The $\beta$-transformation with a hole

This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_{\beta}(x)=\beta x \pmod 1$. Let $\mathcal{J}_{\beta} (a,b) := \{ x \in (0,1) : T_{\beta}^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\beta}$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \[ D_0(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \neq \emptyset \}, \] \[ D_1(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \text{ is uncountable} \}, \] \[ D_2(\beta) = \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \]

These sets were fully described for the doubling map by Glendinning and Sidorov [3] (D 0 (2) and D 1 (2)) and by Hare and Sidorov [6] (D 2 (2)). This work showed that specific 0-1 words known as balanced words are very important in these descriptions. In this paper we dicuss in Section 3 precisely why these words are important, together with how and to what extent the results transfer from the doubling map to the general case. However, for general β balanced words alone do not suffice, and so Section 4 describes this difference and is completely new.

β-expansions and combinatorics on words
Much of the study of these avoidance sets involves combinatorics on words. We therefore include the basic definitions from combinatorics on words here -see [8,Chapter 2] for a more thorough discussion. We will be considering words on the alphabet {0, 1}. Given two finite words u = u 1 . . . u n and v = v 1 . . . v m we denote by uv their concatenation u 1 . . . u n v 1 . . . v m . In particular u k = u . . . u (k times) and u ∞ = lim k→∞ u k . We denote the length of u by |u| and the number of 1s in u by |u| 1 . To compare words we use the lexicographic order : a finite or infinite word u is lexicographically smaller than a word v (that is, u ≺ v) if either u 1 < v 1 or there exists k > 1 with u i = v i for 1 ≤ i < k and u k < v k .
A finite or infinite word w is said to be balanced if for any two factors u and v of w of equal length, we have that ||u| 1 − |v| 1 | ≤ 1. A finite word w is called cyclically balanced if w 2 is balanced.
We also introduce the following notation. Given a finite word w and a subword u of w, we denote by u-max the lexicographically maximal shift of w that begins with the word u. Similarly we denote by umin the lexicographically minimal shift of w that begins with the word u. For example, given w = 10100, we have 0-max = 01010 and 1min = 10010.
In order to use combinatorics on words in the context of the βtransformation, we recall that T β is conjugate to the shift map on a subset of Σ = {0, 1} N . This arises by writing a number x as with x i ∈ {0, 1}, as first studied in [10]. In particular, we consider the greedy β-expansion of x, namely the expansion with x i = ⌊βT i−1 β x⌋. Informally the greedy expansion is given by taking a 1 whenever possible. We denote the set of possible ("admissible") sequences (x i ) ∞ i=1 by X β .
Consider the expansion of 1 given byd i = ⌊βT i−1 β (1)⌋. If this sequence is infinite (i.e. does not end in 0 ∞ ) then set d i =d i . Ifd i is finite then let k = max{j :d j = 0} and set d 1 d 2 · · · = (d 1 . . .d k−1 0) ∞ . This is the periodic quasi-greedy expansion of 1. Then as shown by Parry in [9], we have . . for all j ∈ N}.
We will denote the quasi-greedy expansion of 1 by 1·, akin to a decimal point. This is to avoid confusion between the real number 1, the sequence 10 ∞ , and this quasi-greedy expansion 1·.
Throughout this paper we will refer to a point x ∈ (0, 1) and its expansion ( The only possible ambiguity here is where a point has a finite expansion; that is to say (x i ) = u10 ∞ for some finite admissible word u. Here we naturally have u10 ∞ = u01·. Generally in such cases we will use the finite expansion by default and will specify if this is not the case.
We also make use of the idea of extremal pairs, linked to the study of Lorenz maps through kneading invariants -see [7] and [4].
