Isomorphism of hierarchical structures
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Ergod. Th. & Dynam. Sys. (2001), 21, 1239–1248
Printed in the United Kingdom
c 2001 Cambridge University Press
Isomorphism of hierarchical structures
CHARLES RADIN and LORENZO SADUN
Department of Mathematics, University of Texas, Austin, TX 78712, USA
(e-mail: radin,sadun @math.utexas.edu)
(Received 30 June 1998 and accepted in revised form 18 June 2000)
Abstract. We consider hierarchical structures such as Fibonacci sequences and Penrose
tilings, and examine the consequences of different choices for the definition of
isomorphism. In particular, we discuss the role such a choice plays with regard to matching
rules for such structures.
0. Introduction
This paper concerns certain hierarchical tilings of Euclidean space, of which a well
known example is a ‘kite and dart’ tiling of the plane [Gar]. Such tilings are noted
for two properties: their unusual rotational symmetries (‘ten-fold symmetry’), and the
fact that their global structures derive from local (‘matching’) rules. We will examine
the consequences of different choices for the definition of isomorphism between such
structures. To be more specific, we assume that any tiling of interest is embedded in a
compact metric space of tilings on which the connected component of the Euclidean
group (or sometimes just its translation subgroup) acts in the natural way, and has a
unique translation invariant Borel probability measure. We consider two specific notions
of isomorphism: we say the tiling spaces and are ‘topologically conjugate’ if there is
a homeomorphism between the spaces intertwining the actions of the group; and we say
and
, of measure
and are ‘measurably conjugate’ if there are subsets
zero with respect to the unique translation invariant measure and invariant under the action
onto
intertwining the actions of
of the group, and a bimeasurable map from
the group. We investigate the consequences of assuming one or the other of these notions
of isomorphism for such tilings.
In [RaS] we analyzed the rotational symmetry of such hierarchical structures, and found
a way to associate a useful invariant for measurable conjugacy; the methods used would
not have allowed for topological conjugacy, although perhaps other methods might. In
this paper we analyze hierarchical structures from the point of view of their other main
characteristic, the fact that their global structures derive from local rules, and here we show
that topological conjugacy is definitely not as broadly applicable as measurable conjugacy.
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C. Radin and L. Sadun
The above approach to analyzing tilings is a natural outgrowth of the way symbolic
sequences or arrays are studied, through embeddings in symbolic dynamical systems. The
specific matter we consider for tilings, concerning local rules, is actually better understood
in that older context. So we begin with more detailed descriptions of the relevant ideas
within symbolic arrays.
One result of this paper is the sharpening of a remarkable result of Shahar Mozes [Moz].
Mozes showed that every substitution subshift with 2 action, satisfying mild hypotheses,
is measurably conjugate via an explicit construction to a uniquely ergodic subshift of finite
type. (Such a result is commonly said to provide ‘matching rules’ for the functions in the
substitution subshift.) The map Mozes constructs is continuous (although not uniformly
continuous) after exclusion of appropriate sets of measure zero. A natural question is
whether such exclusion is an artifact of the proof or an essential feature of the theorem:
can Mozes’ theorem be strengthened to give a topological conjugacy? We answer this
question in the negative, exhibiting explicit substitution subshifts with 2 action which are
not topologically conjugate to any subshift of finite type.
After considering the simpler symbolic situation we explore the analogous questions for
actions. Mozes’ proof can probably be extended without
tiling dynamical systems with
major difficulty to subshifts in higher dimensions and to tiling systems in which each
polyhedron only appears, in any single tiling, in only finitely many orientations. However,
more general tiling systems, such as the pinwheel in 2 [Ra1] and quaquaversal tilings in
3 [CoR], require a significant extension of the proof in [Moz]. This was done for the
pinwheel in [Ra1] and then in general in [GoS]. In each case a measurable conjugacy
is constructed, but the map is only continuous after exclusion of certain sets of measure
zero. We demonstrate, as with subshifts, that these theorems cannot be strengthened to the
topological category: there exist substitution tiling systems that are measurably conjugate
to finite type tiling systems but which are not topologically conjugate to any finite type
tiling system.
In both settings, with subshifts and tilings, we give conditions, easily satisfied in natural
examples, for which such a topological conjugacy is not possible. However, the two
constructions are somewhat different for there are important properties of subshifts whose
natural analogues do not hold for tilings. For instance, for subshifts it is well known that
every topological conjugacy is a sliding block code while for tilings we show this to be
false.
