Geometric representation of the infimax S-adic family
We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of substitutions from the family has an asymptotic fixed sequence whose shift orbit closure is an infimax minimal set . The subcollection of substitutions also generates an infinite Bratteli-Vershik diagram with prefix-suffix labeled edges. Paths in the diagram give the Dumont-Thomas expansion of sequences in which in turn gives a projection onto the asymptotic stable direction of the infinite product of the Abelianization matrices. The projections of all sequences from is the generalized Rauzy fractal which has subpieces corresponding to the images of symbolic cylinder sets. The intervals containing these subpieces are shown to be disjoint except at endpoints, and thus the induced map derived from the symbolic shift translates them. Therefore the process yields an Interval Translation Map (ITM), and the Rauzy fractal is proved to be its attractor.
Key words and phrases:S-adic, infimax, geometric representation
2010 Mathematics Subject Classification:Primary 37B10; Secondary 37E05
Substitution morphisms are an integral part of many areas of mathematics including dynamical systems, combinatorics, number theory and formal language theory. The books , , and  give a good sense of the diversity and depth of the field. The natural generalization of a single substitution to the composition of an infinite sequence of substitutions, termed S-adic systems by Ferenczi in , allows the modeling and analysis of a wider range of fundamental structures (see, for example, , , , and ).
In  an S-adic family was found to generate the solutions to the following problem. A one-sided sequence is called maximal if it is larger in the lexicographic order than all its shifts. Let denote all the maximal sequences with asymptotic digit frequency vector . The infimum in the lexicographic order of is called the infimax sequence for and in  it was shown that it can be constructed using a specific S-adic family. In this paper we further study the properties of this infimax S-adic family. The family studied here is indexed by the positive integers with the substitution given by
Note that the substitution is required for the full infimax problem: see Remark 7.4(b) below.
The collection of allowable lists of indices is . Given a list of indices , it is easy to check that the right one-sided sequence
exists. For a constant list of indices , the sequence is a fixed point of . For a general , the sequence is an asymptotic fixed point in the sense that for any one-sided sequence with ,
Returning to the infimax problem, for a given asymptotic digit frequency vector , it was shown in  that a three dimensional continued fraction algorithm using as an input generates a list . The infimax for is then the corresponding as defined in (1.2).
The main dynamical object of study is the orbit closure of under the left shift, or . Since is always dynamically minimal, we call it an infimax minimal set. The study of and its associated substitutions is facilitated by finding a geometric representation of as defined in . This means a concrete, geometrically defined model system so that the dynamics of under the shift are embedded in the dynamics of on in a nice way. Specifically, we seek a map which is a conjugacy, on the image of . In addition, the space is required to have a nice partition by which -orbits are coded so that corresponding sequences are recovered as itineraries. Geometric representations have been found for many substitutions (see  for a summary). Depending on the circumstances, different requirements can be made on the map . From the ergodic theory perspective, one would like to be measure preserving and injective almost everywhere with respect to the appropriate invariant measures. We work here in the topological category and so is required to be continuous and, in fact, will be a homeomorphism onto its image.
The geometric models for the S-adic infimax family given here are elements of a two-parameter family of interval translation maps (ITM). These maps are generalizations of interval exchange transformations in which the images of intervals are allowed to overlap (). The attractor of an ITM is the intersection of the forward iterates of the entire interval. For each , we show in Theorem 11.2 that the symbolic infimax minimal set is conjugate via a map to the attractor of a (slightly extended) ITM. The extension of the ITM is necessary to make it continuous as is often done with interval exchange maps ().
For each , the three symbol infimax minimal set has a geometric representation as the attractor of an Interval Exchange Map (ITM) on three intervals.
In addition, the conjugacy respects order structures: the lexicographic order on the symbolic minimal set is reversed under the conjugacy onto the ITM attractor inside the unit interval.
