On the specification property and synchronisation of unique q -expansions
aa r X i v : . [ m a t h . D S ] J un ON THE SPECIFICATION PROPERTY AND SYNCHRONISATION OFUNIQUE q -EXPANSIONS RAFAEL ALCARAZ BARRERA
In loving memory of Patricia Barrera
Abstract.
Given a positive integer M and q ∈ (1 , M + 1] we consider expansions in base q for real numbers x ∈ [0 , M/q −
1] over the alphabet { , . . . , M } . In particular, we study somedynamical properties of the natural occurring subshift ( V q , σ ) related to unique expansionsin such base q . We characterise the set of q ∈ V ⊂ (1 , M + 1] such that ( V q , σ ) has thespecification property and the set of q ∈ V such that ( V q , σ ) is a synchronised subshift.Such properties are studied by analysing the combinatorial and dynamical properties of thequasi-greedy expansion of q . We also calculate the size of such classes as subsets of V givingsimilar results to those shown by Blanchard [11] and Schmeling in [36] in the context of β -transformations. Introduction
Since their introduction in R´enyi’s [35] and Parry’s [34] seminal papers, the theory of expan-sions in non-integer bases, colloquially known as β -expansions or q -expansions , has receivedmuch attention by researchers in many areas of mathematics, most notably ergodic theory,fractal geometry, number theory and symbolic dynamics.Let us remind the reader the setting of q -expansions. Let M ∈ N and setΣ M = ∞ Y i =1 { , , . . . , M } equipped with the product topology. Given q ∈ (1 , M + 1] for every x ∈ I q,M = [0 , M/ ( q − x = x x . . . ∈ Σ M satisfying(1.1) x = π q ( x ) := ∞ X i =1 x i /q i The sequence x is called an expansion of x in base q (or simply a q -expansion of x ).The greedy q -shift (most commonly known as β -shift ), emerges from the set of expansionsgenerated by the greedy algorithm for x ∈ [0 , q -shift is the subshift (Σ q , σ )given by Σ q = { x ∈ Σ M : σ n ( x ) α ( q ) for every n ≥ } Date : 9th June 2020.2010
Mathematics Subject Classification.
Primary 37B10, 11A63; Secondary 37B40, 68R15.
Key words and phrases.
Expansions in non integer bases, specification property, synchronised systems,Hausdorff dimension.Research of R. Alcaraz Barrera was sponsored by CONACYT-FORDECYT 265667. where α ( q ) stands for the quasi-greedy expansion of 1 in base q , that is, the lexicographicallylargest infinite q -expansion of 1. The properties of (Σ q , σ ) have been studied extensively. Forexample, it is widely known that the topological entropy of (Σ q , σ ) is log( q ). The followingtheorem summarises the results regarding the symbolic dynamics and the size of certainclasses of q -shifts —see [10], [19], [36] and [38]. Theorem 1.1.
Let q ∈ (1 , M + 1] and (Σ q , σ ) be the corresponding greedy q -shift. Then: i ) (Σ q , σ ) is topologically mixing for every q ∈ (1 , M + 1] . ii ) (Σ q , σ ) is a subshift of finite type if and only if α ( q ) is periodic. Moreover, C = { q ∈ (1 , M + 1] : (Σ q , σ ) is a subshift of finite type } is a countable and dense subset of (1 , M + 1] . iii ) (Σ q , σ ) is a sofic subshift if and only if α ( q ) is eventually periodic. Moreover, C = { q ∈ (1 , M + 1] : (Σ q , σ ) is a sofic subshift } is a countable subset. iv ) (Σ q , σ ) has the specification property (see Definition 2.1 iii ) ) if and only if α ( q ) does notcontain arbitrarily long strings of consecutive ’s. Moreover C = { q ∈ (1 , M + 1] : (Σ q , σ ) has the specification property } is an uncountable