Definition 2.1 (Extremal pairs). A pair (s, t) of finite {0, 1} words is said to be an extremal pair if the following inequalities do not hold for any k, ℓ > 0: In our context we will always have that t is a cyclic permutation of s, and so these two inequalities combine into one. Notice that we do not require that s 1 = 0 and t 1 = 1 as in [3]. However as an immediate consequence of the definition and the cyclic permutation requirement we have that s = u0-max and t = u1-min for some word u. The intuitive description is that given an orbit, we take two neighbouring points of that orbit: the rest of the orbit cannot fall in between these two points.
Given an extremal pair (S, T ), we naturally have S, T ∈ {0, 1} n and so denote more fully the pair as (S(0, 1), T (0, 1)). However, it is entirely possible to take another extremal pair (s, t) and use this pair as an alphabet to gain the pair (S(s, t), T (s, t)). These words then belong to {s, t} n . We call such a pair (S(s, t), T (s, t)) a descendant of (s, t). It is shown in [3, Proposition 2.1] that all such descendants are themselves extremal pairs.
We further define an extremal pair (s, t) to be maximal if firstly there does not exist any point x such that the orbit of x is contained in one of either [0, s ∞ ) or (t ∞ , 1), and secondly there does not exist a distinct extremal pair (s,t) such that (s ∞ , t ∞ ) ⊂ (s ∞ ,t ∞ ). Equivalently, an extremal pair (s, t) is maximal if J β (s ∞ , t ∞ ) = {σ n s ∞ : n ∈ N}.

Admissible Sturmian sequences for the β-transformation.
For the case β = 2 the maximal extremal pairs are formed from balanced (Sturmian) words, as shown in [3]. Hence, the first issue is to establish which Sturmian sequences are admissible for which β. A detailed exposition on Sturmian sequences may be found in the book by Lothaire [8, §2] and the survey paper by Vuillon [12]. There are several ways of defining Sturmian sequences. We will do so using the Farey tree as the tree structure allows for easier proofs later.
Definition 2.2 (Farey tree). We construct the Farey tree inductively. Take 0 and 1 as initial sequences, with associated fractions 0/1 and 1/1 respectively. These initial sequences are said to be neighbouring, and more generally a sequence in the tree is considered to be a neighbour to its parents (and children).
For rational γ = p/q, we obtain a finite cyclically balanced word w γ of length q, and we may define X p/q to be the finite set These sets X γ has been well studied in [2]. For rational γ the word w γ given by the Farey tree turns out to be the maximal cyclic shift; that is to say w ∞ γ is the maximal element of X γ . The analogous definition of X γ for irrational γ begets an infinite set of Hausdorff dimension 0.
Each rational γ gives rise to two distinct infinite balanced words. Given w γ = w γ 2 w γ 1 , these words are w ∞ γ and w γ 2 w ∞ γ . Naturally if γ 1 < γ 2 then w γ 1 ≺ w γ 2 . Therefore we may define a function γ(β) to be the maximal admissible Sturmian word for a given β. This is non-decreasing, with the effect that for a given β, w ∞ γ is admissible if and only if γ ≤ γ(β).
The exact description of this function is as follows: Proof. This follows immediately from the fact that w γ is the maximal element of X γ : any β with 1· ≺ w ∞ γ must therefore have w ∞ γ inadmissible and γ(β) < γ. Similarly, if 1· ≻ w γ 2 w ∞ γ then in order to be a valid greedy expansion of 1 we must have that 1· begins w 2 γ 2 , which implies that γ(β) > γ. Therefore As can be seen in Figure 1, γ(β) is a devil's staircase: it is continuous, non-decreasing, and has zero derivative almost everywhere. The same function arises when considering digit frequencies for β-expansions, as described by Boyland et al. in [1].
We can associate an extremal pair with each rational γ. Given the word w γ , take s = 0-max and t = 1-min. These pairs may themselves be constructed using a tree structure: given two neighbouring pairs (s γ 1 , t γ 1 ) and (s γ 2 , t γ 2 ) with associated rationals 0 < γ 1 < γ 2 < 1, their child is (s γ 2 s γ 1 , t γ 1 t γ 2 ). 1 We remark that these pairs satisfy the following: . This is shown for balanced words in [6,Lemma 3.2]. We remark that the result is actually a property of the tree construction method, not specifically of the words themselves: we simply take the tree for balanced words and map 0 and 1 to the left and right roots of our alternative tree, and provided the left root is less than the right root, the result will hold.
