directed graphs and substitutions

DIRECTED GRAPHS AND SUBSTITUTIONS

CHARLES HOLTON AND LUCA Q. ZAMBONI

ABSTRACT. A substitution naturally determines a directed graph with an ordering of the edges inci-dent 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 sub-stitution minimal shift are in bijective correspondence with one-sided infinite paths on its associateddirected graph. Using this correspondence, we show that primitive substitutive sequences in the sub-stitution minimal shift are precisely those sequences represented by eventually periodic paths. We usedirected graphs to show that all measures of cylinders in a substitutionminimal shift lie in a finite unionof geometric sequences, confirming a conjecture of Boshernitzan. Our methods also yield sufficientconditions for a geometric realization of a primitive substitution to be “almost injective”.

1. INTRODUCTION

Every substitution has an associated prefix automaton, a directed graph whose vertices are theletters of the substitution’s alphabet and whose edges are labeled with the strict prefixes of the imagesof the letters under the substitution. In this paper we consider the directed graph obtained by reversingthe directed edges in the prefix automaton. This graph is more natural from the point of view ofiterated 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 betweensubstitution minimal systems and stationary adic transformations, the underlying spaces of which areessentially the infinite path spaces on the associated directed graphs: every substitution minimal shiftis metrically isomorphic to a stationary adic transformation ([24]). Forrest ([17]) and Durand, Hostand Skau ([14]) proved a converse, that every stationary proper minimal Bratteli-Vershik system isconjugate 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 onits directed graph. The induced dynamical system on the infinite path space of the graph is justthe continuous extension of the Vershik map. Other constructions of this kind exist but are morecomplicated (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 primitivesubstitutive sequences. As part of the characterization, we show that a point in the minimal shiftarising from a primitive substitution is primitive substitutive if and only if it is represented by aneventually periodic path in the graph associated to the substitution. We also show that all primitivesubstitutive 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 geo-metric sequences[6]. This is an unsurprising reflection of the self-similarily inherent in substitutiondynamical systems.Many substitutions admit geometric realizations equipped with piecewise translationmappings (see

[19]). Ei and Ito gave a characterization of those substitutions on two letters whose geometric real-ization 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 dynam-ical systems admit an “almost injective” geometric realization in the sense of Queffelec [31]. In thelast 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 TexasAdvanced Research Program. We are grateful to B. Host and F. Durand for their helpful commentsand suggestions. We also thank A. Forrest for useful e-mail exchanges.

2. SUBSTITUTIONS

By an alphabet we mean a finite nonempty set of symbols. We refer to the members of an alphabetas letters. A word of length in the alphabet is an expression where each is aletter of We write for the length of a word It is convenient to define the empty word havinglength and satisfying the concatenation rule for every word Let be the set of allwords of positive length in the alphabet and setA morphism defined on alphabet is a map where is an alphabet. The length ofis defined to be We say is a letter-to-letter morphism if

and is called nonerasing if A morphism determines by concatenation a mapalso denoted A substitution is a nonerasing morphism A substitution on

alphabet determines a map which we also denote by With this slight abuse of notationwe define for on , , and in the obvious way.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 positiveinteger such that for each letter the word contains every letter of Equivalently, isprimitive if and only if is a primitive matrix. If is primitive then

as(1)

A sequence is a periodic point of if for some positive integer we haveEvery primitive substitution has a periodic point. Denote by the set of periodic points ofFor and we write for the word (such words

are called factors of ). Define and analogously, with the convention thatSimilarly, if and then we set and use the obvious

interpretations for andThe 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 andThe subshift of spanned by the language is

for all

It follows from (1) that if is primitive then is an infinite set and is nonempty. Thecylinder determined by a word is The shift map on isdefined by It is easy to see that is a closed subset of is a homeomorphism

and if is primitive then the dynamical system is minimal, i.e., every pointhas a dense orbit under the shift map (see [31]).

DIRECTED GRAPHS AND SUBSTITUTIONS 3

!"#$%&'(!!

""!"#$%&'(####

FIGURE 1. Directed graph associated to the substitution in Example 3.1.