subset of Lebesgue measure zero and dim H ( C ) = 1 . v ) (Σ q , σ ) is synchronised (see Definition 2.1 iv ) ) if and only if the orbit of α ( q ) is notdense in Σ q . Moreover C = { q ∈ (1 , M + 1] : Σ q is synchronised } is a meagre set in (1 , M + 1] . vi ) The set C = (1 , M + 1] \ C is a residual set. Perhaps the most noticeable feature of expansions in non-integer bases is the fact thatthey are not always unique. In fact, for any k ∈ N ∪ {ℵ } ∪ { ℵ } and any M ∈ N , there is q ∈ (1 , M + 1] and x ∈ I M,q such that x has precisely k different q -expansions—see [16, 39].Sidorov in [37] describes the generic behaviour of the set of expansions; that is, for any q ∈ (1 , M + 1) and for Lebesgue-almost-every x ∈ I M,q has a continuum of q -expansions. Thesituation described above is of course completely different to the usual expansions in integerbases where every number has a unique M -expansion except for a countable set of exceptions,the M -adic rationals, that have precisely two.Another particularly well-studied topic in expansions in non-integer bases is the set ofnumbers in I M,q with a unique q -expansion. The properties of this set have received lotsof attention recently, see for example [2, 4, 8, 14, 22, 23, 29] and references therein. Let usremind the reader of this setting. For q ∈ (1 , M + 1] the univoque set on base q is U q := { x ∈ [0 , M/ ( q − x has a unique q -expansion } . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 3 We consider U q := π − q ( U q ) ⊂ Σ M to be the corresponding set of expansions. The pair ( U q , σ ) is not necessarily a subshift [14,Theorem 1.8]. However, the set(1.2) V q := n x ∈ Σ M : α ( q ) σ n ( x ) α ( q ) for every n ≥ o where α ( q ) = ( M − α i ( q )), is a closed, forward- σ -invariant and non-empty subset of Σ M forevery q ∈ [ q G ( M ) , M + 1] where q G ( M ) is the generalised golden ratio introduced by Bakerin [8] (see also [2, Lemma 2.4]). We will refer to ( V q , σ ) as the symmetric q -shift . A naturalquestion to ask is when the symmetric q -shift satisfies similar properties to the properties ofthe greedy q -shift. As we will see along the paper, the behaviours of the subshifts (Σ q , σ )and ( V q , σ ) are completely different. In this direction, it was shown in [26] (see also [3])that the entropy function, i.e. H : (1 , M + 1] → [0 ,
1] given by H ( q ) = h top ( V q ), is a devilstaircase. In [2] a full description of the plateaus of the entropy function, as well as a fulldescription of its bifurcation set, was given. Also, in [2, Theorem 1] the set of q ’s such thatthe symmetric q -shift is a transitive subshift was characterised using the symbolic propertiesof the quasi-greedy expansion of 1 in base q . It is also worth mentioning that in [2] it wasshown that the sets T = { q ∈ [ q G ( M ) , M + 1] : ( V q , σ ) is topologically transitive } and N T = { q ∈ [ q G ( M ) , M + 1] : ( V q , σ ) is not topologically transitive } both have positive Lebesgue measure. Furthermore, for every q ∈ ( q G ( M ) , q T ( M )), thesubshift ( V q , σ ) is not transitive. We will give a brief explanation of the constants q G ( M )and q T ( M ) in Section 2.The objective of the paper is to solve some questions similar to the ones posed by Blanchardin [11] in the context of symmetric q -shifts and to develop a similar result to Theorem 1.1—see [36, p. 693]. For this purpose, inspired by [11] we introduce the following classes ofsubshifts: Definition 1.2.
