An Algebraic Invariant for Substitution Tiling Systems 21 CHARLES
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Geometriae Dedicata 73: 21–37, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
21
An Algebraic Invariant for Substitution
Tiling Systems
CHARLES RADIN and LORENZO SADUN
Mathematics Department, University of Texas, Austin, TX 78712, U.S.A.
e-mail: radin@math.utexas.edu sadun@math.utexas.edu
(Received: 3 January 1997; revised version: 14 October 1997)
Abstract. We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings
made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism
invariant related to a subgroup of rotations and compute it for various examples. We also extend our
analysis to more general dynamical systems.
Mathematics Subject Classifications (1991): 52C22, 52C20.
Key words: tiling, dynamics, invariant.
1. Introduction, Definitions and Statement of Results
This paper concerns tilings of Euclidean spaces by polygons or polyhedra, more
specifically, tilings made by a ‘substitution process’. Given a substitution rule,
the set of resultant tilings is a topological space with an action of the Euclidean
group, hence a dynamical system. We develop here an algebraic invariant that helps
determine when two tiling systems are equivalent as dynamical systems. In this
introduction we define the notions of ‘substitution tiling system’ and of equivalence
between two such systems, and state what the invariant is. In subsequent sections
we analyze the invariant, in particular we show its use in distinguishing between
substitution tilings.
Our eventual goal is to associate certain groups to substitution tilings of Euclid, are generated by the relative
ean -space. These groups, subgroups of SO
orientations of tiles in the tilings, depend on the specific tiling and on some
. Although
specific choices (indexed by an integer ), and are denoted
depends on and , the dependence is quite controlled. If and are different
tilings with the same substitution rule we will show that, under some mild hypotheand
are conjugate as subgroups of SO
. Even without the mild
ses,
) to subgroups of one another, a condition
hypotheses, they are conjugate (in SO
we call ‘c-equivalence’. So we can associate to a substitution tiling system the
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Research supported in part by NSF Grant No. DMS-9531584.
Research supported in part by NSF Grant No. DMS-9626698.
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22
CHARLES RADIN AND LORENZO SADUN
Figure 1. Two ‘pinwheel’ tiles.
Figure 2. The substitution for pinwheel tilings.
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common conjugacy class (or -equivalence class) of the groups associated to the
tilings in the system.
What will remain, then, is to show that this conjugacy (or -equivalence) class
can be considered an invariant in a natural sense. That is, we will show that two
substitution tiling systems that are equivalent as dynamical systems have the same
class of groups. We will do this by finding a dynamical description of the class.
using dynamical
0 and each tiling we will define a group
For each
information only. For sufficiently small, and for almost every , we show that
is conjugate to (or -equivalent to)
for some, and hence all, choices
. The class of
is thus the same as the class of
. Since
is defined
is a
using data that is preserved by dynamical equivalence, the class of
dynamical invariant.
depends only on the geometry of the tiling . Since
Note that the group
the class of
is the same for every tiling with the given substitution rule, we
can obtain information about a substitution tiling system by looking at any single
and
tiling in it. So if two substitution tilings and give rise to groups
that are not conjugate (or -equivalent), then and
cannot belong to
equivalent substitution tiling systems.
Before defining substitution tiling systems in general, we present an example.
Hopefully, the general definitions will be clearer with this example in mind. The
‘pinwheel’ tiling of the plane [Ra1] is made as follows. Consider the triangles of
Figure 1. Divide one of them into five small triangles as in Figure 2 and expand the
figure about the origin by a linear factor of 5, producing 5 triangles congruent to
the originals.
Repeat this two-step procedure simultaneously for all the triangles of the
figure, then again, an infinite number of times, producing a (pinwheel) tiling of
the plane, a portion of which appears in Figure 3. Such tilings have a hierarchical
structure which is of interest for various reasons; in particular it leads to interesting
behavior of the relative orientations of tiles within a tiling [Ra3]. For background
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23
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
Figure 3. Part of a pinwheel tiling.
on related recent work see [AnP, CEP, DwS, G-S, Kel, Ken, LaW, Min, Moz, Ra3,
Rob, Sad, Sch, Sen, Sol, Tha] and references therein.
Substitution Tiling Systems
With the pinwheel example in mind, we now address substitution tiling systems in
general. Let be a nonempty finite collection of polyhedra in (typically 2 or
be the set of all tilings of Euclidean space by congruent
3) dimensions. Let
copies, which we will call tiles, of the elements of (the ‘alphabet’) . We label the
‘types’ of tiles by the elements of . We endow
with the metric
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of the boundaries
is the Hausdorff metric on compact sets defined as follows.
of all tiles in .