2.2. Level n "balanced" words. For describing D 0 (β), the normal balanced words are sufficient. However, to describe D 1 (β) one must consider higher level "balanced" words. These are defined in [3] and for completeness' sake we repeat the discussion here. These words are not themselves balanced but are derived from balanced words.
Consider p/q ∈ (0, 1). Define a function ρ p/q : where s = 0-max and t = 1-min are the Sturmian extremal pair associated to p/q as defined in the previous section. These are thus the descendants of Sturmian pairs. Then for r ∈ (Q ∩ (0, 1)) n we may define as follows: By taking limits this definition may be extended to r ∈ (Q ∩ (0, 1)) N and to r ∈ (Q n−1 × R) ∩ (0, 1) n . We define a function γ(β) to be the maximal admissible higher level "balanced" word for a given β ∈ (1, 2). This will be a vector: The function γ(β) defined above corresponds to (γ(β)) 1 . Essentially on each plateau of γ(β), we define a new devil's staircase giving (γ(β)) 2 . Each plateau of this will then give rise to a further devil's staircase for (γ(β)) 3 , and so the process continues. Throughout this text we shall refer to (γ(β)) 1 = γ(β): we will need the vector when discussing D 1 (β), but only the scalar is needed for D 0 (β) and D 2 (β).
We will describe descendants of balanced and level n "balanced" extremal pairs as being Farey descendants. The vector r then functions almost as a coordinate system, telling you which pair you are descended from.
2.3. Bad n. As in [6], we say that a natural number n is bad for (a, b) if every n-cycle for T β intersects the hole (a, b).
In the β = 2 case, it is natural to discard n = 2 as there is only one 2-cycle, making for an uninteresting definition. As β decreases, gradually each n will have fewer cycles and thus once we have only one n-cycle remaining we wish to discard this n. For even n = 2k this occurs when 1· = (10 k−2 10 k ) ∞ and for odd n = 2k + 1 this occurs at 1· = (10 k−1 10 k ) ∞ . For each β ∈ (1, 2) let N β denote the least n such that there exists at least two n-cycles for β.
Then let B β (a, b) denote the set of n > N β such that n is bad for T β . Then we define

Transfer of results from the doubling map
In this section we transfer what results we can from the case of the doubling map as studied by Glendinning and Sidorov in [3] and Hare and Sidorov in [6].
Proof. As for β = 2, write a = 10 k 1 . . . for some k ≥ 0. The consider the following subshift: Then A ⊂ J β (a, b). A contains periodic orbits (10 m ) for any m > k +1, thus (a, b) has only finitely many bad n.
By definition we have that The restriction for small b is more different to the β = 2 case as it involves γ(β).
Proof. We show this result for γ(β) ∈ Q; the case where γ(β) ∈ R then follows by taking limits. Write inf X γ(β) = u ∞ and let u γ 1 denote the smallest shift of the left Farey parent of γ(β). Then b = u K v for some K and some v ≺ u. Consider the following shift: Clearly B K ⊂ J β (a, b). As |u| and |u γ 1 | are coprime and we are allowed any k > K, this shift B will contain periodic orbits of any suitably long length. Thus (a, b) has finitely many bad n.
We have the following corollaries: , it is clear that any point strictly below 0 inf X γ(β) must fall into the hole (0 inf X γ(β) , inf X γ(β) ). Then the result follows from the above lemma by considering the limit of B K as K → ∞.
Proof. This hole is a preimage of (0, inf X γ(β) ) which by the previous corollary has avoidance set X γ(β) .