Let V = V ( M ) = n q ∈ (1 , M + 1] : α ( q ) σ n ( α ( q )) α ( q ) for every n ≥ o . We define the following classes of symmetric q -shifts: C ′ = C ′ ( M ) = { q ∈ V : ( V q , σ ) is a subshift of finite type } C ′ = C ′ ( M ) = { q ∈ V : ( V q , σ ) is a strictly sofic subshift } C ′ = C ′ ( M ) = { q ∈ V : ( V q , σ ) is a subshift with the specification property } C ′ = C ′ ( M ) = { q ∈ V : ( V q , σ ) is a synchronised subshift } C ′ = C ′ ( M ) = V \ C ′ . RAFAEL ALCARAZ BARRERA
From [14, Theorem 1.7] it is not difficult to see that for Lebesgue-almost-every q ∈ (1 , M +1], the subshift ( V q , σ ) is a subshift of finite type. However, we can ask about the size of theclasses C ′ , C ′ , C ′ , C ′ and C ′ and their topological structure as subsets of V . Also, we want tounderstand what the symbolic properties of α ( q ) are when q belongs to one of the consideredclasses.Using the results obtained by De Vries and Komornik in [14, Theorem 1.7, 1.8] (see also[32, Theorem 1.3]) it is immediate that the class C ′ is countable and dense in V . Moreover, q ∈ C ′ if and only if α ( q ) is periodic. As a consequence of [24, Proposition 2.14], Kalle andSteiner characterised the class C ′ ; namely, q ∈ C ′ if and only if α ( q ) is eventually periodic.It is not difficult to check that C ′ is a countable set and that C ′ ⊂ U = U ( M ) = n q ∈ (1 , M + 1] : α ( q ) ≺ σ n ( α ( q )) ≺ α ( q ) for every n ≥ o ∪ { M + 1 } . The main result of our work is the following.
Theorem A.
Let M ∈ N , q ∈ [ q G ( M ) , M + 1] and consider ( V q , σ ) the symmetric q -shift.Then: i ) ( V q , σ ) has the specification property if and only if α ( q ) is a strongly irreducible sequence(see Definition 4.1) and there exists K ∈ N such that d ( σ k ( α ( q )) , α ( q )) ≥ / K for every k ∈ N . ii ) ( V q , σ ) is synchronised if and only if α ( q ) is an irreducible sequence and α ( q ) , α ( q ) arenot dense in V q . Moreover dim H ( C ′ ) = 1 . iii ) dim H ( C ′ ) = 1 . The structure of the paper is the following. In Section 2 we recall the relevant concepts ofsymbolic dynamics and unique q -expansions needed to develop our investigation. In Section3 we introduce the natural approximation from below of a symmetric subshift ( V q , σ ). Usingthis approximation we prove that every transitive symmetric subshift is mixing and coded.In Section 4 we will characterise the elements of the class C ′ . In Section 5 we will studythe classes C ′ and C ′ . Finally, in Section 6 we will calculate the Hausdorff dimension of theclasses C ′ , C ′ and C ′ . 2. Preliminaries
In this section, we recall some basic tools and definitions used in our study. We will adoptmost of the notation used in [2] and [23] and in Subsection 2.3 we summarise the results inthose papers relevant for this work.We refer the reader to [33] for a thorough exposition of symbolic dynamics. A standardreference for the theory of Hausdorff dimension is [18]. Finally, detailed works on uniqueexpansions in non-integer bases are [13, 14, 25, 28].2.1.
Symbolic Dynamics.
We recall some basic notions of symbolic dynamics firstly. Fix M ∈ N . We call the set { , . . . , M } an alphabet and its elements are called symbols . A word ω = w . . . w n is a finite string of symbols. We denote the length of a word ω by | ω | . Given twowords ω = w . . . w n and ν = v . . . v m their concatenation is the word ων = w . . . w n v . . . v m . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 5 Also, ω k is the word obtained by concatenating ω with itself k times and ω ∞ is the infiniteconcatenation of ω with itself. We denote the empty word by ǫ .We consider Σ M = ∞ Y n =1 { , , . . . , M } , i.e. Σ M is the set of infinite one-sided sequences whose symbols belong to { , . . . , M } . It isa well known fact that Σ M is a compact space with the product topology. Also, the producttopology on Σ M is equivalent to the topology induced by the distance given by(2.1) d ( x , y ) = (cid:26) − j if x = y , where j = min { i : x i = y i } ;0 otherwise.The one-sided shift map σ : Σ M → Σ M is given by σ ( x ) = σ (( x i )) = ( x i +1 ). We callthe pair (Σ M , σ ) a full one-sided shift . We call a sequence x ∈ Σ M periodic if there exist m ∈ N such that σ m ( x ) = x . The smallest m satisfying this property is called the period of x and the word x . . . x m is called the periodic block of x . A sequence x ∈ Σ M is said to be eventually periodic if there exist m ∈ N such that σ m ( x ) is a periodic sequence and σ n ( x ) isnot a periodic sequence for every 0 ≤ n ≤ m − ω = w . . . w n is called a factor of a sequence x ∈ Σ M if there exists k ∈ N suchthat x k +1 . . . x k + n = w . . . w n . If k = 1 we say that ω is a prefix of x . Note that the notionsof factor and prefix can be defined on finite words in a similar fashion. Then, if a word ω of length n is a factor of a word ν and w . . . w n = v m − n +1 . . . v m where | ν | = m we call ω a suffix of ν of length n .Given a sequence x ∈ Σ M we define its reflection as x = ( x i ) = ( M − x i ) . For a word ω wedefine the reflection of ω similarly. In this case ω = w . . . w n = w . . . w n . If ω is a word oflength n such that w n < M we write ω + = w . . . w + n = w . . . w n − w n + 1and if ω is a word of length n with w n > ω − = w . . . w − n = w . . . w n − w n − . Throughout this work, we use the lexicographic order . Given x , y ∈ Σ M we say that x ≺ y if there exist n ∈ N such that x i = y i for every i ≤ n −
1, and x n < y n . Also, we write x y if x = y or x ≺ y . We can define the lexicographic order between words of the same lengthin a similar way.A subshift ( X, σ ) is a pair where X is a non-empty, closed and forward- σ -invariant subsetof Σ M and σ is understood to be σ | X . From [33, Theorem 6.1.21] we have that for everyclosed, non-empty and forward- σ -invariant subset X of Σ M there is a set of finite words F with symbols in { , . . . , M } such that X = X F where X F = { x ∈ Σ M : ω is not a factor of x for any ω ∈ F } . If F can be chosen to be finite we say that ( X, σ ) is a subshift of finite type . We say that asubshift (
X, σ ) is a sofic subshift if it is a factor of a subshift of finite type, i.e. there existsa subshift of finite type ( X ′ , σ ) (not necessarily in the same alphabet) and a semi-conjugacy RAFAEL ALCARAZ BARRERA h : X ′ → X , i.e. a continuous and surjective map such that h ◦ σ | X ′ = σ | X ◦ h . Alternatively,a subshift ( X, σ ) is sofic if there is a labelled graph G = ( G , E ) which represents ( X, σ ).For a subshift (
X, σ ) and n ∈ N , the set of admissible words of length n is given by B n ( X ) = { ω : ω is a factor for some x ∈ X and | ω | = n } . For n = 0, B n ( X ) = { ǫ } . We define the language of X by L ( X ) = ∞ [ n =0 B n ( X ) . Given M ∈ N and a subshift ( X, σ ) of Σ M we define the topological entropy of ( X, σ ) by h top ( X ) = lim n →∞ (1 /n ) log( B n ( X ))where log = log M +1 and X, σ ) and ω ∈ L ( X ) we define the follower set of ω to be F X ( ω ) = { ν ∈ L ( X ) : ων ∈ L ( X ) } ;and the prefix set of ω to be P X ( ω ) = { υ ∈ L ( X ) : υω ∈ L ( X ) } . Given m ∈ N and a word ω ∈ L ( X ) we denote by F mX ( ω ) = { υ ∈ F X ( ω ) : | υ | = m } . Definition 2.1.