,
max
,
Given two compact subsets and of
where
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with
denoting the usual Euclidean norm of . Although the metric depends
on the location of the origin, the topology induced by is translation invariant.
A sequence of tilings converges in the metric if and only if its restriction to
converges in
. It is not hard to show [RaW] that
every compact subset of
(which is automatically nonempty in our applications) is compact and that
on
,
the natural action of the connected Euclidean group
, is continuous.
A ‘substitution tiling system’ is a closed subset
satisfying some
additional conditions. To understand these conditions we first need the notion of
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24
CHARLES RADIN AND LORENZO SADUN
3
Figure 4. The substitution for pinwheel variant tilings.
4
‘patches’. A patch is a (finite or infinite) subset of an element
; the set
of all patches for a given alphabet will be denoted by . Next we need, as for the
pinwheels, an auxiliary ‘substitution function’ , a map from
to , with the
following properties:
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and
,
(i) There is some constant
, where
is the conjugate of by the similarity of
Euclidean space consisting of stretching about the origin by .
(ii) For each tile
and for each
1, the union of the tiles in
is
congruent to
, and these tiles meet full face to full face.
(iii) For each tile
,
contains at least one tile of each type.
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Condition (ii) is quite strong. It is satisfied by the pinwheel tilings only if we
add additional vertices at midpoints of the legs of length 2, creating boundaries of
4 edges. A similar (minor) adjustment is needed for other examples in this paper.
Even with such adjustments however, condition (ii) is not satisfied by the kite and
dart tilings [Gar], or those which mimic substitution tilings using so-called edge
markings [G-S, Moz, Ra3]. It is to handle such examples that we introduce the
general development of Section 3.
t
DEFINITION. For a given alphabet
of polyhedra and substitution function
, where
is the
the ‘substitution tiling system’ is the pair
compact subset of those tilings with the property that every finite subpatch of
for some
0 and some
, and is the
is congruent to a subpatch of
on
. (For simplicity we often refer to
as a substitution
natural action of
tiling system.)
One planar example of a substitution tiling system is based on the pinwheel
substitution of Figure 2. A slight variant of the pinwheel is defined by the 1-3- 10
right triangle and its reflection, and the substitution of Figure 4.
Two further special conditions which we will occasionally impose are:
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(iv) A tiling in
can only be tiled in one way by supertiles of level , for any
1.
(v) For every
, there exists
0 such that
contains a tile , of the
same type as , and parallel to .
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are called
We note here that with the convention that patches of the form
‘supertiles’ of ‘level’ and ‘type’ , it is easy to show by a diagonal argument that,
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25
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
@
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0, each tiling is tiled by supertiles of level
for each
level 4 for the pinwheel is shown in Figure 3.
Finally, let
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[Ra3]. A supertile of
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We call such a family of sets a ‘local contracting direction (at )’.
Our goal is to define a notion of equivalence for substitution tiling systems, and
an invariant for that equivalence. For the equivalence we use:
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2
and
are ‘equivDEFINITION. The substitution tiling systems
1
2
, invariant under
and of measure zero
alent’ if there are subsets
, and
with respect to all translation invariant Borel probability measures on
1
2
a one-to-one, onto, Borel bimeasureable map
1
2 , such that:
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We call such a map an ‘isomorphism’.
This notion of equivalence is stronger than simply intertwining the actions of
. This is appropriate; it has been known at least since [CoK] that substitution
subshifts show almost none of their richness if considered merely as subshifts. So
in classifying tilings that have a hierarchical structure we make some feature of
that hierarchical structure part of our notion of equivalence.
To define an invariant we extract information from the local contracting directions. Since the local contracting directions are preserved by equivalence, such
information is manifestly invariant. We define here the invariant. In later sections
) and demonstrate its use
we relate it to directly computable quantities (the
in distinguishing between tiling dynamical systems.
as the semidirect product of SO
with
, with
Consider
denoting a rotation about the origin followed by a translation . Then consider,
and
0:
for any substitution tiling system
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Now let
be the subgroup of SO
generated by
. The corollary to
is independent of and
Theorem 2 shows that the conjugacy class of
(when small enough) for substitution tiling systems satisfying (iv) and (v). The
is therefore an invariant of the tiling dynamical system,
conjugacy class of
not just a feature of the individual tiling .