These results describe what are essentially the easy cases, where a ≥ 1/β or b ≤ inf X γ(β) . The interesting behaviour that is more difficult to describe thus occurs within this region for (a, b): Notice that as β → 2, I β approaches as expected the (1/4, 1/2) × (1/2, 3/4) region seen for the doubling map. However as β → 1, 3.2. Extremal pairs. We now commence to transfer results from the doubling map. Essentially, if an extremal pair is admissible for a given β, then all results involving that extremal pair will still hold for that β. We formalise this as follows.
is an extremal pair such that {s, t} N is admissible for β and (s ∞ , t ∞ ) ∈ I β . Let u and v be words such that s = uv and t = vu. Then for any ǫ > 0, we have Proof. These results are shown for the case of balanced pairs (or n-th level balanced in the place of Farey descendants) in [3] and [6]. We collect them here in a bid to make clearer precisely what combinatorial property of words each result is relying upon, and so for the sake of clarity we repeat the arguments here and alter them as necessary to encompass the general case. Let (s, t) be an extremal pair. Item (1) -that {σ n s ∞ : n ≥ 0} ⊆ J β (s ∞ , t ∞ ) -follows immediately from the definition.
For item (2), to show that J β (s ∞ , ts ∞ − ǫ) is uncountable, we follow [3, Lemma 2.2]. Let N ∈ N and define W N = {σ i w : w is composed of blocks ut m with m > N}.
Because {s, t} N is admissible, we know that W N is admissible for all N. Furthermore W N is shift invariant and has positive entropy (and therefore positive Hausdorff dimension). For any ǫ > 0, there exists an N such that ts ∞ − ǫ < ts N . Then we claim W N ⊂ J β (s ∞ , ts ∞ − ǫ). To see this, notice that ut m = s m u. By extremality, the only shifts we need be concerned about are those beginning with s or t. Any shift beginning s will be of the form s i u < s ∞ , so avoids the hole. Any shift beginning t either has multiple ts and so avoid the hole, or begins ts m u for some m > N. This therefore also avoids the hole for large enough N.
The case J β (st ∞ + ǫ, t ∞ ) is similar, using shifts with t m v = vs m . Either of these shifts will then avoid (st ∞ + ǫ, ts ∞ − ǫ). This leads immediately to item (3), following [6,Theorem 3.6]. Notice that the orbit (ut m ) has period mq + j. If j and q are coprime, then for every ℓ there exists k such that ℓ ≡ kj mod q. So by considering points of the form for sufficiently large m i > N, we can create orbits of any sufficiently large length which avoid the hole. Thus whenever |u| and |s| are coprime, we have that (s ∞ , ts ∞ − ǫ), (st ∞ + ǫ, t ∞ ) and (st ∞ + ǫ, ts ∞ − ǫ) have finitely many bad n. Item (4) is immediately clear from the definition of maximal extremality combined with (1), and therefore in fact functions as an equivalent definition of maximal extremal.
For (5), we follow [3, Lemma 2.12] and use induction to show that J β (s ∞ , t ∞ ) is countable for Farey descendants of maximal pairs. The result clearly holds for 0th level descendants; that is to say for the maximal pair itself. Assume the claim holds for all kth level descendants (s k , t k ) = (s (r 1 ,...r k ) , t (r 1 ,...r k ) ). We show it must then hold for the (k + 1)st level. Write r k+1 = p k+1 /q k+1 . Note firstly that as J β (s ∞ k , t ∞ k ) is countable, we wish to show that all but countably many points of (s ∞ k , t ∞ k ) must fall into (s ∞ k+1 , t ∞ k+1 ). Any word (s k+1 , t k+1 ) is by definition a balanced word on the alphabet {s k , t k }, with length in this alphabet q k+1 . The only shifts of s k+1 that fall into (s ∞ k+1 , t ∞ k+1 ) are those beginning with s k or t k . Label these (in order) as x 1 , . . . , x q k+1 . Balanced words correspond to ordered orbits as discussed in [2] and [5]. This means that any interval [x i , x i+1 ] will be mapped by σ q k+1 to some other interval [x j , x j+1 ], and by repeatedly applying σ q k+1 we will cycle through all possible j ∈ {1, . . . , q k+1 − 1}. One of these intervals is [s ∞ k+1 , t ∞ k+1 ]. Therefore all but countably many points in (x 1 , x q k+1 ) will fall into (s ∞ k+1 , t ∞ k+1 ).
