For a finite alphabetΣ we denote byΣ∗ ≔ {u ∈ Σn : n ∈ N} the set of all finite words inΣand byΣNall infinite words. A semigroup homomorphismσ: Σ→ Σ∗ onΣ∗ or ΣN is called substitution. The name semigroup homomorphism is from the fact thatσ(u)↦→ σ(u0)σ(u1)σ(u2). . .is well-defined, for any finite of infinite wordu. As just indicated finite sequencesu = (ui)n−1i=0 ∈ Σn for somen ∈N may
2.2. SYMBOLIC SPACES 25 also be denoted byu0u1u2. . .un−1or may even be functionsu: {0, . . . ,n−1} → Σn and the same goes for infinite sequences. The length of anyu ∈Σ∗∪ΣNis given by|u| ≔ n, if u = u0. . .un−1 ∈ Σn, while |u| = ∞, if u ∈ ΣN. Moreover for any v∈Σ∗we define
|u|v ≔ |{n∈N:∀0≤i≤ |v| −1,un−i = vi}|,
to be theoccurences of v in u. As already used and known from sequences, letters of uare adressed by un for 0 ≤ n ≤ |u| −1, factorsof uare finite words of the formu[n,n+m] ≔(ui)n≤i≤n+mandsubwordsofuare factors, but may also be infinite, hence of the form u[n,∞]. The prefix of u of length n is defined to be the first n letters ofuand is denoted byu|n ≔u[0,n], while thesuffixof lengthnofuis given byu[|u|−n,|u|−1]and is only defined for finite words.
Remark 2.2.1. Take note thatN= {0,1,2,3, . . .}, whileN+ ={1,2,3,4, . . .}. Also we make use of the conventionΣ0 ≔{∅}, while the empty word is also denoted by ε.
Definition 2.2.2. Foru ∈ ΣN,v ∈ Σ∗, if it exists, the frequency of vinuis given by
fv(u)≔ lim
n→∞
|u|n|v n .
Further information for the frequency can be found in e.g. [70, Ch. 1.2.4], [71, Ch. 5.3,5.4].
Definition 2.2.3. A wordu ∈ ΣNis calledperiodic, if it exists av ∈Σ∗such that for all m ∈ N, u|(m|v|) = vm , where vm ∈ Σm|v| is the unique word that satisfies (vm)j = v(j mod|v|)for 0 ≤ j ≤ m|v|. The worduis calledultimately periodic, if it exists an infinite periodic subword ofu. Ifuis not ultimately periodic it is called aperiodic.
Definition 2.2.4. Let u ∈ Σ∗∪ΣN. The complexity function p ≔ pu: N → N counts all different factors of lengthn∈Ninu
p: n↦→ |{v∈Σn:vis a factor ofu}|.
Definition 2.2.5. ForΣ = {0,1}a wordu∈Σ∗∪ΣNis calledSturmian of level m in the casem≔sup{k∈N: p(l+1)= p(l)+1∀l<k}is a natural number, while form= ∞it is calledSturmian.
Lemma 2.2.6([70, Prp. 1.1.1]). A sequence u is ultimately periodic if and only if p is bounded, especially there is an n ∈Nsuch that p(n)≤ n.
For further details on the complexity function see also [70, Ch. 1.1.2]. With that factors of sequencesubecome interesting in itself and we denote byL(u)≔ {v∈Σ∗:vis a factor ofu}thelanguageofu. The language of Sturmian sequences is special in the sense that for alln ∈Nthere is only one word v∈ Ln(u) ≔ {w ∈ L(u) : |u| = n}such that v0,v1 ∈ Ln+1(u); vis then calledright special. The set of all right special words is denoted by LR(u) ≔ {v ∈ L(u) : vis right special}.
