DIRECTED GRAPHS AND SUBSTITUTIONS
CHARLES HOLTON AND LUCA Q. ZAMBONI
A BSTRACT. A substitution naturally determines a directed graph with an ordering of the edges incident at each vertex. We describe a simple method by which any primitive substitution can be modified
(without materially changing the bi-infinite fixed points of the substitution) so that points in the substitution minimal shift are in bijective correspondence with one-sided infinite paths on its associated
directed graph. Using this correspondence, we show that primitive substitutive sequences in the substitution minimal shift are precisely those sequences represented by eventually periodic paths. We use
directed graphs to show that all measures of cylinders in a substitution minimal shift lie in a finite union
of geometric sequences, confirming a conjecture of Boshernitzan. Our methods also yield sufficient
conditions for a geometric realization of a primitive substitution to be “almost injective”.
1. I NTRODUCTION
Every substitution has an associated prefix automaton, a directed graph whose vertices are the
letters of the substitution’s alphabet and whose edges are labeled with the strict prefixes of the images
of the letters under the substitution. In this paper we consider the directed graph obtained by reversing
the directed edges in the prefix automaton. This graph is more natural from the point of view of
iterated function systems.
Vershik introduced the notion of an adic transformation defined on the infinite path space of a
Bratteli diagram as a means of studying operator algebras. Livshitz outlined a connection between
substitution minimal systems and stationary adic transformations, the underlying spaces of which are
essentially the infinite path spaces on the associated directed graphs: every substitution minimal shift
is metrically isomorphic to a stationary adic transformation ([24]). Forrest ([17]) and Durand, Host
and Skau ([14]) proved a converse, that every stationary proper minimal Bratteli-Vershik system is
conjugate to either a substitution minimal system or a stationary odometer system.
In this paper we describe a simple but completely general way of modifying a primitive substitution
so that the substitution minimal shift is homeomorphic to the space of one-sided infinite paths on
its directed graph. The induced dynamical system on the infinite path space of the graph is just
the continuous extension of the Vershik map. Other constructions of this kind exist but are more
complicated (see, for example, [14, 17, 18]).
A primitive substitutive sequence is, by definition, an image under a morphism of a fixed point
of a primitive substitution (see [13, 20]). Directed graphs give a new way to characterize primitive
substitutive sequences. As part of the characterization, we show that a point in the minimal shift
arising from a primitive substitution is primitive substitutive if and only if it is represented by an
eventually periodic path in the graph associated to the substitution. We also show that all primitive
substitutive sequences in any minimal shift can be completely described by a single directed graph.
As another application of the directed graph construction, we establish the following conjecture of
Boshernitzan: the measures of cylinders in a substitution minimal shift lie in a finite union of geometric sequences[6]. This is an unsurprising reflection of the self-similarily inherent in substitution
dynamical systems.
Many substitutions admit geometric realizations equipped with piecewise translation mappings (see
[19]). Ei and Ito gave a characterization of those substitutions on two letters whose geometric realization is an interval exchange ([15]). In such cases the substitution minimal system is a symbolic
1
2
CHARLES HOLTON AND LUCA Q. ZAMBONI
representation for the interval exchange transformation. It is natural to ask which substitution dynamical systems admit an “almost injective” geometric realization in the sense of Queffélec [31]. In the
last section we use the graph directed construction to obtain some partial results.
ACKNOWLEDGEMENTS
The second author was supported in part by NSF grant INT-9726708 and a grant from the Texas
Advanced Research Program. We are grateful to B. Host and F. Durand for their helpful comments
and suggestions. We also thank A. Forrest for useful e-mail exchanges.