The only remaining possibilities are points in (s ∞ k , x 1 ) and points in (x q k+1 , t ∞ k ). Applying σ q k+1 to these intervals maps them to (s ∞ k , x i ) and (x j , t ∞ k ) respectively for some i > 1 and j < q k+1 . Thus again by applying σ q k+1 repeatedly we see that all but countably many points must fall into (s ∞ k+1 , t ∞ k+1 ). Therefore item (5) holds and J β (s ∞ , t ∞ ) is countable for Farey descendants of maximal pairs.
The final item (6) is more complex, with a degree of subtlety as to why the result holds only for Farey descendants, not for either all extremal pairs or only maximal extremal pairs. 2 We follow [3, Theorem 2.13]. Suppose (s, t) is a Farey descendant of a maximal extremal pair (u, v). Consider J β (s ∞ , t ∞ ). This is a countable subset of {u, v} N .
Then take any point in [ts ∞ , t ∞ ]. By applying σ q repeatedly we can see that all but countably many points in this interval must fall into Similarly, consider any point in [s ∞ , sts ∞ ]. Apply σ q repeatedly, and we see that all but countably many points must fall into (sts ∞ , ts ∞ ). Therefore we have that J β (sts ∞ , ts ∞ )\J β (s ∞ , ts ∞ ) must be countable. Hence J β (sts ∞ , ts ∞ ) is countable.
The cases with s and t reversed are similar.
Remark 3.6. It is key to note that this result relies very strongly on J β (s ∞ , t ∞ ) being countable, which holds only by item (5). This is where the proof fails if the pair (s, t) is not a Farey descendant of a maximal extremal pair.
Remark 3.7 (Continuity of boundaries). Notice that the boundaries of D 0 (β), D 1 (β) and D 2 (β) are continuous in (a, b), in the sense that if we have a sequence (a i , b i ) ∈ ∂D j (β) converging to a limit (a, b), then this limit point belongs to ∂D j (β) also. This allows us to extend results to include the points arising as limits of maximal extremal pairs.
We therefore wish to establish which extremal pairs are maximal for a given β. The above results will then combine to delimit the boundaries of D 0 (β), D 1 (β), and D 2 (β), with minor modifications for cases where for example s ∞ is admissible but st ∞ is not. Each set has a continuous boundary consisting of a countable set of plateaus given by [s ∞ , st ∞ ] in the case of D 0 (β) and D 2 (β) and by [s ∞ , sts ∞ ] in the case of D 1 (β), as shown in Figure 2. Notice that given a maximal pair (s, t), we have -up to a set of measure zero given by the limit points -that Figure 2. Lemma 3.5 for a maximal extremal pair (s, t). The dark grey shows points in D 2 (β), the light grey shows points in D 0 (β), and the white region shows where the Farey descendants of (s, t) will lie, so these points are in D 0 (β) and may or may not be in D 1 (β).
where (s r , t r ) are the Farey descendants of (s, t). This follows easily from Lemma 2.10 of [3] with minor modifications for general β. This ensures that once we have the correct maximal pairs, D 1 (β) is well defined.
If it is the case that a pair (s, t) is such that s ∞ is admissible but st ∞ is inadmissible, the above results are not significantly disrupted. Any inadmissible sequence must be replaced by the largest admissible sequence that is less than the intended inadmissible sequence. The results showing that a point is not in D i (β) for some i will clearly still apply as we have less admissible sequences meaning J β will if anything be smaller than previously shown. The difficulty is when we want to show that J β is large, as we must ensure the inadmissibility has not removed too much of J β .