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2. The Group of Relative Orientations
The group
generated by
is not directly computable. In this section
we remedy this by constructing, for a substitution tiling system, a more easily
26
CHARLES RADIN AND LORENZO SADUN
3
Figure 5.
related to the relative orientations of the tiles in the single
computable group
is then shown to be conjugate to
.
tiling . The group
SO
be the set
Given a tiling and some tile of type in it, let
of relative orientations with respect to of the tiles of type in ; that is,
is the set of rotations of which bring a tile of type parallel to (the fixed) .
is easily seen to be generated by the relative
The group generated by
orientations between all pairs of tiles of type in ; in particular it is independent
. Furthermore,
of , and we denote it by
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LEMMA 1. If has a tile of type parallel to the tile in , then
. For any ,
is conjugate to
.
Proof. First note that
is generated by the relative orientations between
and all other tiles of type in . So consider the generator of
which is the
,
relative orientation of a tile with respect to in . We will show that
. By symmetry, we would then have
from which it follows that
, and hence
.
From the definition of substitution tilings, the tiles and can be thought of
as belonging to some supertile of level (although not all of need exist in
of level in
). Since is tiled by supertiles of level , there is a supertile
containing a pair of tiles,
and , which have the same positions relative to
as do and relative to . See Figure 5.
1
with respect to . Then
Let be the relative orientation of
where is the relative orientation of
with respect to . But is then also the
with respect to , which is the same as that of
with
relative orientation of
respect to , and
is thus an element of
. But then is an element of
, as claimed.
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27
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
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and
are conjugate by an element of SO
If is any tiling at all,
namely a rotation which makes a tile of type in parallel to one in .
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DEFINITION. Two subgroups of SO
are ‘ -equivalent’ if each is conjugate (in
) to a subgroup of the other. (Note that in SO(2) -equivalence is the same
SO
as identity.)
LEMMA 2. For any tilings
and tile types and ,
is -equivalent
.
to
Proof. By Lemma 1 it is sufficient to show that
and
are -equivalent.
Consider any two tiles and of type in , and let be the relative orientation of
with respect to . After one substitution and give rise to tiles and of type
in the tiling . The relative orientation of with respect to is again , since
takes each part of onto the corresponding part of . Applying this construction to
all the generators of
, we see that
is a subgroup of
. Similarly,
is a subgroup of
. But
and
are conjugate to
and
, respectively, so
and
are conjugate to subgroups of each
other.
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LEMMA 3. Assume satisfies (v). Then
.
Proof. Let
. Since the tilings defined by
are the same as those
1, so that
condefined by we can, without loss of generality, assume
and
tains tiles parallel to every tile of . Then, by Lemma 1,
. But we have shown that
and
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To summarize: From Lemmas 1 and 2 we can associate a subgroup of SO
to
any substitution tiling system, uniquely defined up to -equivalence. If the substitution tiling system satisfies (v), Lemma 3 shows that the group is uniquely defined
up to conjugacy.
Before we can use these groups as an invariant for equivalence of substitution
tiling systems we must refer to the relative orientations in a more fundamental
way. Our next goal is to connect this group with the invariant introduced at the
end of Section 1. The essential observation is that, if tilings
agree in some
and
will agree in a
neighborhood of the origin in Euclidean space, then
.
larger neighborhood of the origin, so we typically expect
We are thus led to a quantity introduced earlier. For each in the substitution tiling
and for each
0, consider:
system
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28
CHARLES RADIN AND LORENZO SADUN
THEOREM 1. Assume a substitution tiling system
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(a) Given any
0 there exists
0 such that
implies
.
0 such that, for every
,
and every tile
(b) There exists
that meets the origin, there is a tile
that exactly coincides with .
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Proof. (a) is immediate from the form of the metric. The proof of (b) requires
the following two lemmas.
1
0 and every neighborhood
of the identity in
LEMMA 4. For every
there exists
0 with the following property: Let
be any two tilings
, and let be a tile of that is contained in
. Then contains
with
a tile of the form
where
.
Proof. Let
0 be such that for each
some ball of diameter lies in
0 3 and define the heart
of
as
the interior of . Fix some
for all
. By the corridor
of a tiling we mean
. Let be the largest of the diameters of all
.
the complement of
.