Later we want to see which sequences can be approximated by the language of an infinite word. The intuitive idea to say that two sequences are equal if they are the same for arbitrary prefixes can be put into action with the following metric, let u,v∈ΣN
d(u,v)≔ |u∨v|−c,
is a metric onΣN, whereu∨v ≔{w ∈Σ∗∪ΣN :wis a prefix ofuandv}denotes the longest prefix uand v share and 0 < c, ∞−c ≔ 0. It may be extended onto Σ∗∪ΣNby using that|u∨v|= inf{inf{n∈N:un ≠vn},|u|,|v|}. With that at hand a topology can be defined forΣNviad. Another way to define a topology is given via cylinder sets [w] ≔ {u ∈ ΣN : u|n = w} and both ways generate the same topology onΣN, which can be seen from [w]={u∈ΣN:d(v,u) (|w| −1)−c}for anyv ∈[w] and from{u∈ ΣN : d(v,u) < (|w| −1)−c}= {u∈ ΣN : d(v,u) ≤ |w|−c} we see that each cylinder is a clopen set anddis an ultra-metric. On this occasion we would like to introduce one of the most important maps onΣN, the left shift
S: ΣN→ ΣN, (ui)i∈N↦→(ui+1)i∈N.
It may also be defined forΣ∗and in this case we setS(ε)= εfor the empty word ε. AsS[w]= [S w] for anyw∈Σ∗it is clear thatS is continuous.
Definition 2.2.7(Subshift). A subshiftX ⊆ ΣNis a closed shift invariant set, that isS(X)⊆ X. Thelanguage of a subshift is given byL(X) ≔ {L(u) : u ∈ X}and the language of right special words is defined byLR(X) ≔ {LR(u) : u ∈ X}. For a sequenceu ∈ ΣN, the subshift of uis the set Xu ≔ {Sn(u) :n∈N}. Xu may be calledSturmian subshift (of level n) ifuis Sturmian (of leveln). Either, Xu or u are calledminimal, if every factor of uoccurs infinitely often inu with bounded gaps. That is, for every factorvof uexists anrv ∈ Nsuch that for all n ∈ N the wordvis a factor ofun. . .un+rv. In this caseumay also be calledrecurrentinstead of minimal.
For a minimal subshift Xu it follows for anyv ∈ Xu that Xv = Xu and hence L(Xu)= L(u)=L(v)=L(Xv), [70, Prp. 5.1.10]. One can then say that a subshift X is minimalifXu = Xfor allu ∈X. Furthermore, the complexity function is the same for all elements of a minimal subshift. In this sense all the information of
2.2. SYMBOLIC SPACES 27 a minimal subshift or its language is stored in every of its elements and notions like complexity and periodicity may be used for the subshift instead of stating the notion with respect to every element of the subshift.
Remark 2.2.8. In [70, Ch. 6.1] one can find that every Sturmian subshift is min-imal. By definition, the languageL(X) of a Sturmian subshift contains a unique right special word per length and for any w ∈ LR(X) it follows Sk(w) is a right special word for allk∈ {1,2, . . . ,|w|}.
Definition 2.2.9. Foru∈ΣNthetopological entropyis given by h≔h(u)≔ lim
n→∞
log|Σ|(p(n))
n .
Since p is monotone and p(n) ≤ |Σ|n the limit in the topological entropy is well-defined. Sequences with topological entropy 0 are said to be deterministic.
For a primitive substitutionζ(compare Definition 6.6.1), there is a constantC> 0 such that p(n) ≤ Cn for all n ∈ N. With that one can deduce that for every fixed pointuofζ anyv ∈ Xu is deterministic, [71, Prp. 5.12, 5.7]. Although the following definition are given for subshifts, they will only find use for minimal subshifts in this work.
Definition 2.2.10. Therepetitive function R: N+ →N+of a subshiftX maps any nto the smallestn′ such that any element ofL(X) with lengthn′ has all elements ofL(X) with lengthnas factors.
In other words let X be a subshift,n′ ≔ R(n) andv ∈ L(X), |v| = n′, then for anyw∈ L(X) with|w|= nit follows thatwis a factor ofv.