Considered the balanced pairs (s, t) = (0-max, 1-min) discussed in the previous section. As shown by Glendinning and Sidorov, these pairs (when admissible) are maximal extremal. Each pair is admissible when the associated γ is less than γ(β). Notice that in the context of this problem, we consider these particular pairs because they are maximal extremal: that they are balanced is a side effect, not the reason for interest.
For β = 2, the holes formed from these pairs will customarily have two distinct preimages, formed by appending either a 0 or a 1 to both endpoints of the hole. As β decreases, the preimage formed by appending a 1 becomes inadmissible and so a particular hole may have a unique preimage. Because J β (a, b) is invariant under T β , this means that all results pertaining to the original hole will also apply to its unique preimage. This leads us to the following conclusion: Let (s, t) = (0-max, 1-min) be the maximal extremal balanced pair corresponding to γ < γ(β). Then the pairs (0 k -max, 0 k−1 1-min) are also maximal extremal.
Notice that when γ(β) ∈ [ 1 n+1 , 1 n ), we have that 1· begins with 10 n−1 . This gives the correct range of 0 < k ≤ n to ensure a unique preimage. Also note that inf X γ(β) begins with 0 n , so as one would expect these pairs will fall into the region This means that by considering all suitable γ and k, balanced pairs will cover the range (0 n 1, 01·) × (0 n−1 1, 1 inf X γ(β) ). However these are all available balanced pairs, and so we cannot expect the remaining region R β = (0 inf X γ(β) , 0 n 1) × (inf X γ(β) , 0 n−1 1) to involve balanced pairs. Figure 3 shows the balanced pairs giving D i (β) for I β \ R β , with β ≈ 1.427. Note that D 1 (β) is shown by the dark grey and the white areas "between" the light and dark grey. These white areas do exist but are so small as to be barely visible, therefore the inset image shows a magnification as indicated. Notice how the overall image has the same section repeated three times at different scales. This corresponds to the shifting of the balanced words as in Lemma 3.8 above. Furthermore there are vertical intervals that appear to be jumps, at a = 1/β k , such that ∂D 2 (β) = ∂D 1 (β) = ∂D 0 (β). This corresponds to where s ∞ is admissible but st ∞ is inadmissible. 4. The region R β = (0 inf X γ(β) , 0 n 1) × (inf X γ(β) , 0 n−1 1) The previous sections have described D 0 (β), D 1 (β) and D 2 (β) for a ≤ 0 inf X γ(β) and for a ≥ 0 n 1. In countably many cases the remaining region R β is empty. This occurs precisely when 1· = w ∞ γ for γ = k (n + 1)k − 1 ∈ (1/(n + 1), 1/n], with k, n ∈ N. For these values of β, the description of the D i (β) is already complete and needs only balanced pairs. The doubling map is one of these exceptional cases (n = k = 1), as is the golden ratio β = (1 + √ 5)/2 (n = 2, k = 1). For the remaining β we hence need to find the maximal extremal pairs that fall into the region R β .
We begin by showing that if the pairs defined above fall into R β , then they must be maximal. We do this by induction, exploiting the tree structure of the definition.
Remark 4.2. Following the above proof it is also easy to see that pairs (s, t) = (0u0 k−1 , u0 k ) must also be maximal extremal: notice that in this case it is simpler as t ∞ = σs ∞ . Proof. This has been shown for the tree of balanced words in [11,Lemma 2.5] and again is more a property of the tree construction than of the specific words. We repeat the proof here for completeness' sake. To show maximality of (s, t), we aim to show that J β [s ∞ , t ∞ ] = ∅. Consider x ∈ (0, 1). We know by maximal extremality that We know by the tree construction that s ∞ 1 < s ∞ and t ∞ 1 < t ∞ , so we may restrict to x ∈ [s ∞ 1 , s ∞ ). Then σ |s 1 | (x) ∈ [s ∞ 1 , t ∞ 1 ], so restrict again. Continuing this process, we see that the only possible point avoiding [s ∞ , t ∞ ] must be s ∞ 1 . But then this shifts to t ∞ 1 ∈ [s ∞ , t ∞ ]. The above two lemmas combine to imply that if (s, t) is a suitable admissible extremal pair as described, then (s, t) is maximal extremal.