Without loss of generality, we can assume
we have
With this notation we note that if
and
. So if
0 each tile in in
is closely approximated by some tile in and vice versa. In particular it now follows that for small
then the tiles
must be of the same type as the tiles
enough , if
they approximate, and in fact satisfy
with
.
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LEMMA 5. For each
0 there is a neighborhood
of the identity in
with the following property: If is a tile in
and 1 and 2 are distinct elements
, then 1 and 2 overlap but are distinct. In particular, it is impossible
of
for 1 and 2 to both be tiles in the same tiling.
on tiles, and the fact
Proof. This follows from the continuity of the action of
that polyhedra do not admit infinitesimal symmetries.
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We now return to the proof of Theorem 1. Pick
and let
be as in
of the identity of
Lemma 5. Pick a smaller bounded neighborhood
with the property that
implies
. Then pick small enough
that Lemma 4 applies.
Let be a tile of containing the origin. By Lemma 4 there is a tile in of
the form
with
. We will show that
0 implies that, for
, while
0,
0 implies that lim
some ,
0. This will complete the proof.
. If
0, pick such that
is outside the
Note that
. Let be a tile of
containing the origin. Then there
neighborhood but in
is a tile
in
. But by Lemma 5 this means there cannot be a tile
in with
. By Lemma 4 this means that
.
of the form
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29
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
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0
0 then
. If
0, for every tile
containing the
If
overlapping it and with relative orientation , which
origin there is a tile
implies that the distance between
and
will not go to zero.
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SO
1
D
there exists such that
O
Let
be the group generated by
1(b), we see that every
respect to a corresponding tile of
, and if
region of containing
includes the relative orientation of
Consider the following property.
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. Assuming small enough for Theorem
is the relative orientation of a tile of with
near the origin. By Theorem 1(a), if is a
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to .
±
¼
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À
Recall the following quantity from Section 1
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PROPERTY F. The subset of tilings , for which every fixed finite ball of Euclidean space is contained in some supertile of finite level in , is of full measure for
every translation invariant measure on .
Ô
Ò
Ñ
Õ
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×
Ø
We will prove that Property F holds for a large class of interesting systems, at
least those satisfying condition (iv). This assumption, which implies that is a
, is satisfied by all known nonperiodic examples. In fact it
homeomorphism on
is automatically true for a system that contains nonperiodic tilings and in which the
tiles only appear in finitely many orientations in any tiling [Sol].
If a tiling contains two or more regions each tiled by supertiles of level for all
0, we call these regions supertiles of infinite level. Recall that any tiling is tiled
by supertiles of any finite level . If a ball in a tiling fails to lie in any supertile
of any level , then is tiled by two or more supertiles of infinite level, with the
offending ball straddling a boundary. (One can construct a pinwheel tiling with
two supertiles of infinite level as follows. Consider the rectangle consisting of two
1 orient
supertiles of level
1 in the middle of a supertile of level . For each
such a rectangle with its center at the origin and its diagonal on the -axis, and fill
by periodic extension. By compactness
out the rest of a (non-pinwheel) tiling
this sequence has a convergent subsequence, which will be a pinwheel tiling and
which will consist of two supertiles of infinite level.)
We now use the above to prove:
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LEMMA 6. For a substitution tiling system satisfying (iv), let be the set of tilings
in which some ball does not lie within a supertile of any level . has zero measure
with respect to any translation invariant measure on .
Proof. We only give the proof for dimension
2. Note first that the boundary
of a supertile of infinite level must be either a line, or have a single vertex, since
it is tiled by supertiles of all levels and therefore cannot contain a finite edge.
ï
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30
CHARLES RADIN AND LORENZO SADUN
Furthermore, for a given substitution system there is a constant
such that no
tiling in it contains more than vertices of supertiles of infinite level; specifically,
one can take
2
where is the smallest angle of any of the vertices of the
tiles.
Next we fix some orthogonal coordinate system in the plane and decompose
into disjoint subsets as follows. Let
0 1
0 1 be the ‘half open’ unit edge
square in 2 . Let be the translate of by the vector . Let be the subset of
consisting of tilings containing vertices of supertiles of infinite level. For
we
choose a vertex
by lexicographic order: we choose that vertex which in the
given coordinate system has the largest first coordinate; if there is more than one
with that coordinate we choose the one with the largest second coordinate. Then
, where
if
. It is easy to
we decompose
2
is measurable, and that they are translates of one another so they
see that each
must have zero measure with respect to any translation invariant measure.
contain two supertiles of infinite level, each occupying a
The tilings
where
if the
half plane. Next we decompose
boundary between the supertiles of infinite level crosses the first axis in
1,
if the boundary between the supertiles of infinite level is parallel
and
1 . Note that all sets
are
to the first axis and crosses the second axis in
translates of one another, and all sets are translates of one another, so
has
zero measure with respect to any translation invariant measure.