Previously Bruin and Troubetzkoy have shown that each is isomorphic to the attractor of an ITM (, see also Section 5 in ). We extend these results obtaining a full homeomorphic conjugacy and in addition study the two-sided infimax minimal set. The starting point in  was the natural renormalizations of a family of ITM. The substitutions (1.1) then arise as the symbolic descriptors of this process. On the other hand, motivated by the infimax problem, we begin here with the substitutions and sequences themselves. Using generalizations of methods commonly used for single substitutions we find the ITM geometric representation as a direct and natural consequence of the structure of the generalized Rauzy fractal and its induced transformation under the shift.
The methods we use have their origin in Rauzy’s classic papers (, , ) and their subsequent development by many authors. Most significant and relevant here is the process laid out by Canterini and Siegel (, ) and from a somewhat different angle by Holton and Zamboni (, ). The paper  provides an excellent exposition of the process and related constructions for a very important special case. We adopt these single substitution methods to the S-adic case. While often the generalizations are reasonably straightforward, they differ in enough detail that an independent, self-sufficient treatment is required.
The main idea in this geometric representation process is projection onto the stable subspace of the Abelianizations of the substitution. For the family (1.1) the Abelianizations are
Each has two eigenvalues outside the unit circle and one inside (proof of Lemma 52 in ) and so each substitution (1.1) is inverse-Pisot, unimodal and primitive. It is shown in Theorem 7.3 below that for each the limit
has a well defined asymptotic (or generalized) one-dimensional stable direction . A similar argument shows the existence of an asymptotic two-dimensional unstable subspace. However, if the grow sufficiently fast there is not an asymptotic one-dimensional strongest unstable direction (Theorems 11 and 12 in  and Theorem 27 in ). This implies that in these cases the asymptotic digit frequency vector of does not exist and, in addition, is not uniquely ergodic ().
To achieve the projection, Rauzy’s original method was to embed the Abelianization of the sequence in and then project its vertices down to the stable subspace and take the closure. We adapt the alternative route developed by Canterini and Siegel (, ) and use the machinery of the Prefix-Suffix Automaton. This automaton is a particularly useful way of labeling and ordering a Bratelli diagram based on a single substitution. The sequence of edge labels in an infinite path naturally yield both the Dumont-Thomas prefix-suffix expansion of a symbolic sequence (, ) and a map to the stable subspace of the Abelianization. In the natural generalization to the S-adic case given in Example 3.5 of  and used below, each level of the diagram corresponds to a substitution from the list . The sequence of edge labels in an infinite path then yields the S-adic version of the Dumont-Thomas prefix-suffix expansion as in formula (5) in 
With the appropriate alterations for the S-adic situation, this yields a map from the path space of the Bratelli diagram to as well as a projection onto the asymptotic stable subspace of the infinite composition of the Abelianizations in (1). The inverse of the first map composed with the second map yields the map from to the real line given in Theorem 1.1 above. The image of is sometimes called a generalized Rauzy fractal, and in our case it is always a Cantor set embedded in an interval
The final ingredient in this geometric representation process is the subdivision of the Rauzy fractal into pieces corresponding to the images of symbolic cylinder sets under . The importance of these subpieces is that there is a natural translation induced on them by the shift map on . Thus the process yields a geometric representation as long as the subpieces are disjoint almost everywhere and so proving this disjointness is often the central problem in this process of geometric representation. For the infimax family, these subpieces are again Cantor sets and we show in Theorem 9.1 that their convex hulls (obviously intervals) can only intersect at their endpoints. The induced map on these interval convex hulls is the representing Interval Exchange map (ITM) and we then show that the Rauzy fractal is, in fact, the attractor of this ITM.