We now discuss which γ are associated with which β. We describe this in two ways: firstly, by giving the set of correct β for a particular γ and secondly by giving the correct γ in terms of a particular β.
Lemma 4.4. Let w and u denote the maximal and minimal cyclic shifts of the balanced word associated to γ ∈ Q with γ = 1/n and Farey parents given by γ 1 < γ 2 . Then for , we have that the admissible pairs (s, t) from the Farey tree formed by 0 and u γ are maximal extremal pairs.
Proof. Firstly, notice that the given interval is precisely the region where we have γ(β) ∈ [γ 1 , γ). Therefore at least part of the Farey tree generated by 0 and u γ will be admissible and give pairs (s, t) satisfying (s ∞ , t ∞ ) ∈ R β . Outside of these values of γ(β) we have that either the entirety of the tree will be inadmissible or the sequences will fall below R β .
It remains to explain why these intervals cover almost all of R β . To see this, suppose that (1) 1· ∈ [(w γ 1 ) ∞ , w ∞ γ ). Then for almost every β in this range there exists a maximal pair (s, t) in the tree from 0 and u γ such that s ∞ is admissible and st ∞ is inadmissible. Then consider the greatest admissible sequence in [s ∞ , st ∞ ]. This will end in 1· so may be rewritten as a finite sequence.
Lemma 4.6. The greatest admissible finite sequence for γ described above is equal to 0u γ 2 .
Proof. s ∞ is admissible so clearly the sequence st ∞ becomes inadmissible with the very first t. Therefore, to be admissible we should truncate from the maximal shift of s and replace the preceding 0 with a 1. We know that s begins 0u γ and must be the maximal shift beginning this way, so consider u γ = u 1 . . . u q = u γ 1 u γ 2 . There exists k such that σ k u γ begins w γ 2 , which will be the point at which to truncate the sequence. Then because u γ is balanced, we have that u 1 . . . u k−2 1 = u γ 2 . 3 This proves the lemma.
The descendants of the above maximal pairs will be given by taking an n-th level Sturmian pair (s r , t r ) and applying the map m : 0 → 0, 1 → u γ . This completes the description of D 1 (β). Figure 4 shows an approximation to D i (β) in the region R β for β ≈ 1.427. For this value of β, the region R β is very small, and D 1 (β) is once again too small to see distinctly.

Summary
We summarise the results in the following theorem.
If a maximal extremal pair (s, t) has s ∞ admissible and st ∞ inadmissible, then the above results hold with any inadmissible sequences replaced by the greatest admissible sequence in [s ∞ , st ∞ ].
In summary, the boundaries of D i (β) consist of a countable set of plateaus which are closely linked to the set of maximal extremal pairs for β, as explained in Section 3. The maximal extremal pairs are mainly balanced words, but for almost every β ∈ (1, 2) there is a small region where there are no admissible balanced words. In this region the maximal extremal pairs are formed by taking certain inadmissible balanced words and adding a 0 to make them admissible, as described in Section 4. We include some pictures of D i (β) for different values of β; see Figures 5 and 6.
We note the following result.  Proof. Each C i (β) is given by sup{b − a : (a, b) ∈ D i (β)} = max{b − a : (a, b) is a corner of D i (β)}.
5.1. Acknowledgements. The author would like to thank their supervisor Nikita Sidorov for his initial suggestion of the problem and his ongoing support, and Kevin Hare for kindly providing his code as a basis to help produce the figures.