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THEOREM 2. For any substitution tiling system
satisfying (iv), there exists
0 such that for all
0 0 , and for almost every tiling
,
0
is -equivalent to
for some (and therefore any) . Up to conjugacy,
is independent of . Furthermore, if satisfies (v) then
for some
(and therefore any) .
is contained in
Proof. From Theorem 1(b) it follows that, for small ,
, where is the type of any of the tiles of which meet the origin. On the
other hand, let correspond to in Theorem 1(a). By Lemma 6, for almost every
there is some such that the tiles which intersect
are contained in some supertile of level in . Let be the type of . It follows from Theorem 1(a) that
is a subgroup of
, where
. But
and
are -equivalent,
is conjugate to a subgroup of
, and therefore is conjugate to a
so
. So
and
are -equivalent. By Lemma 1,
is,
subgroup of
.
up to conjugacy, independent of . If satisfies (v), then
,
.
Since
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COROLLARY 1. For each substitution tiling system satisfying (iv) the group
is uniquely defined up to -equivalence, for almost all tilings , and all small
enough , thus the -equivalence class of the group is an invariant for equivalence.
Furthermore, among substitution tiling systems also satisfying (v), the conjugacy
is an invariant for equivalence.
class of this subgroup of SO
A
Õ
K
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31
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
N
3. Abstract Substitution Systems
In going from Lemma 2 to Theorem 2 we see that we can associate with each substitution tiling system a -equivalence class of subgroups of SO
in a reasonably
fundamental way. We are now ready to relax the hypotheses.
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DEFINITION. A ‘substitution (dynamical) system’ is a quadruple
consisting of a compact metric space
on which there is a continuous action
of
and a homeomorphism
such that
for all , where
is the conjugate of by the
1.
similarity of Euclidean space consisting of stretching about the origin by
Substitution tiling systems are special cases of substitution systems. The map
since, for tiling systems,
is not intrinsic to the substitution tiling system
and
lead to the same set of tilings; so equivalence of such systems should not
,
and
be required to intertwine the actions of the maps . The objects
are well-defined in our abstract setting. Motivated by the last section, we use
the following notion of equivalence.
S
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§
DEFINITION. The substitution systems 1 1 1 1 and 2 2 2 2
are ‘equivalent’ if there are subsets
, invariant under
and of measure
, and
zero with respect to all translation invariant Borel probability measures on
1
2
,
a one-to-one, onto, Borel bimeasureable ‘isomorphism’
1
2 such
1
2
. Furthermore, must respect the ‘local contracting directions’
that
1
. Respecting the local contracting directions means that, for each
1,
0 and
0, there exist
0 and
0 such that
, and
1
.
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It is easy to see that for the special case of substitution tiling systems this notion
of equivalence reduces to that previously defined. We will now introduce an invariwe found for
ant for equivalence which reduces to the class of subgroups of SO
substitution tiling systems. We note that this allows us to generalize our discussion
of substitution tiling systems to include tiling systems which do not quite fit the
conditions of Section 2. In particular, our analysis applies to the various versions
of Penrose tilings of the plane, such as the kite and dart tilings, both the substitution
version and the version with edge markings [Gar, Ra3], and to the various tilings
discussed in [G-S, Moz].
We will need to introduce a few more definitions. Given two subgroups 1 and
we write 1
)
2 of SO
2 if
1 is conjugate (by an element of SO
to a subgroup of 2 . The binary relation lifts in an obvious way to a partial
ordering on the set of -equivalence classes. We denote by ‘lower bound’ to a set
of subgroups of SO
any -equivalence class of groups each of which satisfies
if
. For each
for all
. It is almost immediate that
we define
as the set of all lower bounds of the family
0;
is an invariant for substiit is nonempty since it contains . Note that the set
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32
CHARLES RADIN AND LORENZO SADUN
for almost every . For
tution systems – if is an isomorphism then
have unique greatest elements which are
substitution tiling systems, the sets
constant for almost every with respect to every translation invariant measure. In
has an almost everywhere constant greatest element, we
the latter case, where
. Note that
is a -equivalence class,
denote this greatest element by
, which is a specific group. We have thus generalized the analysis of
unlike
substitution tiling systems to the more general setting.