Lemma 2.2.11. For any subshift X and u∈X, one has pu(n)≤R(n)for all n∈N and if X
Proof. Let r = pu(n), then a word w can be constructed, which has r different factors of lengthn. That is|w|= n+rwith factorsw[i,i+n], where 0≤i≤ r−1 and w[i,i+n] ≠ w[j,j+n] fori ≠ j. One may believe that wbelongs to the shortest words
with this property and henceR(n)≥ n+r.
Definition 2.2.12(α-repetitive). LetXbe a subshift andα≥ 1, set Rα ≔lim sup
n→∞
R(n) nα . X is calledα-repetitiveifRαis finite and non-zero.
Take note that if 1 ≤ α < β and 0 < Rβ < ∞, then Rα = ∞. Similarly, if 0< Rα <∞, thenRβ = 0. The notionα-repetitive has been used before, compare [26] and we remark that the definition given in [26] will never be used in this work.
Remark 2.2.13.Letu∈ΣN, a subshiftXis said to be linearly repetitive orlinearly recurrent, if and only if, there exists a positive constantC, such thatR(n)≤Cn, for alln ∈N+. As aperiodicity of a subshift guarantees that the complexity function p(n) > n, for all n ∈ N+, Lemma 2.2.6 and p(n) ≤ R(n), Lemma 2.2.11, this yields that linearly repetitive or linearly recurrent and 1-repetitive are equivalent for aperiodic subshifts.
Definition 2.2.14(α-repulsive/-finite). LetX be a subshift andα≥1. Set ℓα≔ lim inf
n→∞ Aα,n, where for anyn≥ 2
Aα,n ≔inf
{︃|W| − |w|
|w|1/α : w,W ∈ L(X),wis a prefix and suffix ofW, |W|= nandW ≠w≠∅
}︃.
and ifℓαis finite and non-zero, then we say thatXisα-repulsive. Forn≥1 set Q(n)≔ sup{p∈N+: there existsW ∈ L(X) with|W|=nandWp ∈ L(X)}
and the subshiftXisα-finiteif the value Qα ≔lim sup
n→∞
Q(n) nα−1
is non-zero and finite. Also, for ease of notation, for a given wordv ∈ L(X), we letQ(v) denote the largest integer psuch thatvp ∈ L(X), in the case that no such pexists, we setQ(v)≔∞.
Remark 2.2.15. In a similar fashion toα-repetitive, if 1≤α < βand 0< ℓβ< ∞, thenℓα = 0 and if 0 < Qβ < ∞, then Qα = ∞. Whether for 0 < ℓα < ∞, then ℓβ = ∞ and if 0 < Qα < ∞, then Qβ = 0. To see this it is enough to check the properties for ℓα. Suppose that 0 < ℓβ < ∞. Thus, for n ∈ N+ sufficiently large, there exist wordsw,W ∈ L(X) withwa prefix and suffix ofW,|W|=nand W ≠w≠ ∅, so that
ℓβ
2 ≤ |W| − |w|
|w|1/β ≤2ℓβ.
2.2. SYMBOLIC SPACES 29 Hence,|w| ≥n(2ℓβ+1)−1, and
ℓβ|w|1/β−1/α
2 ≤ |W| − |w|
|w|1/α ≤ 2ℓβ|w|1/β−1/α. Therefore, we have thatℓα= 0.
Theorem 2.2.16([35, 28]). Forα ≥ 1, we have that X isα-repulsive if and only if it isα-finite.
Proof. Letα≥ 1 be fixed and letX beα-repulsive. Suppose thatQα = ∞. In this case there exist sequences of natural numbers (nk)k∈N+and (pk)k∈N+satisfying
1. (nk)k∈N+is increasing with pkn1−αk > k, and
2. there existsW(k)∈ L(X) with|W(k)|=nk andW(k)pk ∈ L(X).
Thus, we have that pk > 1, for allksufficiently large. SinceW(k)pk−1 is a prefix and a suffix ofW(k)pk we have that
|W(k)pk| − |W(k)pk−1|
|W(k)pk−1|1/α
= |W(k)|
|W(k)|1/α(p(k)−1)1/α = nk
nk1/α(pk −1)1/α ≤ 21/αnk(α−1)/α
pk1/α < 21/α k1/α, for allksufficiently large. Therefore, we have thatℓα =0.