While the construction of an infimax minimal set requires infinitely many choices to designate the list of defining substitutions in (1.2), one of the striking features of the geometric representation is that it faithfully describes an infimax minimal set by specifying just two parameters in the family of ITM. The geometric representation has additional consequences. For example, Boshernitzan and Kornfeld note in §7 of  that after using the defining intervals of the ITM to code orbits it is straightforward to show that the number of distinct words of length in itineraries of orbits can grow at most at a polynomial rate. This implies that the ITM have zero topological entropy. Thus using the geometric representation all the infimax minimal sets also have zero entropy (this also follows from more general, more recent results; see Theorem 4.3 in ). Boshernitzan and Kornfeld ask whether this factor complexity growth rate is actually linear. In the special case of what are called infimax minimal sets here, Cassaigne and Nicolas showed that the factor complexity satisfies ().
In many of the geometric representations of substitutions in the literature the symbolic minimal set is represented by either an interval exchange map or a toral translation. This provides a great deal of information about the symbolic minimal set, in particular, about its spectrum. Compared to interval exchange maps, ITM are poorly understood and there is little known about their spectrum. The ITM in the representing family here have a rich variety of behaviours like non-unique ergodicity and thus provide a good model problem for future development: one can work jointly with the ITM and the symbolic, S-adic description.
There are a number of important features that are specific to the family studied here. The first is that a list of substitutions when it acts on bi-infinite sequences yields asymptotic period-two points rather than a fixed point. As a consequence, the resulting Bratteli-Vershik diagram is not properly ordered: it has two maximal elements and one minimal element (for background on Bratteli-Vershik diagrams and adic transformations see  and for their use with substitutions  and ). This complicates the reading off the Dumont-Thomas expansion of a sequence from the labels on edges of the maximal paths and necessitates the eventual use of a map from back to the path space in Section 5.
Also, the improper order implies that the Vershik map on the path space cannot be globally defined. As done in  and many subsequent papers, it can be defined almost everywhere but since we are working in the topological category, we require all maps to be globally defined. Our main objective is a map from to , and so the path space is a convenient intermediate structure, but we never need to consider the dynamics on it. Thus the order on the Bratteli diagram and the Vershik map are not utilized here. However, the labeling of edges in the diagram by prefixes and suffixes is of crucial importance and so we have adapted the terminology “infinite prefix-suffix automaton” (IPSA) for the labeled Bratteli diagram corresponding to a list of substitutions indexed by .
Another special feature is the central role of the lexicographic order on the symbol space and its relation under the representation map to the usual linear order on the real line. Note that each substitution preserves the lexicographic order (Lemma 2 in ) and that the family arose as the solution to the infimax question which depends fundamentally on the lexicographic order. In the final analysis it is the relation of the orders on sequences and the reals which yields the fundamental fact of the disjointness of the subpieces of the Rauzy fractal.
Finally, in contrast to much of the existing work on S-adic systems, the infimax family contains infinitely many substitutions and further, our results hold for all sequences rather than just a large, say full measure, subcollection.
While our main results concern the one-sided shift space, a number of steps in the representation process are technically simpler using two-sided infinite sequences. This has the added benefits of yielding useful results about the relationship between the two-sided version and the one-sided version of the infimax minimal set. For example, in Theorem 6.1 we show that the projection is injective except on the forward orbits of the asymptotic period two points, or informally, the one-sided version is obtained by collapsing a single pair of orbits of the two-sided version. This in turn implies that the left shift has unique inverses in except on the defined in (1.2).
This paper deals primarily with the geometric representation of the S-adic family with symbols: more detailed results about the infimax minimal sets and their languages are saved for a later paper. We remark on the case in the last section.
We start with some basic definitions about words, sequences and substitutions. The alphabet here will always be . The length of a finite word is denoted , and the empty word has . A bi-infinite sequence is an element of and is written with a decimal point between the zeroth and minus first symbols, . A right infinite sequence has the form and we use an under-arrow to indicate it , and a left infinite sequence is written . The collection of right infinite sequences is denoted .
The spaces and are given the topology induced by the metric where . The left shift acting on is and acting on is . In the dynamics literature is usually used for the shift and in the substitutions literature usually denotes a substitution. To avoid confusion we refrain from using altogether.