2. S UBSTITUTIONS
By an alphabet we mean a finite nonempty set of symbols. We refer to the members of an alphabet
as letters. A word of length in the alphabet is an expression
where each is a
letter of
We write
for the length of a word It is convenient to define the empty word having
length and satisfying the concatenation rule
for every word Let
be the set of all
words of positive length in the alphabet and set
A morphism defined on alphabet is a map
where is an alphabet. The length of
is defined to be
We say is a letter-to-letter morphism if
and is called nonerasing if
A morphism
determines by concatenation a map
also denoted A substitution is a nonerasing morphism
A substitution on
alphabet determines a map
which we also denote by With this slight abuse of notation
on ,
, and
in the obvious way.
we define for
Associated to a substitution on is the
matrix
with th entry equal to the number
of occurrences of the letter in the word
The substitution is primitive if there exists a positive
the word
contains every letter of
Equivalently, is
integer such that for each letter
is a primitive matrix. If is primitive then
primitive if and only if
(1)
as
is a periodic point of if for some positive integer we have
A sequence
Every primitive substitution has a periodic point. Denote by
the set of periodic points of
For
and
we write
for the word
(such words
and
analogously, with the convention that
are called factors of ). Define
Similarly, if
and
then we set
and use the obvious
interpretations for
and
The topology of
is metrizable as the countable product of the discrete space
Specifically, we
define a metric on
by setting
if is the greatest integer for which
The language of a substitution is the set
of all factors of the words
The subshift of
spanned by the language
is
and
for all
It follows from (1) that if is primitive then
is an infinite set and
is nonempty. The
The shift map on
is
cylinder determined by a word is
It is easy to see that
is a closed subset of
is a homeomorphism
defined by
and if is primitive then the dynamical system
is minimal, i.e., every point
has a dense orbit under the shift map (see [31]).
DIRECTED GRAPHS AND SUBSTITUTIONS
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3
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F IGURE 1. Directed graph associated to the substitution in Example 3.1.
3. D IRECTED
GRAPHS ASSOCIATED WITH A SUBSTITUTION
A directed graph consists of a finite nonempty set of vertices
and a finite set
of edges together with maps
A walk or path on is a finite or infinite sequence of
edges
such that
for all relevant indices A finite walk
starts at its
initial vertex
and ends at its terminal vertex
We say that is strongly connected if for
there is a finite walk on starting at and ending at A strongly
every ordered pair of vertices
connected directed graph is said to be aperiodic if there are two walks of relatively prime lengths with
the same initial vertex and the same terminal vertex.
The adjacency matrix of is the
matrix
with th entry
equal to the
is the number
number of edges starting at vertex and ending at vertex It is easy to see that
of paths of length starting at and ending at Thus, is strongly connected if and only if
is
irreducible, and is aperiodic if and only if
is primitive.
the space of (one-sided) infinite walks on
viewed as a subspace of the countable
Denote by
product of the discrete edge set. With this topology
is compact.
We associate to a directed graph
with
Let be a substitution on an alphabet
vertex set
and the edge set
and
where
and
These graphs were independently studied by Canterini and Siegel
is equal to the incidence matrix
Thus is primitive if and
in[8]. The adjacency matrix
only if
is aperiodic.
Example 3.1. The Thue-Morse substitution has alphabet
associated graph is shown in Figure 1.
and
Our construction naturally determines a partial ordering on
and only if they have the same terminal vertex. Specifically, for
in which edges are comparable if
(2)
The
and
Conversely, a directed graph having at least one edge ending at each vertex, together with an
on alphabet
ordering on the set of edges ending at each vertex, determines a substitution
given by
where
are the edges ending at
It is readily verified that
and
Suppose is a primitive substitution on alphabet and is the associated directed graph with
partial ordering on the edge set given by (2). We also denote by the partial ordering on
defined by
and
An element of
is minimal (maximal) if it is minimal (maximal) with respect to this partial ordering. Note that for each vertex and for each positive integer there is a unique walk of length
ending at and consisting entirely of minimal edges (ditto for maximal). Consequently,
Proposition 3.2. Minimal and maximal elements always exist and are periodic (as sequences of
of each.
edges), and there are, at most,
4
CHARLES HOLTON AND LUCA Q. ZAMBONI
Every nonmaximal element has a unique successor, and we write for the successor map
maximal walks
That is, if
is not maximal and is the least index for
is not a maximal edge then
which
(3)
where
4. W ORDS
AND WALKS
Fix a primitive substitution on an alphabet such that
is infinite. Our first order
of business is to give a method for coding points of by infinite walks on
The coding
is suggested by the following lemma which states that all primitive substitutions have the unique
admissable decoding (UAD) property of [25].