As with substitution tiling systems, we can avoid the use of -equivalence classes
for systems with a special property.
_
1
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PROPERTY P. For almost every
then
.
there exists an
0 such that, if 0
Ò
,
Ù
Note that, by Theorem 2, any substitution tiling system that satisfies (v) also
satisfies Property P. If a substitution system satisfies Property P, we can define
for sufficiently small. If the conjugacy class of
is almost
to be
to be that conjugacy class. The previously
everywhere constant, we define
0
is, of course, the -equivalence class of
.
defined
0
1
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2
THEOREM 3. Suppose
and
are equivalent
1
1
1
1
substitution systems, with the notation of the definition. Then if
2
2
2
2
satisfies Property P so does
. Furthermore, for almost every
1
. In particular, if
is almost everywhere constant up to con,
2
is almost everywhere constant up to conjugacy and
jugacy then
0
1
.
0
Proof. Let be a generic point of 1 . Since 1 1 1 1 has Property P
0
. From the
we can find 0 0 such that, for 0
0,
0
. Now let 0
. We will
equivalence we can find such that
0
show that, for any 0
. From this it will
0,
has Property P and that
.
follow that
is an isomorphism there exists
0 such
Fix any 0
0 . Since
0
0
0
that
. But
. If
then all the inclusions must be equalities, and we are done. So it suffices to show
0
. If
0 this follows from the definition of 0 . But if
0
0
0
then
, so
.
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4. Examples and Analysis of the Invariant
For pinwheel
is the group generated by rotations by
2 and 2
0
1
arctan 2 ; for the variant of pinwheel
is the group generated by rotations
0
by 2 and 2 arctan 13 [RaS]. It is clear that these are distinct, so the substitution
tiling systems are not equivalent.
Rotations appear in more interesting ways in 3-dimensional tilings, for example
the quaquaversal and dite and kart substitution tiling systems, defined in [CoR] and
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33
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
[RaS] respectively. These systems both satisfy (v) and therefore Property P. Let
be a rotation about the axis by an angle , with similar notation for other axes.
2
2
If we denote by
the subgroup of SO(3) generated by
and
, it
can be shown [CoR, RaS] that
is
the
conjugacy
class
of
6
4
for
the
0
is the conjugacy class of 10 4 for the dite
quaquaversal tilings and
0
&
and kart tilings. We shall see that 6 4 and 10 4 are not conjugate (indeed
not even -equivalent) by using the following obvious fact: if the groups and
are conjugate (or -equivalent) and one of them has an element of order (finite
or infinite) then the other must have an element of order .
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Structure Theorem for G(p,q) [RaS]
(a) If
E
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,
3 are odd, then
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is isomorphic to the free product
`
(5)
(b) If
0
a
b
4 is even and
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(c) If
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has the presentation
j
(6)
e
{
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2
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3 is odd, then
c
3 odd, then
2
has the presentation
(7)
(d) If 4 divides both and , then
the least common multiple of and .
4 has the presentation
(e) If 4 divides , then
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In cases (a), (b) and (c), the isomorphism between the abstract presentation and
2
2
is given by
,
. In case (e) the isomorphism is
similar.
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THEOREM 4. (a) If 4 does not divide both and then the orders of elements of
are factors of
factors of ; (b) If 4 divides both and
finite order in
then the orders of elements of finite order in
are factors of
3.
ô
õ
ù
÷
ø
ö
ú
÷
û
ü
ý
þ
ÿ
÷
2
ù
÷
COROLLARY 2. If and
or , then
and
÷
÷
$
÷
+
,
-
.
/
%
&
'
(
)
are not both divisible by 4, and
are not -equivalent.
*
0
1
2
is not a factor of
#
!
"
÷
3
COROLLARY 3. The quaquaversal and dite and kart systems are not equivalent.
Proof of the theorem. (a) Assume
know can be expressed in one of the forms
1
1
2
1,
1
1
2
or
k
4
5
6
7
8
9
:
;
<
has finite order
1
O
?