A pointed word is word with decimal point placed between two of its symbols or at the beginning or end of the word. The shift acts on pointed words as long as its action does not move the decimal point beyond the beginning or end of the word. A pointed one-sided sequence has the form or . The empty symbol is included to indicate the end or beginning of the pointed one-sided sequence. The shift also acts on pointed one-sided sequences again with the proviso that the decimal point cannot move beyond the end.
A substitution is specified by assigning a nonempty word to each symbol . It acts on sequences, words and pointed objects yielding another object of the same type by respecting the decimal point, so, for example,
For a homeomorphism , the full orbit of
a point is
and its forward orbit is . When is not injective, is defined in the same way.
3. The Infinite Prefix-Suffix Automaton
Fix a sequence . Our main object of study is the sequence of substitutions . For each , let
For a subcollection of indices write , and so .
We now define the principal tool in this paper, the Infinite Prefix-Suffix Automaton (IPSA). A related automaton for a single substitution was contained in Rauzy’s classic papers (, , ) and similar constructions were present in other seminal works in other fields (see page 218 of  for some history). The automaton was formalized, extended and utilized in  and  and independently in a slightly different form in . The version we use here is the S-adic generalization described in Example 3.5 of .
The Infinite Prefix-Suffix Automaton (IPSA) or Bratteli-Vershik diagram associated with is an infinite directed graph built in levels. Each level of the graph contains three nodes or states 1, 2, 3 and the levels are indexed by . There is a directed edge from state on level to state on level if and only if for some perhaps empty words and . This edge is then labeled . Note that that level zero of our diagram has three states and not a single root state as is common in the literature. See Figure 1.
It is easy to check that an infinite path in the IPSA is uniquely specified by its sequence of labels and we write or where for all , . Note that the sequence of edges is indexed by while the sequence of indices is indexed by . The collection of all infinite paths in the IPSA generated by an is denoted or just . The space is a compact metric space under the metric where .
3.2. The Dumont-Thomas expansion; the word and sequence maps
A finite path in the IPSA is said to have length and the collection of all such length paths is denoted . We now define the maps which assign a Dumont-Thomas expansion (, ) of a word or sequence to each finite or infinite path.
For each the word map assigns a finite pointed word to a length path via
Given an infinite path , for each let be the length path . The sequence map assigns a pointed sequence to each infinite path via
In most cases the sequence will be bi-infinite, but in certain special cases studied in Section 3.3 it will be infinite in only one direction.
The first lemma describes in more detail how the word map gives a correspondence between paths in the IPSA and symbolic words. As an example, let be a length two path that terminates in the node labeled . Thus from the definitions and keeping track of the decimal points we have and for some . Evaluating the word map we have
Thus for any length two path which terminates at the node , its word map image is a shift of . In fact, Lemma 3.1 and Theorem 3.4 together will show that each shift of corresponds to one and only one length two path terminating at . In general, let be the subcollection of length- paths which terminate at node and so . The first lemma shows that for any path in its word map image is a shift of . Theorem 3.4 will show that the correspondence is injective. Part (b) of the lemma shows how to replace the inner portion of a sequence map image with the form of the image of an initial segment of the path given in part (a).
For all ,
if , then for some .
If , then for all there exists so that
We prove (a) by induction on , with the case following from the definition of the IPSA. So assume the result is true for length paths. Given with , we first show that has the required form. Form the length path in the IPSA of . Using the inductive hypothesis on , the assignment associated with , we have a with
for some where since we know that is a finite, pointed word, . But since from the IPSA, is impossible, in fact , finishing (a).
For (b) it follows immediately from the definition that for any ,
and so (b) then follows from (a). ∎
Letting , we also have
3.3. Some special paths and sequences
The sequence map given in (3.2) yields a two-sided sequence for most paths in . The paths for which is a pointed one-sided sequence require individual consideration and fall into three classes. Recall that two paths and are said to be tail equivalent if there exists a so that for all . This obviously yields an equivalence relation on the path space .
The set will consist of the tail equivalence class of
with and even.