Lemma 4.1. Let
some integer
Then there is a unique point
Moreover, the map
such that
is continuous.
for
Proof. The original idea for the proof is probably due to Martin in [26]. The result can also be inferred
from work of Mossé in [30]. Also see [14] for more details.
We record a useful fact whose proof is straightforward.
is a walk on
Lemma 4.2. If
and
with
then
where
We define a map
Thus
starts at vertex
which we call the coding map of
by
if and only if
Proposition 4.3. The coding map
is a continuous surjection satisfying
(a)
(b)
whenever the latter is defined.
is minimal if and only if
(c)
(d)
is maximal if and only if
(e) The infinite walk
begins in
if and only if
Proof. The image of
is dense in
for if
is a finite walk on and
is any point with
then
is a point of and
begins in
Surjectivity follows from compactness of
continuity of
being obvious.
To prove (b) suppose
Assertions (a) and (e) follow immediately from the definition of
and
is not maximal. Let be the least index for which
is not maximal. Let
be the unique point of for which
and
For any
(4)
we have
DIRECTED GRAPHS AND SUBSTITUTIONS
5
and repeated application of this identity yields
Putting
we see from (a) that
which is, according to (3), the successor of
To prove (c) we observe that
is minimal if and only if
Assertion (c) now follows from the equality
is in the image of
for all
(5)
To prove (d) we note that by (4)if
is maximal then
is in the image of for every positive
must be in
Conversely, if
is not maximal then (b) implies that
integer by (5),
is a successor and hence not minimal, whence it follows from (c) that
is not in
We wish to extend to a continuous map on all of
so that becomes a conjugacy. However,
the map need not be one-to-one. If, for instance, has more periodic points than
has minimal
elements, then part (c) of Proposition 4.3 shows that
cannot be one-to-one. The substitution of
Example 3.1 illustrates this possibility; it has four fixed points but its graph has only two infinite
minimal walks. One can show that is bijective if and only if
minimal elements of
maximal elements of
To extend the successor map we shall modify the substitution so that the resulting coding map
is one-to-one. Let
be the subset of
consisting of the words of length 3. We define
a substitution
on
by setting for
Note
sequence in the letters of
Write
for the shift map on
with th entry
Proposition 4.4. With the above notation,
Moreover
and
Denote by
and set
then
must satisfy
is primitive and
For
denote by
It is easy to see that
the
is a conjugacy
the coding map of
Let
By Lemma 4.2 and (e) of Proposition 4.3, if
so, by Proposition 4.4,
Any
and therefore
This together with (e) of Proposition 4.3 implies
Thus,
is seen to be a homeomorphism and the successor map
initially defined on the nonmaximal members of
may be extended to a continuous homeomorphism (also denoted
)
on all of
In this way,
sends each maximal path in
onto a minimal path. We
summarize this discussion as follows.
6
CHARLES HOLTON AND LUCA Q. ZAMBONI
Proposition 4.5. If
conjugacy
conjugacy.
is a primitive substitution then there exists a primitive substitution
such that
and the coding map
and a
is a
For the remainder of the paper we shall assume that is a primitive substitution on the alphabet
and that
is an infinite set conjugate to
via the coding map
5. C HARACTERIZING
LEFT SPECIAL SEQUENCES
Let be the set of all pairs
for which
we show that points which appear in an ordered pair of
eventually periodic paths in the graph.
Proposition 5.1. If
eventually periodic.
are distinct points of
and
In this section
are special in that they are represented by
with
Proof. Suppose
Then
such that
must be a nonnegative integer
using Lemma 4.1 that
and
we must have
some integers
then
and
and
are
by Lemma 4.1, so there
It can be shown
defines a permutation of and thus for
From (a) and (b) of Proposition 4.3 we obtain
(6)
is in the orbit of a minimal or maximal element, then
is eventually periodic by PropoIf
sition 3.2. If
is not in the -orbit of a minimal or maximal element of
then
and
differ in only finitely many coordinates; this together with (6) shows that
is an
eventually periodic sequence of edges.