@
A
B
D
1
E
F
G
2
H
H
H
>
=
I
J
C
U
V
V
V
W
X
Y
Z
U
c
c
c
d
e
f
B
R
S
T
_
[
\
]
^
`
a
b
g
h
i
j
1
L
1
M
P
Q
1. We
,
, with all
K
k
l
1
U
m
n
2
o
o
o
p
q
r
N
34
s
CHARLES RADIN AND LORENZO SADUN
0
, 0
and
1. Within the class of
which are conjugate to with respect to
, we assume that is minimal.
Assume
2; we will obtain a contradiction. Since one could conjugate with
1 , the form
1
1
2
can be exchanged for 1 2
and since
can be exchanged for
one could conjugate with 1 , the form 1 1 2
1
2
, and the form 1 1 2
for 1 1 2
. So we as1
1
2
. If
2 (resp.
2) for any
sume that the form is
2
2
2
2
then we could use the relation
(resp.
) to
(or ) cannot occur. But then, by
reduce the value of ; thus these values of
the structure theorem, has infinite order, which is a contradiction. Thus must
equal 1, and can be assumed to be of the form 1 , 1 or 1 1 . Consider1
1 , the only way
1
1 could have finite order is if
ing 1 1 1 1
2 or 1
2, in which case has order 2, and 2 is a factor of or .
1
Finally, the elements 1 can have as orders any factor of and the elements 1
can have as orders any factor of .
4 , so we consider
(b) If and are divisible by 4 then
4 with divisible by 4. Using the presentation (8), we can put any
2 4
2
1
2
4 in the form
with
,
,
0,
4 and with both
and in the cube group
4 4 . Assume has
finite order 1 and that in its conjugacy class (which of course all have the same
order), the smallest value of in the above representation is 2. (We will obtain a
contradiction to this.) By conjugation we eliminate from .
2 4
Now 4 4 can be partitioned: 4 4
, where
1
1
1
and 1 is the 8 element subgroup generated by 2 and . In detail,
t
u
v
t
w
í
x
y
z
x
|
}
{
~
U
U
¤
¤
¤
¥
¦
§
B
í
¢
£
¡
¬
¨
´
©
ª
­
®
¯
°
°
°
±
²
³
«
µ
¶
·
¸
¸
¸
¹
º
»
¾
¼
Ð
Ó
À
Á
Â
Â
Â
Ã
Ä
Å
È
½
Æ
U
Ò
¿
Ô
Ö
Ö
Ö
×
Ø
É
Ê
Ë
Ì
Ì
Ì
Í
Î
Ï
Ç
Ù
Õ
Ñ
Ú
Û
Ü
Ý
Þ
ß
à
á
â
ã
ä
ý
®
è
å
æ
é
ê
ë
ì
í
î
ï
ð
ô
ç
ñ
ò
õ
ö
÷
ø
ù
ú
û
ü
ó
ý
þ
ÿ
k
®
B
%
k
%
B
!
&
"
#
*
$
'
(
-
,
+
)
í
.
/
0
1
2
í
4
í
3
E
F
H
5
6
7
8
9
:
;
<
=
>
?
@
A
B
C
D
¶
I
G
J
K
L
N
O
P
U
Z
Z
Z
[
\
]
^
_
c
W
X
`
a
d
i
R
S
T
U
k
l
V
f
,
I
í
p
j
b
Y
e
o
M
s
Q
I
g
z
I
h
m
n
k
{
w
q
r
s
t
u
v
x
y
|
~
}
B
2
,
í
í
2
1
1
¡
¢
£
¤
£
Some power of
µ
¶
·
´
¸
¼
1
¹
º
»
3
¦
§
¨
2
©
ª
2
«
¬
­
2
®
2
¯
°
½
½
¾
¿
À
Á
Â
Ã
Ä
¾
¿
À
1
¼
Å
Æ
È
2
Ç
È
È
É
Ô
Õ
Ö
ã
à
Ó
ñ
ò
ó
ô
ø
1
õ
ö
÷
1
á
ù
ù
2
3
±
Ê
We consider the three cases: (i)
1 ; (ii)
(i) The factor in (10) is of the form 2
We alter (10) to
Ó
2
(9)
²
³
equals the identity element
½
2
¥
¤
ù
ú
û
2
ü
ý
þ
1
ÿ
2
ã
1
Ì
×
Í
Î
Ø
ä
å
æ
Ï
È
È
È
Î
Ð
Ñ
Ü
Ú
Ý
1
Þ
Û
s
(10)
Ò
; (iii)
with
0 1 and
1
Ù
ç
â
2
Ë
è
.