This last point is both surprising and profound. The continuous nature of the
actions allows for two kinds of small changes to a tiling. Tiles may be changed near
infinity, or a small Euclidean motion may be applied to the entire tiling. Neither notion
is topologically invariant since, as we show, a topological conjugacy can encode pattern
information from distant regions as small Euclidean motions near the origin. This
mechanism, which has no analogue in the theory of subshifts (where the translation group
acts discontinuously), indicates that the nature of topological conjugacies of tilings is much
subtler than topological conjugacy of subshifts.
In §1 we discuss subshifts and show that many substitution subshifts are not
topologically conjugate to finite-type subshifts. In §2 we explore topological conjugacy
in tiling systems, giving an example of a conjugacy that is not a sliding block code. This
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Isomorphism of hierarchical structures
1241
example also demonstrates that local finiteness, a property automatic for subshifts, is not a
topological invariant for tilings. In §3 we return to the question of Mozes’ theorem, only
now for tilings, and show there exist substitution tilings that are measurably conjugate, but
not topologically conjugate, to finite-type tilings.
4
C
1. Notation and results for subshifts
0 and a finite abstract alphabet
For
onto
restriction of functions in
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That is,
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for some
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. We say the subshift is ‘of finite type’ if
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Given any second alphabet and a ‘block map’
by
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‘sliding block code (of code size )’
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it is a topological conjugacy. An equivalent description of a sliding block code of size
actions, such that, whenever
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T HEOREM 1. (Curtis–Lyndon–Hedlund [LiM]) Any topological conjugacy
subshifts and is a sliding block code.
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implies
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The following theorem seems to be widely known but does not appear to be in the
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subshift
T HEOREM 2. Assume the
subshift . Then is of finite type.
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is of finite type and topologically conjugate to the
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Proof. Assume
1
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both and
such that
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we can extend to
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We next define substitution subshifts and show that certain substitution subshifts are not
of finite type. By Theorem 2 they cannot be topologically conjugate to subshifts of finite
type, but by Mozes’ theorem they are measurably conjugate to subshifts of finite type.
¯
1242
C. Radin and L. Sadun
we define the set of ‘words’ as
We begin in dimension one. Given an alphabet
. We assume we are given a ‘substitution function’
such
1
for
some
.
A
word
of
the
form
is
said
to
be
a
‘letter
of
that
2
the set of all such words. Finally, we define
level ’ for any
0, and we denote by
the substitution subshift associated with the substitution as
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(2)
It is a simple fact that every function in such a subshift can simultaneously be considered a
for each value of
0. In interesting cases functions in a
function with values in
substitution subshift can be considered functions with values in letters of each higher level
in only one way; such subshifts are said to be ‘uniquely derivable’. (This is the origin of
the term ‘hierarchical’ for these structures. The same hierarchical phenomenon exists for
tilings, and in a manner easier to understand, which we illustrate in Figure 2.)
subshifts as long as the images of
The above has a straightforward generalization to
the substitution function fit together geometrically, so that one can iterate the substitution.
substitution subshifts that are products of substitution subshifts,
This is automatic for
which we illustrate by an example below.
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C OROLLARY 1. If a
contains a function such that
substitution subshift
2 , and
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some
is topologically conjugate to a
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1
1
finite-type subshift, then contains a periodic function.
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defines a word
Proof. The condition on
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consider the periodic function defined by the condition
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1 0
1
for all
. Since
is topologically conjugate to a subshift of finite type, by
. Taking
large enough it follows
Theorem 2 there is some
1 such that
that contains the periodic function .
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C OROLLARY 2. There exists a 2 substitution subshift that is measurably conjugate to a
finite-type subshift, but is not topologically conjugate to any finite-type subshift.
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Proof. Consider the uniquely derivable symbolic Fibonacci substitution defined, using the
alphabet
, by the substitution function
; let 1 be
2
be the
the strictly ergodic substitution subshift thus determined. Let 2
substitution subshift which is the product of 1 with itself:
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It follows [Moz] that 2 is uniquely derivable. Mozes’ theorem states that under rather
which is uniquely
general conditions (satisfied by
2 ) a 2 substitution subshift
derivable is measurably conjugate to some finite-type subshift, so
2 is measurably
conjugate to a finite-type subshift.