The set will consist of the tail equivalence class of
with and odd.
The set will consist of the tail equivalence class of
Finally, . Note that if and only if there exist arbitrarily large with and there exist arbitrarily large with . Thus consists of those paths for which is a two-sided infinite sequence.
As mentioned in the introduction, we don’t make explicit use of the substitution induced order on the path space but its description will clarify what follows. On a path space derived from a general diagram, Vershik defines a partial order which restricts to a total order on each tail equivalence class. It is built from a total order on the outgoing edges at each vertex. Specifically, for two paths in the same tail equivalence class, the lesser one is the one with the lesser outgoing edge at the last vertex they disagree. The Vershik map on the path space sends each non-maximal path to its successor.
In the particular case of a Bratteli-Vershik diagram which is the Prefix-Suffix Automaton for a single substitution , it was noted in  and  that there is a natural total order on the outgoing edges from a vertex. If the vertex is labeled by the letter , each outgoing edge is labeled where , and so declare if . When the path space is mapped onto the substitution minimal set via the analog of (3.2), the Vershik map on the path space will conjugate (on a large set) with the shift map on the substitution minimal set.
For the IPSA of the infimax S-adic family considered here, one may define a partial order on the path space exactly as in the single substitution case. In this order the path is the maximal element and the paths and are minimal elements. Thus the sets just defined are the tail equivalence classes of the maximal and minimal elements. We therefore expect from Vershik’s construction that each set would correspond to an orbit of sequences under the shift. This is the content of Lemma 3.3. The corresponding sequences are built from the following:
It is clear that the limits exist. Let and . These are the S-adic analogs of the period two point in the single substitution case in the sense that
With Lemma 3.3 in mind, now define
3.4. Injectivity of the word and sequence maps
The maps takes the prefix-suffix labels along a path and generates a word or sequence via the Dumont-Thomas expansion. The next theorem states that this assignment is injective.
This theorem coupled with later results may be be viewed as a desubstitution or recognizability result. As in the proof of Lemma 3.1, given and a path in , let and form the path . If is the sequence map associated with the path space , then directly from the definitions of the sequence maps
Thus if is a bi-infinite sequence (i.e. ), it is desubstituted under by . By Theorem 3.4, the assignment is injective, so the desubstitution is unique. To obtain a full result, we need in addition that (Theorem 5.3) and for , we must extend to a bi-infinite sequence in as described above Theorem 5.3. Note that a sequence in is desubstituted under by a sequence in . One may continue, desubstituting under a sequence in by one in , where , etc. Fisher studies this situation in  and constructs the S-adic analog of the subshift of finite type corresponding to a single substitution (see Section 7.2 in ).
The proof of Theorem 3.4 requires more detailed information on the word and sequence maps . Since we are considering words and sequences of different types we need a stronger notion of inequality. Given words or one-sided or two-sided sequences and , they are said to be strongly unequal, , at index if , , and . Thus is not equal to but it is not strongly unequal.
The following is easy to check from the IPSA, but very useful. If is bi-infinite, only the following pairs can occur in the sequence : , , , , and . The same restrictions hold inside any or , and in addition, a right-one sided sequence or finite word can only start with a while a left-one sided sequence or finite word can only end with a or a .
For all :
If in or in , then and .
If in then at an index except for the case of pairs of the form
We prove (b) first. The proof is by induction on , the smallest index with . To start assume that . Now if , then at the index and so we may assume . Since just one edge emerges from state , the case is impossible, so we are left with two remaining cases.
Now if and , the only possibilities are, say, which can only be followed by , and which can only be followed by with or . In either case, and , and so at index .
Now assume and we consider various subcases. First assume that and both terminate at state . Thus and where, say, . Thus and with or . Since , at index .
Now assume that and both terminate at state . Thus and where, say, . If for all then we are in the excluded case (3.8). Thus for some , with . Thus and since begins with either a or a , at index .
For the last subcase, assume that terminates at and terminates at . Thus and . Of necessity with