. If S is a primitive tiling
substitution then the corresponding tiling
dynamical system is uniquely ergodic.
Unique ergodicity has a natural interpretation in
terms of tilings.
Let P(x,y) be the set of all t so that Tty occurs
in x
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This means every patch in every tiling has a well
defined frequency.

• Theorem 6.1 follows from Theorem 6.3 by routine
  arguments in ergodic theory.
• Theorem 6.3 follows from the Perron-Frobenius
  Theorem applied to A.

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7. Eigenvalues and mixing

• Quasicrystals are a new form of matter which
• have enough order to have distinct spots in their
  X-ray diffraction pattern.
• have a non-periodic atomic structure.

Quasicryatals can have symmetries forbidden to
ordinary crystals by the crystallographic
restriction e.g., Penrose tilings have 5-fold
rotational symmetry
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• A standard result in ergodic theory is that ? is
  a subgroup of .
• The case where the eigenfunctions f span L2 is
  called pure point spectrum.

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Eigenvalues essentially correspond to the spots
in the X-ray diffraction pattern.

• Theorem. (Corollary 8.6.)
• For the Penrose tilings ? is generated by the 5th
  roots of unity.
• For the octagonal tilings ? is generated by the
  8th roots of unity
• For the chair tilings ? is generated by the
  dyadic rational vectors All these examples
  have pure point spectrum.

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Weak mixing
Defintion. A uniquely ergodic system (X,T) is
weakly mixing if the only eigenvalue is w 0.

• Theorem 7.1. (Solomyak) Let ?d be the Perron-
  Frobenius eigenvalue for the matrix A associated
  with a self similar tiling substitution S
• If ? is not a Pisot number then (X,T) is weakly
  mixing.

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For the binary tiling
Binary tilings are weakly mixing.
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Tilings in a weakly mixing system look more
disordered than in one with pure point
spectrum
Penrose tiling v.s. binary tiling
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Some additional comments.

• Substitution tiling systems never have the
  strong mixing property.
• Substitution tiling systems have entropy zero.
• This means that (Q(n)) grows sub-exponentially
  in n.
• Uniquely ergodic finite type tiling systems have
  entropy zero.

Thus these tilings are, at best, only weakly
disordered
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ConclusionMethods for constructing examples

• Entropy zero methods (quasicrystals)
• Substitutions
• Uniquely ergodic finite type
• Quasiperiodic tilings (i.e., projections see the
  notes).
• Positive entropy (high temperature)
• Zd shifts of finite type.
• Random tilings (Gibbs measures)