Remark 5.2. Related to
on given by
write
is the number
for the equivalence class of
defined as follows. Let
be the equivalence relation
and set
Queffélec showed that this sum is finite [31], but no sharp bound is known. We conjecture that
The conjecture holds when
and in this case there is a simple algorithm for
finding all pairs of The question seems to be open for
6. P RIMITIVE
SUBSTITUTIVE SEQUENCES
In [13] Durand adapted the notion of automatic sequences to the context of primitive substitutions:
a one-sided infinite sequence is called primitive substitutive if and only if it is the image of a fixed
point of a primitive substitution under a letter-to-letter morphism. Durand [13] showed that primitive
substitutive sequences are precisely those minimal sequences which admit only finitely many derived
sequences on prefixes (see Theorem 6.5 below). This description was generalized by Holton-Zamboni
in [20] to obtain a characterization of primitive substitutive subshifts. In this section we extend the
notion of primitive substitutive to bi-infinite sequences in the natural way, and derive an alternative
characterization in terms of the associated directed graph. A two-sided infinite sequence is called
primitive substitutive if it is the image of a fixed point of a primitive substitution under a letter-toletter morphism. The main result of this section is
DIRECTED GRAPHS AND SUBSTITUTIONS
7
Theorem 6.1. Let be a primitive substitution, and a morphism taking
to an infinite minimal
shift space
Then, for every
the sequence
is primitive substitutive if and only if
is eventually periodic.
We begin with a few technical remarks. Durand showed that the hypothesis that the morphism be
letter-to-letter in the definition of primitive substitutive may be relaxed (see [13] Proposition 3.1) to
arbitrary nonerasing morphisms. Durand’s proof extends immediately to the bi-infinite case:
Lemma 6.2. Let
letter occurring in
be primitive substitutive. If
then
The next lemma characterizes the sequences in
The proof is straightforward and left to the reader.
Then
Lemma 6.3. Let
-orbit for some positive integer
is a morphism such that
is primitive substitutive.
for some
represented by eventually periodic paths in
is eventually periodic if and only if
and
lie in the same
Our proof of Theorem 6.1 requires several lemmas. We begin with some terminology adapted from
[13]. Let be a minimal shift space and let be a factor of some (hence every) sequence of
A
return word to in is a nonempty word such that
is a factor of some sequence of
if and only if
Denote by
the set of return words to in It follows from minimality of that
is
finite. Durand showed that primitive substitutive shifts are uniformly recurrent:
is infinite and contains a primitive substitutive
Lemma 6.4. (F. Durand, [13] Theorem 4.5) If
sequence then there exists a positive constant such that for all every return word
satisfies
Fix, for each a bijection
written uniquely as a concatenation
If
with
then
can be
each
and we obtain the derived sequence of
with respect to
Derived sequences are unique up to permutation of their alphabets, and Durand’s result holds for
bi-infinite sequences:
Theorem 6.5. (F. Durand, [13] Theorem 2.5) Let
primitive substitutive if and only if
be a minimal shift space. A sequence
is finite.
is
Remark 6.6. In case
has only finitely many derived sequences, Durand’s proof gives a method
and a letter-to-letter morphism
for constructing a primitive substitution fixing a point
mapping
onto
with
In view of Theorem 6.1 this is all one needs to identify all
primitive substitutive sequences in
A bi-infinite sequence is primitive substitutive if and only if
is primitive substitutive (see
Lemma 2 in [21]). We introduce the notion of derived orbit to reflect this fact. Let be a minimal
shift and let
and
be as above. If
then (by minimality of ) there is a point in the
and we set
equal to the shift orbit of the sequence
shift orbit of such that
Clearly the derived orbit
depends only on the orbit of
The following lemma, together with Lemmas 6.2 and 6.3, establishes the ‘if’ part of Theorem 6.1.
Lemma 6.7. If
is such that
lies in the shift orbit of
then
is primitive substitutive.