3. DIRECTED GRAPHS ASSOCIATED WITH A SUBSTITUTION

A directed graph consists of a finite nonempty set of vertices and a finite setof edges together with maps A walk or path on is a finite or infinite sequence ofedges such that for all relevant indices A finite walk starts at itsinitial vertex and ends at its terminal vertex We say that is strongly connected if forevery ordered pair of vertices there is a finite walk on starting at and ending at A stronglyconnected directed graph is said to be aperiodic if there are two walks of relatively prime lengths withthe same initial vertex and the same terminal vertex.The adjacency matrix of is the matrix with th entry equal to the

number of edges starting at vertex and ending at vertex It is easy to see that is the numberof paths of length starting at and ending at Thus, is strongly connected if and only if isirreducible, and is aperiodic if and only if is primitive.Denote by the space of (one-sided) infinite walks on viewed as a subspace of the countable

product of the discrete edge set. With this topology is compact.Let be a substitution on an alphabet We associate to a directed graph with

vertex set and the edge set and whereand These graphs were independently studied by Canterini and Siegel

in[8]. The adjacency matrix is equal to the incidence matrix Thus is primitive if andonly if is aperiodic.

Example 3.1. The Thue-Morse substitution has alphabet and Theassociated graph is shown in Figure 1.

Our construction naturally determines a partial ordering on in which edges are comparable ifand only if they have the same terminal vertex. Specifically, for

and(2)

Conversely, a directed graph having at least one edge ending at each vertex, together with anordering on the set of edges ending at each vertex, determines a substitution on alphabetgiven by where are the edges ending at

It is readily verified that andSuppose 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 ondefined by

and

An element of is minimal (maximal) if it is minimal (maximal) with respect to this partial order-ing. Note that for each vertex and for each positive integer there is a unique walk of lengthending 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 ofedges), and there are, at most, of each.


Every nonmaximal element has a unique successor, and we write for the successor mapmaximal walks That is, if is not maximal and is the least index forwhich is not a maximal edge then

(3)

where

4. WORDS AND WALKS

Fix a primitive substitution on an alphabet such that is infinite. Our first orderof business is to give a method for coding points of by infinite walks on The codingis suggested by the following lemma which states that all primitive substitutions have the uniqueadmissable decoding (UAD) property of [25].

Lemma 4.1. Let Then there is a unique point such that forsome integer Moreover, the map is continuous.

Proof. The original idea for the proof is probably due to Martin in [26]. The result can also be inferredfrom work of Mosse in [30]. Also see [14] for more details.

We record a useful fact whose proof is straightforward.

Lemma 4.2. If is a walk on and with then

where

We define a map which we call the coding map of by

Thus starts at vertex if and only if

Proposition 4.3. The coding map is a continuous surjection satisfying(a)(b) whenever the latter is defined.(c) is minimal if and only if(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 andis any point with then is a point of andbegins in Surjectivity follows from compactness of continuity of

being obvious.Assertions (a) and (e) follow immediately from the definition of To prove (b) suppose

and is not maximal. Let be the least index for which is not maximal. Letbe the unique point of for which and

For any we have

(4)


and repeated application of this identity yields

Putting we see from (a) that

which is, according to (3), the successor ofTo prove (c) we observe that is minimal if and only if is in the image of for all

Assertion (c) now follows from the equality

(5)

To prove (d) we note that by (4)if is maximal then is in the image of for every positiveinteger by (5), must be in Conversely, if is not maximal then (b) implies that

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 minimalelements, then part (c) of Proposition 4.3 shows that cannot be one-to-one. The substitution ofExample 3.1 illustrates this possibility; it has four fixed points but its graph has only two infiniteminimal walks. One can show that is bijective if and only if

minimal elements of maximal elements ofTo 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 definea substitution on by setting for

Note Write for the shift map on For denote by thesequence in the letters of with th entry It is easy to see that

Proposition 4.4. With the above notation, is primitive and is a conjugacyMoreover and

Denote by the coding map of Letand set By Lemma 4.2 and (e) of Proposition 4.3, if

then so, by Proposition 4.4, Anymust satisfy 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 non-maximal 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. Wesummarize this discussion as follows.


Proposition 4.5. If is a primitive substitution then there exists a primitive substitution and aconjugacy such that and the coding map is aconjugacy.