ß
ê
s
é
ë
ì
í
0 1 2 3
î
ï
ð
!
"
#
#
#
$
%
&
(11)
'
or
B
(
)
*
1
E
í
1
+
F
G
,
H
-
I
.
2
1
J
/
K
/
L
/
0
2
M
1
M
2
M
3
N
4
5
O
6
7
P
Q
R
4
=
8
S
9
:
T
;
U
<
1
V
W
>
X
?
Y
@
Z
A
2
B
4
[
C
\
D
]
2
^
_
`
a
a
a
b
c
(12)
d
and we know [RaS] this cannot be the case. So we cannot have
e
f
g
1.
35
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
2
,
(ii) is now of the form
alter (10) to
h
í
z
{
Using
¤
ª
2
~
¥
©
}
¤
¦
§
¨
¡
Ì
Ì
¬
­
Ì
2
®
Í
Î
Ï
¯
Ð
Ñ
¯
Ò
°
Ó
m
n
o
m
4
2
with
s
p
q
1
0 1 and
r
2
s
0 1 2 3 We now
u
w
x
Ó
s
t
v
y
(13)
, (13) becomes
§
¯
l
¢
1
«
1
k
j
1
|
1
£
i
±
²
1
Ô
Õ
Ö
×
³
´
Ø
Ù
µ
¶
·
1
Ú
Û
¸
¹
º
4
Ü
Ý
»
1
¼
2
Þ
ß
À
½
¾
¿
à
4
á
Á
Â
A
â
2
Ã
â
â
Ä
Å
Ã
À
1
ã
ä
Æ
Ç
È
4
É
Ê
Ë
2
Ì
Ì
Ì
(14)
å
Again, we know [RaS] this cannot be the case. So we cannot have
.
1
2. For if we represent conjugacy by
(iii) We cannot have
1 and
æ
ç
è
é
ê
w
í
ë
ì
ï
ð
í
ñò
î
ó
ôõ
ö
÷
1
ø
1
ú
2
û
ü
ü
ü
2
ý
þ
ÿ
2
1
1
01
ù
2
3
2
!
4
2
5
5
5
6
7
8
9
:
;
<
=
"
#
1
>
$
%
&
'
À
?
@
A
B
4
(
C
4
)
D
E
*
+
1
2
,
1
F
G
-
.
1
/
A
I
1
H
J
K
L
2
M
N
1
O
(15)
P
,
í
and is conjugate to a word with smaller .
0 or
1.
0 means
4 4 , and these have orders
Thus
1
where 1
4 and
4 4.
1 2 3 4.
1 means is of the form
We again consider the three cosets to which may belong. As before we see that
cases (i) and (ii) lead to infinite order for . But in case (iii) is conjugate to
2
2
2
1
, which can have as orders the factors of .
Q
R
s
S
`
a
s
I
T
c
U
V
W
X
Y
Z
[
\
]
^
,
d
I
I
t
I
u
l
m
n
o
i
_
b
e
f
g
h
j
v
p
q
r
s
k
w
x
z
y
{
~
|
}
5. Conclusion
2
¶
®
B
®
¶
v
We have been concerned with substitution tilings of Euclidean spaces, and have
defined an invariant for them related to the group generated by the relative orientations of the tiles in a tiling. This feature is captured in an intrinsic way by means
of a contractive behavior of the substitution. It is unrelated to other features of
tiling systems, such as their topology, and we introduce the notion of substitution
dynamical system to emphasize the features associated with the invariant.
To distinguish examples, for instance to distinguish the quaquaversal tilings
from the dite and kart tilings, requires consideration of 2-generator subgroups of
SO(3), in particular the orders of elements of such subgroups, which we analyze.
36
CHARLES RADIN AND LORENZO SADUN
Acknowledgements
We are pleased to thank Ian Putnam for useful discussions.
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-algebras associated with one-dimensional almost periodic tilings, Comm.
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[CEP]
[CoR]
[CoK]
[DwS]
[Gar]
[G-S]
[GrS]
[Kel]
[Ken]
[LaW]
[Min]
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¡
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[Moz]
¥
¤
[Ra1]
[Ra2]
[Ra3]
¦
§
¨
©
[RaS]
ª
«
[RaW]
¬
[Rob]
[Sad]
­
[Sch]
®
[Sen]
°
¯
37
INVARIANT FOR SUBSTITUTION TILING SYSTEMS
[Sol]
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[Tha]
[Wa1]
[Wa2]