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Isomorphism of hierarchical structures
1243
, and hence all functions in 2 ,
A simple calculation, however, shows that 23
, and a standard argument ([Ra2, Ra3]) using the unique
contain the block
derivability of 2 implies that 2 contains no periodic functions. Corollary 1 then
implies that 2 is not topologically conjugate to any finite-type subshift.
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2. Conjugacies of Fibonacci tilings
of polyhedra in
Now we consider ‘tiling systems’. Given a finite collection
as the space of all tilings of Euclidean -space by congruent
dimensions, we define
copies of elements of , that is, congruent under the connected Euclidean group. (We
is non-empty.) We put a metric
on
as follows:
assume
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denotes the intersection of two sets: the closed ball
of radius centered
where
at the origin of the Euclidean space and the union
of the boundaries
of all tiles
is the Hausdorff metric on compact sets defined as follows. Given two compact
in .
,
max
, where
subsets and of
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with
denoting the usual Euclidean norm of . (We assume the tiles are small enough
is non-empty for any .) It is not hard to show that with this metric
is
so that 1
compact and that the natural representation of the connected component of the Euclidean
is continuous. Finally, let
be either the group of translations of
or
group on
the connected component of the Euclidean group. A tiling system is a closed -invariant
.
subset of
Two important properties of tiling systems are ‘local finiteness’ and ‘finite type’. For
0 and polyhedron
appearing in the tiling of
consider the set
each
of polyhedra in which intersect the open ball of radius centered at the
center of mass of . Define
to be the union of all such ‘neighborhoods of radius
’ for all
and
. A tiling system is ‘locally finite’ if, for every
0, the
set
is finite up to the action of . In practice, tiling systems are almost always
assumed to be locally finite.
to be the set of all in
such that
Now, in analogy to the subshifts , we define
only has neighborhoods of radius which are in
. A tiling system is said to
for some
0.
be ‘of finite type’ if
We will be considering the analogue among tiling systems of the substitution subshifts.
Traditionally such substitution tiling systems are defined through a substitution function as
for subshifts, whereby for each letter (polyhedron) there is given a way to decompose it
into polyhedra similar to those in the alphabet, all smaller by some factor
1, so that
following the decomposition by a stretch about the origin by 1 one associates to each
letter of a letter of level 1, and so on. For example, Figure 1 shows a substitution for
a ‘chair’ tile, in this case with
1 2, and Figure 2 shows the hierarchical structure
in a piece of a chair tiling. Applying the substitution to all tiles in a tiling gives an
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1244
C. Radin and L. Sadun
F IGURE 1. The chair substitution.
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F I GURE 2. 63 letters of level 0, 14 letters of level 1 and 2 letters of level 2 in part of a chair tiling.
Í
automorphism
. Notice that
commutes with rotations about the origin
, where
is a translation by
.
and that
We use this last property to define a generalized notion of substitution tiling system. For
our purposes, a substitution tiling system is a tiling system with an automorphism and a
1 such that commutes with any rotations about the origin in and such
constant
for all
.
that
A simple example of a substitution tiling is the standard Fibonacci tiling in one
dimension. In this tiling the alphabet consists of two tiles (intervals), and , of length
5 2. The substitution consists of expanding about the
1 and
1
origin by a factor of , and then replacing each expanded (of size ) with a , and
) with an and a . The resulting space of
replacing each expanded (of size 2 1
tilings is precisely the set of tilings by ’s and ’s arrayed in the sequence of the Fibonacci
subshift 1 .
This last characterization allows us to define general Fibonacci tilings, based on any
two tiles and , as the set of tilings in which the pattern of ’s and ’s is the same as
a sequence in the Fibonacci subshift. If
this is not a substitution tiling in the
is not an expansion by a constant
traditional sense, as the substitution
factor. However, it is a substitution tiling system in the extended sense of this paper.
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Isomorphism of hierarchical structures
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1245
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and
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T HEOREM 3. Let and be Fibonacci tiling systems, with tiles
then and are topologically conjugate.
respectively. If
Moreover, the conjugacy is not a sliding block code.