For the remainder of the paper we shall assume that is a primitive substitution on the alphabetand that is an infinite set conjugate to via the coding map

5. CHARACTERIZING LEFT SPECIAL SEQUENCES

Let be the set of all pairs for which and In this sectionwe show that points which appear in an ordered pair of are special in that they are represented byeventually periodic paths in the graph.Proposition 5.1. If are distinct points of with then and areeventually periodic.Proof. Suppose Then and by Lemma 4.1, so theremust be a nonnegative integer such that It can be shownusing Lemma 4.1 that defines a permutation of and thus forsome integers and we must have

From (a) and (b) of Proposition 4.3 we obtain(6)If is in the orbit of a minimal or maximal element, then is eventually periodic by Propo-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 aneventually periodic sequence of edges.Remark 5.2. Related to is the number defined as follows. Let be the equivalence relationon given by

write for the equivalence class of and set

Queffelec showed that this sum is finite [31], but no sharp bound is known. We conjecture thatThe 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. PRIMITIVE 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 fixedpoint of a primitive substitution under a letter-to-letter morphism. Durand [13] showed that primitivesubstitutive sequences are precisely those minimal sequences which admit only finitely many derivedsequences on prefixes (see Theorem 6.5 below). This description was generalized by Holton-Zamboniin [20] to obtain a characterization of primitive substitutive subshifts. In this section we extend thenotion of primitive substitutive to bi-infinite sequences in the natural way, and derive an alternativecharacterization in terms of the associated directed graph. A two-sided infinite sequence is calledprimitive substitutive if it is the image of a fixed point of a primitive substitution under a letter-to-letter morphism. The main result of this section is


Theorem 6.1. Let be a primitive substitution, and a morphism taking to an infinite minimalshift 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 beletter-to-letter in the definition of primitive substitutive may be relaxed (see [13] Proposition 3.1) toarbitrary nonerasing morphisms. Durand’s proof extends immediately to the bi-infinite case:

Lemma 6.2. Let be primitive substitutive. If is a morphism such that for someletter occurring in then is primitive substitutive.

The next lemma characterizes the sequences in represented by eventually periodic paths inThe proof is straightforward and left to the reader.

Lemma 6.3. Let Then is eventually periodic if and only if and lie in the same-orbit for some positive integer

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 Areturn word to in is a nonempty word such that

is a factor of some sequence ofif and only if

Denote by the set of return words to in It follows from minimality of that isfinite. Durand showed that primitive substitutive shifts are uniformly recurrent:

Lemma 6.4. (F. Durand, [13] Theorem 4.5) If is infinite and contains a primitive substitutivesequence then there exists a positive constant such that for all every return wordsatisfies

Fix, for each a bijection If with then can bewritten uniquely as a concatenation

eachand we obtain the derived sequence of with respect to

Derived sequences are unique up to permutation of their alphabets, and Durand’s result holds forbi-infinite sequences:

Theorem 6.5. (F. Durand, [13] Theorem 2.5) Let be a minimal shift space. A sequence isprimitive substitutive if and only if is finite.

Remark 6.6. In case has only finitely many derived sequences, Durand’s proof gives a methodfor constructing a primitive substitution fixing a point and a letter-to-letter morphismmapping onto with In view of Theorem 6.1 this is all one needs to identify allprimitive substitutive sequences in

A bi-infinite sequence is primitive substitutive if and only if is primitive substitutive (seeLemma 2 in [21]). We introduce the notion of derived orbit to reflect this fact. Let be a minimalshift and let and be as above. If then (by minimality of ) there is a point in theshift orbit of such that and we set equal to the shift orbit of the sequenceClearly the derived orbit depends only on the orbit ofThe 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.


Proof. Replacing with if necessary and using Lemma 4.2 we may assume thatfor some integer If or then or is fixed by in whichcases is primitive substitutive. Thus we consider Put let be theset of return words to in and let be a bijection. If then

is a factor of which begins and ends in (since ) and thereforecan 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 The proof iscompleted by invoking Lemma 6.2 with

We now turn our attention to proving the ‘only if’ part of Theorem 6.1. We show

is primitive substitutive is primitive substitutive(7)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 substitutivesequence then there is a finite set of disjoint minimal shift spaces whose union contains every derivedsequence 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 isa morphism defined on the alphabet of such that is infinite then is bounded-to-one.