Suppose that Qα = 0. Forn ∈ N+letV(n),v(n) ∈ L(X) be such that|V(n)| = n, v(n) ≠V(n)is a prefix and suffix ofV(n)and
|V(n)| − |v(n)|
|v(n)|α1 = Aα,n.
Since 0 < ℓα < ∞, this means that there exists a sequence (nk)k∈N+ of natural numbers such that 2|v(nk)| > |V(nk)|, for allk ∈ N+. Thus, for eachk ∈ N+, there exists aqk ≥2 such that
v(nk) = u(k)u(k)· · ·u(k)
⏞ˉˉˉˉˉˉˉˉˉˉ⏟⏟ˉˉˉˉˉˉˉˉˉˉ⏞
qk−1
z(k) and V(nk) =u(k)u(k)· · ·u(k)
⏞ˉˉˉˉˉˉˉˉˉˉ⏟⏟ˉˉˉˉˉˉˉˉˉˉ⏞
qk
z(k),
whereu(k),z(k) ∈ L(X) with 0<|z(k)|<|u(k)|. Hence, it follows that
⎛
⎜⎜
⎜⎜
⎜⎝
|V(nk)| − |v(nk)|
|v(nk)|α1
⎞
⎟⎟
⎟⎟
⎟⎠
α
= (|V(nk)| − |v(nk)|)α
|v(nk)| (2.2)
≥ |u(k)|α
qk|u(k)| = |u(k)|α−1 qk
≥ |u(k)|α−1
Q(u(k)) ≥ |u(k)|α−1
Q(|u(k)|), (2.3)
where the lengths of theu(k)are unbounded, as otherwise lim supk→∞Q(u(k))= ∞.
However, since by assumptionQα =0, we have lim inf
n→∞
nα−1 Q(n) =∞.
This together with (2.2) yields thatℓα = ∞. For the other direction suppose that Qα is non-zero and finite. This means there is a sequence of tuples ((nk,pk))k∈N+
so that the sequence (nk)k∈N+ is strictly monotonically increasing such that for the limit 0 < lim
k→∞pkn1−αk = Qα < ∞, and for each k ∈ N+ there exists a word W(k) ∈ L(X) with|W(k)|= nkand
W(k)W(k)· · ·W(k)
⏞ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏟⏟ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏞
pk
∈ L(X).
For a fixedk ∈N+, setting
W = W(k)W(k)· · ·W(k)
⏞ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏟⏟ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏞
pk
and w= W(k)W(k)· · ·W(k)
⏞ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏟⏟ˉˉˉˉˉˉˉˉˉˉˉˉˉ⏞
pk−1
,
we have that
|W| − |w|
|w|1/α = n1−1/αk (pk −1)1/α =
(︄ pk
pk−1 nα−1k
pk
)︄1/α .
This latter value converges toQ−1/αα , and so, we have thatℓαis finite.
By way of contradiction, suppose ℓα = 0. This implies there is a strictly increasing sequence of integers (nm)m∈N+, so that there exist W(nm),w(nm) ∈ L(X) withW(nm) ≠w(nm),|W(nm)|= nm,w(nm)is a prefix and suffix ofW(nm)and
|W(nm)| − |w(nm)|
|w(nm)|1/α < 1 m.
This means the two occurrences of w(nm) in W(nm) overlap. Thus, there exist p = pnm ∈N+so that
w= u u⏞ˉˉˉ⏟⏟ˉˉˉ⏞· · · u
p−1
v and W =u u⏞ˉˉˉ⏟⏟ˉˉˉ⏞· · · u
p
v,
whereu = u(nm),v = v(nm) ∈ L(X) with 0 < |v| < |u|. Combining the above gives p|u|1−α >mα, and so,Qα = ∞, contradicting the assumption thatQα is finite.
2.3. SUBSTITUTIONS OF ROTATION TYPE 31