8
CHARLES HOLTON AND LUCA Q. ZAMBONI
Proof. Replacing with
if necessary and using Lemma 4.2 we may assume that
for some integer
If
or
then or
is fixed by in which
Put
let
be the
cases is primitive substitutive. Thus we consider
set of return words to in
and let
be a bijection. If
then
is a factor of which begins and ends in (since
) and therefore
can be written uniquely as a concatenation of return words. We may thus define a
substitution on
in the following way: if
then we may write
each
and we set
We leave it to the reader to verify that is a primitive substitution fixing
completed by invoking Lemma 6.2 with
The proof is
We now turn our attention to proving the ‘only if’ part of Theorem 6.1. We show
(7)
is primitive substitutive
is primitive substitutive
has only finitely many derived orbits
and
are in the same shift orbit.
We begin by paraphrasing a result from [20].
Theorem 6.8. ([20], Theorem 1.3) If is a minimal shift space containing a primitive substitutive
sequence then there is a finite set of disjoint minimal shift spaces whose union contains every derived
sequence of every sequence of Each of these shift spaces contains a primitive substitutive sequence.
To establish the first implication of (7) we shall need the following key fact.
Proposition 6.9. If is a minimal shift space containing a primitive substitutive sequence and is
a morphism defined on the alphabet of such that
is infinite then
is boundedto-one.
Proof. The hypotheses imply that is infinite. Let be as in Lemma 6.4, so that every word of length
in the language of occurs in every word of length
in the language of Since some letter in
for
the alphabet of is not sent by to the empty word, we must have
every word in the language of
Suppose that
are distinct sequences of
all sent to the same sequence by
such that
For all large enough the words
are distinct. Fix such
For any two of the words
be the shortest of these words. Likewise
is a prefix of each of these words. Put
Now let be a word in the language of
Each of the words
one must be a suffix of the other; let
the shortest of the words
Then
of length
Then
is a word of length
must occur in and if
then
is a suffix of
and
is a prefix of
Since
there must
be at least occurences of the word in
Let be the union of the first
shift images of
Then is an infinite minimal shift space which by Lemma 6.2 contains a primitive substitutive
sequence. Denote by
the return constant for guaranteed by Lemma 6.4. Then
contains at
least
return words to in and by Lemma 6.4 we have
Thus
DIRECTED GRAPHS AND SUBSTITUTIONS
Finally, if
and is the least nonnegative integer for which
the th shift image of then
It follows that
to-one.
9
then
is at most
and if
is
-
The next proposition is a partial converse of Lemma 6.2.
Proposition 6.10. Let be a minimal shift space containing a primitive substitutive sequence and
suppose is a morphism such that
is infinite. If
is such that
is primitive substitutive
then is primitive substitutive.
Proof. Let
and
be as in the proof of Proposition 6.9. For all sufficiently large
if
then
may be written uniquely as a concatentation of elements of
This
having the property
defines a morphism
that
By Lemma 6.4
for any
This implies that
for
from which we obtain the crude bound
valid for
any
It follows from the bound on the
that there are only finitely many distinct maps
By Durand’s theorem, there are only finitely many different derived sequences
and each is primitive substitutive. By Theorem 6.8, there is a finite disjoint collection of minimal
shift spaces
each containing a primitive substitutive sequence, with the property that
any derived sequence of a sequence in is contained in one of them. If
then
is
defined on
and takes it to an infinite subset of the closure of the shift orbit of
By
Proposition 6.9, there are only finitely many sequences in
which are sent to
by
and
is one of them. Thus we see that there can be only finitely many distinct sequences
Lemma 6.11. A sequence belonging to a minimal shift space
if
a factor of
is finite.
is primitive substitutive if and only
Proof. Suppose first that is primitive substitutive. We will construct finite collection of sequences
be the least
which contains a member of each derived orbit. Let be any factor of and let
positive integer for which is a factor of
Then
ends in and
where is
then
begins in
and therefore
the constant from Lemma 6.4. If is a return word to
begins and ends in Thus
can be written uniquely as a
concatenation of return words to
each
It follows from Lemma 6.4 that
Consider the morphism
given by
One checks easily that
is in
It follows from the bound on the
that
there are only finitely many different possible maps
and by Durand’s theorem there are only
finitely many distinct derived sequences
Thus,
a factor of
is a
finite set containing a representative of each derived orbit.