Proof. The hypotheses imply that is infinite. Let be as in Lemma 6.4, so that every word of lengthin the language of occurs in every word of length in the language of Since some letter in

the alphabet of is not sent by to the empty word, we must have forevery word in the language ofSuppose that are distinct sequences of all sent to the same sequence by

such that For all large enough the words are distinct. Fix suchFor any two of the words one must be a suffix of the other; let

be the shortest of these words. Likewise the shortest of the wordsis a prefix of each of these words. Put ThenNow let be a word in the language of of length Then is a word of length

Each of the words must occur in and ifthen is a suffix of and is a prefix of Since there mustbe 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 substitutivesequence. Denote by the return constant for guaranteed by Lemma 6.4. Then contains atleast return words to in and by Lemma 6.4 we haveThus


Finally, if and is the least nonnegative integer for which then and if isthe th shift image of then It follows that is at most -to-one.

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 andsuppose is a morphism such that is infinite. If is such that is primitive substitutivethen is primitive substitutive.

Proof. Let and be as in the proof of Proposition 6.9. For all sufficiently large ifthen may be written uniquely as a concatentation of elements of This

defines a morphism having the propertythat

By Lemma 6.4 for any This implies that forany from which we obtain the crude bound valid for

It follows from the bound on the that there are only finitely many distinct mapsBy 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 minimalshift spaces each containing a primitive substitutive sequence, with the property thatany derived sequence of a sequence in is contained in one of them. If then isdefined on and takes it to an infinite subset of the closure of the shift orbit of ByProposition 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 is primitive substitutive if and onlyif a factor of is finite.

Proof. Suppose first that is primitive substitutive. We will construct finite collection of sequenceswhich contains a member of each derived orbit. Let be any factor of and let be the leastpositive integer for which is a factor of Then ends in and where isthe constant from Lemma 6.4. If is a return word to then begins in and therefore

begins and ends in Thus can be written uniquely as aconcatenation of return words to

each

It follows from Lemma 6.4 thatConsider the morphism given by

One checks easily that is in It follows from the bound on the thatthere are only finitely many different possible maps and by Durand’s theorem there are onlyfinitely many distinct derived sequences Thus, a factor of is afinite set containing a representative of each derived orbit.Now suppose a factor of is finite. We may assume that is not periodic. Then

we can find positive integers for which and every return word tois longer than the longest return word to Every return word to can be written


uniquely as a concatenation of return words to and thus we may define a substitutionby 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 withcompletes the proof that is primitive substitutive.

We recall a well-known inequality (see [31]). Since is primitive the matrix is also primitiveand the Perron-Frobenius theorem guarantees a positive eigenvalue strictly greater in absolute valuethan any other eigenvalue of

Lemma 6.12. There exist positive constants such that forevery word and for every positive integer

The next lemma completes the proof of Theorem 6.1.

Lemma 6.13. A sequence has only finitely many distinct derived orbits if and only if thereexists a positive integer such that is in the shift orbit of

Proof. If is in the shift orbit of then by Lemma 6.7 is primitive substitutive, whence it hasonly finitely many distinct derived orbits, by Lemma 6.11.Now suppose is primitive substitutive. Let be any factor of and consider the array

......

... . . .

There are only finitely many distinct entries in the first column, by Lemma 6.11.If then begins in and is a factor of It follows that is a factor of

which begins and ends in and thus may be written uniquely as a concatenation of wordsof This induces a morphism defined for

by

whereeach

is the unique representation of as a concatenation of return words to One mayverify that for any

Thus, for each we have

By Lemmas 6.4 and 6.12 we have for each independentof and This means there are only finitely many distinct morphisms It follows that the en-tire array has only finitely many distinct entries. In particular, for some we must have


from which we see that and lie in the same shiftorbit. Thus, is in the shift orbit of

7. MEASURES OF CYLINDERS

It is well known that if is primitive, then is uniquely ergodic [31]. Denote by the -invariant probabilitymeasure on By the Perron-Frobenius theorem, admits a simple positiveeigenvalue greater in absolute value than any other eigenvalue of and a unique strictlypositive (right) eigenvector normalized so that For a finite walkdenote by the cylinder set for all and set

(8)

One readily verifies the consistency requirement

so by Kolmogorov’s theorem extends to a probability measure on

Proposition 7.1. The measure is -invariant and

Proof. The first assertion is proved by considering sets of the form while thesecond follows from unique ergodicity.