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Proof. We define a map
as follows. A tiling
defines an element of
1 (up to translation), which in turn defines a tiling
, up to translation. To fix
the translation we pick a tile
. This tile is a letter of level 0, which sits inside a
letter 1 of level 1, which sits inside a letter 2 of level 2, and so on. We approximate the
placement of the tiles in by lining up the middle of the letter in that corresponds to
with the middle of . Since
, the letters of level in the two
. The adjustment in translation between the
tilings differ in size by a constant times
th approximant and the (
1)th is then exponentially small, and we can take the limit as
to obtain
. Loosely speaking, the map lines up the corresponding letters of
infinite level.
We must show that this map is well defined, in that it does not depend on the choice
is another tile, then there are two possibilities. The more common
of tile . If
, so the approximants to
based
possibility is that there is a level at which
on are eventually the same as the approximants based on . Alternatively, there may be
a point between and which is the endpoint of letters of arbitrarily high level. It is not
hard to check that, working from either or , aligns this special point in with the
corresponding special point in
.
The map is clearly a bijection which commutes with translations. It is continuous
since a small change to a tiling , either via a small translation or by changing the tiles
near infinity, results in a small change to
. A small translation in turns into a small
, while changing the distant tiles of changes the distant tiles of
,
translation in
and also causes a small translation.
This last point means that is not a sliding block code. If two tilings and agree
exactly on a large neighborhood of the origin but disagree near infinity, then the sequence
will agree near the origin, but the placement of the tiles will
and
of tiles of
differ by a small translation. Thus there is no neighborhood of the origin where
and
agree exactly.
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Let
have some fixed value. There are two special Fibonacci tilings. One is
. The other has
, and thus closely
the standard Fibonacci tiling with
resembles the Fibonacci subshift 1 . By Theorem 3 any Fibonacci tiling is conjugate to
each of these. In particular, we have the following.
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4
C OROLLARY 3. Every Fibonacci tiling system is a substitution tiling system.
Proof. Every Fibonacci tiling system is topologically conjugate to a Fibonacci system
. As noted earlier, is a substitution system with automorphism
with
and scale factor
1 . If is another Fibonacci tiling system and
is the
1
is a substitution automorphism on .
conjugacy of Theorem 3, then
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Unlike a sliding block code the domain of a topological conjugacy
for sufficiently large. Suppose for
of tiling systems cannot always be extended to
1,
,
2 and
example that and are Fibonacci tiling systems with
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1246
C. Radin and L. Sadun
1. For any the system
contains periodic tilings, whose period is an integral
have integer length, there
linear combination of 1 and . Since the tiles of
with period , and so there are no maps
that
are no periodic tilings in
intertwine translations.
Since topological conjugacies of tilings are more general than sliding block codes,
topological invariants of subshifts are not automatically topological invariants of tilings.
For example, we have the following theorem.
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T HEOREM 4. There exist topologically conjugate tiling systems
locally finite and is not.
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Proof. We work in two dimensions, with being the translation group. Given two square
and
of unit size, with edges parallel to the coordinate axes, let be the set
tiles
of all tilings such that the tiles meet full edge to full edge, and such that each (horizontal)
, where
row of the tiling is a (one-dimensional) Fibonacci tiling. Let
is a rectangle of height 1 and width and
is a rectangle of height 1 and width
1.
Let be the conjugacy of Theorem 3, taking the one-dimensional Fibonacci tiling with tile
1. This map is
sizes 1 and 1 to the one-dimensional Fibonacci tiling with sizes and
naturally extended to a map from rows of the tiling to Fibonacci-like rows made up of
. Applying this to every row of a tiling
defines an extended map (also
the tiles of
. Let be the image of this map. Near the origin (or any other
called ) from into
point), the rows of a tiling
will appear shifted relative to one another. Since there
are an infinite number of ways in which this shift can occur, is not locally finite.
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3. Conjugacies of tiling systems of finite type
In this section tiling systems are assumed to be locally finite unless stated otherwise.
A natural question, in the light of Mozes’ theorem, is whether Theorem 2 can be
extended to the category of tiling systems. Certainly the proof breaks down, as conjugacies
.
of tilings are not necessarily sliding block codes and need not extend to spaces
However, there is hope that the statement of the theorem remains true.