a factor of
is finite. We may assume that is not periodic. Then
Now suppose
we can find positive integers
for which
and every return word to
is longer than the longest return word to
Every return word to
can be written
10
CHARLES HOLTON AND LUCA Q. ZAMBONI
uniquely as a concatenation of return words to
and thus we may define a substitution
by putting
It is easy to verify that
is primitive, and if is the integer for which
holds for all then
for all
Lemma 6.7 asserts that
is primitive substitutive, and an application of Lemma 6.2 with
completes the proof that is primitive substitutive.
We recall a well-known inequality (see [31]). Since is primitive the matrix
is also primitive
and the Perron-Frobenius theorem guarantees a positive eigenvalue strictly greater in absolute value
than any other eigenvalue of
Lemma 6.12. There exist positive constants
every word and for every positive integer
such that
for
The next lemma completes the proof of Theorem 6.1.
Lemma 6.13. A sequence
exists a positive integer such that
has only finitely many distinct derived orbits if and only if there
is in the shift orbit of
Proof. If
is in the shift orbit of then by Lemma 6.7 is primitive substitutive, whence it has
only finitely many distinct derived orbits, by Lemma 6.11.
Now suppose is primitive substitutive. Let be any factor of and consider the array
..
.
..
.
..
.
..
.
in the first column, by Lemma 6.11.
There are only finitely many distinct entries
then
begins in and is a factor of
It follows that
is a factor of
If
which begins and ends in
and thus
may be written uniquely as a concatenation of words
This induces a morphism
defined for
of
by
where
each
is the unique representation of
verify that for any
Thus, for each
as a concatenation of return words to
One may
we have
By Lemmas 6.4 and 6.12 we have
for each
of and
This means there are only finitely many distinct morphisms
tire array has only finitely many distinct entries. In particular, for some
independent
It follows that the enwe must have
DIRECTED GRAPHS AND SUBSTITUTIONS
orbit. Thus,
from which we see that
is in the shift orbit of
7. M EASURES
11
and
lie in the same shift
OF CYLINDERS
It is well known that if is primitive, then
is uniquely ergodic [31]. Denote by the invariant probability measure on
By the Perron-Frobenius theorem,
admits a simple positive
eigenvalue
greater in absolute value than any other eigenvalue of
and a unique strictly
normalized so that
For a finite walk
positive (right) eigenvector
denote by
the cylinder set
for all
and set
(8)
One readily verifies the consistency requirement
so by Kolmogorov’s theorem
Proposition 7.1. The measure
extends to a probability measure on
is -invariant and
Proof. The first assertion is proved by considering sets of the form
second follows from unique ergodicity.
while the
The following result was conjectured by M. Boshernitzan [6].
Theorem 7.2. The measures of all cylinders in
lie in a finite union of geometric sequences.
Proof. We shall make explicit use of the directed graph
As in the proof of Proposition 4.5 we
for the shift map on
and
for the conjugacy
given by the
write
coding map. Let
denote the normalized Perron eigenvector of
and
the
associated probability measure as in (8).
as in Lemma 6.12, and let
be such that for each
and
Fix positive constants
each word
the minimum gap between successive occurrences of in is at least
(Indeed the value
of Lemma 6.4 will suffice for
Fix
with
and let
denote the corresponding
word in
The measure of the cylinder
in
is clearly equal to the measure of the cylinder
in
Set
Then
holds for every
If
is a walk on
then by Lemma 4.2
(9)
This implies that
where
either contained in
or disjoint from it, since
Proposition 4.3 we have
and thus
(10)
and
is
By (e) of
12
CHARLES HOLTON AND LUCA Q. ZAMBONI
where the sum is over all walks
on
for which
(11)
the number of walks of length
According to (9), for each vertex
and satisfying (11) is equal to the number of indices
It follows from our estimates that there are at most
indices. This together with (10) gives
on
ending at
for which
such
and
8. G EOMETRIC
REALIZATIONS OF SUBSTITUTIONS
In 1982, Rauzy showed that the subshift
generated by the so-called Tribonacci substitution
and
is a natural coding of a rotation on the two-dimensional torus, i.e., is
each
measure-theoretically conjugate to an exchange of three fractal domains on a compact set in
domain being translated by the same vector modulo a lattice [32]. This example, also studied in great
detail by Messaoudi in [28, 29] and Ito-Kimura in [23], prompted a general interest in the question
of which substitutions admit a geometric realization as in the case of Tribonacci. This question was
made precise by Queffélec in [31]. Since then many partial results have been obtained for various
types of substitutions (see [2, 3, 4, 5, 7, 9, 11, 15, 16, 19, 33] to name just a few).