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 wewrite for the shift map on and for the conjugacy given by thecoding map. Let denote the normalized Perron eigenvector of and theassociated probability measure as in (8).Fix positive constants as in Lemma 6.12, and let be such that for each and

each word the minimum gap between successive occurrences of in is at least(Indeed the value of Lemma 6.4 will suffice forFix with and let denote the corresponding

word in The measure of the cylinder in is clearly equal to the measure of the cylinderin

Set Then holds for everyIf is a walk on then by Lemma 4.2

(9)

where This implies that iseither contained in or disjoint from it, since and By (e) ofProposition 4.3 we have

and thus

(10)


where the sum is over all walks on for which

(11)

According to (9), for each vertex the number of walks of length on ending atand satisfying (11) is equal to the number of indices for which

It follows from our estimates that there are at most suchindices. This together with (10) gives

and

8. GEOMETRIC REALIZATIONS OF SUBSTITUTIONS

In 1982, Rauzy showed that the subshift generated by the so-called Tribonacci substitutionand is a natural coding of a rotation on the two-dimensional torus, i.e., is

measure-theoretically conjugate to an exchange of three fractal domains on a compact set in eachdomain being translated by the same vector modulo a lattice [32]. This example, also studied in greatdetail by Messaoudi in [28, 29] and Ito-Kimura in [23], prompted a general interest in the questionof which substitutions admit a geometric realization as in the case of Tribonacci. This question wasmade precise by Queffelec in [31]. Since then many partial results have been obtained for varioustypes 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 continuous function , a nonzero complex number and a nonzero vectorsuch thatand

for allOne can deduce from the definition that and for eachIn [19] we showed that every nonzero eigenvector of corresponding to a nonzero eigenvalue ofmodulus strictly less than determines a geometric realization ofSuppose that is a geometric realization of Consider the edge weights

for Let and If and is the unique point of for whichand then

(12)

Proposition 8.1. The function given by

is continuous and

Proof. Continuity follows from the fact that The second assertion follows from (12).

For each put and for each finite walk on set


The function above is related to the graph-directed construction of Mauldin andWilliams in [27].Define edge maps by setting Then

(13)

The Mauldin and Williams construction requires that the edge maps be defined on compact subsets ofwith 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 Hausdorffdimension of

Proposition 8.2. The Hausdorff dimension of is no greater than

Proof. We see from (13) that Set For each positiveinteger we have

where the sums are over all walks on 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.Suppose For each and thus

(14)

and therefore

(15)

Writing for the vector whose th entry is we have This implies that in factso 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 generalizationof this argument to longer paths yields:

Proposition 8.3. If and then the restriction of to is one-to-one off a setof -measure zero. Moreover, the restriction of to is a multiple of

Remark 8.4. We have no general method for verifying the hypothesis of Proposition 8.3.

Conjecture 8.5. If then If and then


We do not know whether Proposition 8.3 implies that is one-to-one off a set of -measure zerowhenever One partial result is the following.Suppose are vertices of and are nonnegative integers such that

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 deduceThus, by Proposition 8.3, if and for each pair of vertices there exist nonnegative

integers satisfying A1–A3 above then is one-to-one off of a set of -measure zero.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 pairthere is an integer for which and the number of occurrences of each letter of the alphabetin is the same as that in This condition was originally introduced by Dekking in [10] inthe context of constant length substitutions. It was rediscovered by Host for substitutions defined ontwo-letter alphabets [22], and later generalized by Arnoux and Ito [4] to arbitrary substitutions.

Proposition 8.6. If and has the coincidence property then is one-to-one off of a setof -measure zero.

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CA 94720-3840,USAE-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NORTH TEXAS, DENTON, TX 76203-5116, USAE-mail address: [email protected]

directed graphs and substitutions

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