In this section we prove a somewhat weaker result, the analogue of Corollary 1, thereby
showing that there exist tiling systems that are measurably conjugate to finite-type systems
but not topologically conjugate. The key is showing that topological conjugacies, while not
sliding block codes, do preserve some useful geometrical information. One can model a
topological conjugacy
between tiling systems and —with metrics
and alphabets and —by something close to a sliding block code, as follows.
and
First, we note that is uniformly continuous. So given
0 there is some
0 such
provided
. Now it is known [RaW] that given any
that
it follows that there is a rigid motion
0 there is an
0 such that if
‘agree to distance 1
within distance of the identity in such that the tilings and
around the origin’, in the sense that they match perfectly in their polyhedra which intersect
0 such that if
the ball of radius 1 centered at the origin. Furthermore, there is some
and so the
tilings and in agree to distance around the origin then
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Isomorphism of hierarchical structures
3
tilings
less than
È
agree to distance 1 around the origin for some
and
from the identity. We have proven the following theorem.
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1247
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of distance
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T HEOREM 5. Let be a topological conjugacy of the locally finite tiling system in
onto the locally finite
tiling system . Given
0 there is some
0 such that
around the origin then the tilings
and
if tilings and in agree to distance
agree to distance 1 around the origin for some rigid motion of distance less
than from the identity in .
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be a topological conjugacy of locally finite tiling systems
C OROLLARY 4. Let
0 and a bounded connected subset of
there exists
0
in . Then given
there is a rigid motion of distance less
such that if tilings and in agree on
agree on .
than from the identity in such that
and
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Proof. The proof follows immediately by applying Theorem 5 to all the balls of radius
which intersect , and noting that their overlapping regions lead to consistency
requirements in .
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We say that a tiling in an 2 tiling system with alphabet has a ‘periodic frame’
and
if there is some pair
of linearly independent translations for which
1
agree to distance 1 about the origin for all 0
1 and
1 8, and
1
and
agree to distance 1 about the origin for all 0
1 and
also
1 8.
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T HEOREM 6. If an 2 substitution tiling system contains a tiling with a periodic frame,
and is topologically conjugate to a finite-type tiling system, then contains a periodic
tiling.
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0 is given and the frame is described by the defining notation above.
Proof. Assume
By using the substitution repeatedly if necessary we can assume without loss of generality
that min
is larger than any given number. So we can assume and
where 1
agree on 1
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1
1 8 and that and
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where 2
agree on 2
0
1
1 8 . Applying Corollary 2 we
agree on 1 and that
and
agree on 2 for rigid
see that
and
motions and within distance of the identity of . Note that, because of the overlap
between them, it follows that and are pure translations and so the central lines of 1 ,
is of finite type this implies
1 , 2 and
2 form a parallelogram. And since
that it contains a periodic tiling. But then so does .
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Remark. At the end of §1 we showed that the subshift with 2 action made as a product
of the Fibonacci sequences with itself could not be topologically conjugate to a finitetype subshift. Theorem 6 allows us to apply the same argument to tilings. Starting with
any Fibonacci tiling system we can define the product tiling system with 2 action and
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1248
C. Radin and L. Sadun
show that it contains a tiling with a periodic frame, and therefore cannot be topologically
conjugate to a finite-type tiling system since it does not contain a periodic tiling.
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4. Conclusion
The best known tiling dynamical system is that of the Penrose kites and darts [Ra2, Ra3].
By this, one refers to both a substitution system and a finite-type system, which are in fact
topologically conjugate. The fact that the conjugacy is topological has lead to interesting
work in non-commutative topology [AnP, Con].
To a large extent, the interest in substitution tiling systems is due to the fact that
they are conjugate to finite-type systems. Theorem 6 shows that, when dealing with
substitution tilings and their conjugacy to finite-type tilings, for reasonable generality one
cannot consider the tilings well defined up to topological conjugacy—although of course
measurable conjugacy is sufficient. One issue we have not resolved is whether or not the
notion of finite type is itself a topological invariant among tiling systems.
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Acknowledgements. We thank Klaus Schmidt for showing us the proof of Theorem 2.
After this work was distributed we learned of independent work in progress [Pet], which
has some overlap with ours concerning Curtis–Lyndon–Hedlund-type theorems for tiling
dynamical systems.
Charles Radin was supported in part by NSF grant No. DMS-9531584 and Texas ARP
grant 003658-152. Lorenzo Sadun was supported in part by NSF Grant No. DMS-9626698
and Texas ARP grant 003658-152.
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R EFERENCES
[AnP]
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[Con]
[CoR]
[Gar]
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[GoS]
[LiM]
j
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j
[Moz]
l
[Pet]
[Ra1]
[Ra2]
m
j
n
i
[Ra3]
[RaS]
[RaW]
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