By a complex geometric realization [19] of a primitive substitution on an alphabet we mean
, a nonzero complex number and a nonzero vector
a continuous function
such that
and
for all
One can deduce from the definition that
and
for each
In [19] we showed that every nonzero eigenvector of
corresponding to a nonzero eigenvalue of
modulus strictly less than determines a geometric realization of
Suppose that
is a geometric realization of Consider the edge weights
for
Let
and
and
If
and
is the unique point of
for which
then
(12)
Proposition 8.1. The function
given by
is continuous and
Proof. Continuity follows from the fact that
For each
put
The second assertion follows from (12).
and for each finite walk
on
set
DIRECTED GRAPHS AND SUBSTITUTIONS
13
The function above is related to the graph-directed construction of Mauldin and Williams in [27].
Define edge maps
by setting
Then
(13)
The Mauldin and Williams construction requires that the edge maps be defined on compact subsets of
with nonempty interiors and have nonoverlapping images. Although these conditions need not be
satisfied by our edge maps, the methods can still be applied to give an upper bound on the Hausdorff
dimension of
Proposition 8.2. The Hausdorff dimension of
is no greater than
Proof. We see from (13) that
integer we have
where the sums are over all walks
Set
on
For each positive
This shows
as required.
We are interested in a condition which guarantees that the map be one-to-one off a set of measure zero. The remainder of the section is devoted to some partial results.
For each
and thus
Suppose
(14)
and therefore
(15)
Writing
for the vector whose th entry is
we have
This implies that in fact
so we must also have equality in the first line of (15). It follows from this and (14) that if
and are distinct edges with
then
A straightforward generalization
of this argument to longer paths yields:
and
then the restriction of to
Proposition 8.3. If
to
is a multiple of
of -measure zero. Moreover, the restriction of
is one-to-one off a set
Remark 8.4. We have no general method for verifying the hypothesis of Proposition 8.3.
Conjecture 8.5. If
then
If
and
then
14
CHARLES HOLTON AND LUCA Q. ZAMBONI
We do not know whether Proposition 8.3 implies that is one-to-one off a set of -measure zero
whenever
One partial result is the following.
are vertices of and
are nonnegative integers such that
Suppose
A1)
A2)
A3)
Then there exist walks
starting at
and ending at
respectively, for which
We have
and
and the comments preceding Proposition 8.3 show
from which we deduce
Thus, by Proposition 8.3, if
and for each pair
of vertices there exist nonnegative
satisfying A1–A3 above then is one-to-one off of a set of -measure zero.
integers
There is a simple combinatorial criterion on substitutions which guarantees A1–A3 will be satisfied
for each
The substitution is said to have the coincidence property if for each pair
there is an integer for which
and the number of occurrences of each letter of the alphabet
is the same as that in
This condition was originally introduced by Dekking in [10] in
in
the context of constant length substitutions. It was rediscovered by Host for substitutions defined on
two-letter alphabets [22], and later generalized by Arnoux and Ito [4] to arbitrary substitutions.
Proposition 8.6. If
of -measure zero.
and
has the coincidence property then
is one-to-one off of a set
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D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF C ALIFORNIA , B ERKELEY, CA 94720-3840,USA
E-mail address: [email protected]
D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF N ORTH T EXAS , D ENTON , TX 76203-5116, USA
E-mail address: [email protected]