We say a subshift (
X, σ ): i ) is topologically transitive , if for every ordered pair of words υ, ν ∈ L ( X ) there is a word ω ∈ L ( X ) such that υων ∈ L ( X ). This is equivalent to having a point x ∈ X such that { σ n ( x ) } ∞ n =0 is a dense subset of X ; ii ) is topologically mixing , if for every ordered pair of words υ, ν ∈ L ( X ) there exists N = N ( υ, ν ) ∈ N such that for every n ≥ N there is a word ω ∈ B n ( X ) such that υων ∈ L ( X ); iii ) has the specification property , or simply has specification , if there exist S ∈ N such thatfor any υ, ν ∈ L ( X ) there exist ω ∈ B S ( X ) such that υων ∈ L ( X ), i.e. every two words υ and ν can be connected by a word ω of length S ; iv ) has the almost specification property , or simply A-specification , if there exist S ∈ N suchthat for any υ, ν ∈ L ( X ) there exists ω ∈ L ( X ) such that υων ∈ L ( X ) and | ω |≤ S ; v ) is a coded system , if X = ∞ [ n =1 X n , where ∞ S n =1 X n denotes the topological closure of ∞ S n =1 X n , each ( X n , σ ) is a transitivesubshift of finite type, and X n ⊂ X n +1 for every n ∈ N - see[20, Theorem 2.1] and [30]. N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 7 It is well known that transitive subshifts of finite type and transitive sofic subshifts arecoded. Also, observe that every subshift (
X, σ ) with specification is topologically transitive,and in fact mixing. It is easy to show that the almost specification property coincides withthe specification property if (
X, σ ) is topologically mixing. Finally, we would like to remarkthat all coded systems are topologically transitive [11].We now state the specification property in a more convenient way for our purposes. Givena subshift (
X, σ ) we define s n = s n ( X ) = inf { k ∈ N : for every υ, ν ∈ B n ( X ) there exists ω ∈ B k ( X )such that υων ∈ L ( X ) } . Then (
X, σ ) has specification if and only if lim n →∞ s n < ∞ . If such limit exists we call it thespecification number of ( X, σ ) and we denote it by s X .Given a transitive subshift ( X, σ ) a word ω ∈ L ( X ) is said to be intrinsically synchron-ising if whenever υω and ων ∈ L ( X ) we have υων ∈ L ( X ) . A transitive subshift (
X, σ ) is synchronised if there exists an intrinsically synchronising word ω ∈ L ( X ).Finally, we list the following list of implications stated in [12] for topologically mixingsubshifts:(2.2)Finite Type ⇒ Sofic ⇒ Specification ⇔ A-specification ⇒ Synchronised ⇒ Coded.2.2.
Hausdorff Dimension.
Let A be a subset of a metric space X . Recall that a collectionof subsets of X , U = { U λ } λ ∈ Λ is an open cover of A if each U λ ∈ U is an open set and A ⊂ [ λ ∈ Λ U λ . If U = { U i } ∞ i =1 is a countable open cover of A and diam( U i ) ≤ δ for a given δ > U is a δ -cover of A .Fix s ≥
0. For δ > H sδ ( A ) := inf ( ∞ X i =1 diam( U i ) s : { U i } ∞ i =1 is a δ − cover of A ) . The s -dimensional Hausdorff measure of A is defined to be H s ( A ) := lim δ → H sδ ( A ) . The
Hausdorff dimension of A is given bydim H ( A ) = inf { s : H s ( A ) = 0 } = sup { s : H s ( A ) = ∞} . We will only consider X = R with the usual topology in this work.Given a subset A ⊂ R and x ∈ A we define, following [40], the local Hausdorff dimensionof A at x to be dim locH ( A, x ) = lim δ → dim H (( x − δ, x + δ ) ∩ A ) . The following standard result is useful to compute the Hausdorff dimension of a subset of R . RAFAEL ALCARAZ BARRERA
Lemma 2.2.
Let X be a metric space and A ⊂ X be a compact subset. Then, dim H ( A ) = sup x ∈ A n dim locH ( A, x ) o . Moreover, if A ⊂ X satisfies that its topological closure A is compact, then dim H ( A ) = sup x ∈ A n dim loc H ( A, x ) o if the map x dim locH ( A, x ) is continuous on A . Expansions in non-integer bases.
Let us bring up to mind the properties of q -expansions used in our study.Fix M ∈ N and let q ∈ (1 , M + 1]. The greedy q -expansion of x ∈ I M,q = [0 , M/q −
1] is thelexicographically largest q -expansion of x , and the quasi-greedy q -expansion of x ∈ I M,q \ { } is the lexicographically largest q -expansion of x with infinitely many non-zero elements. Wedenote by β ( q ) = ( β i ( q )) the greedy q -expansion of 1, and the quasi-greedy q -expansion of 1is denoted by α ( q ) = ( α i ( q )). It is not hard to check that if β ( q ) is not a finite sequence, then β ( q ) = α ( q ) and if β ( q ) is a finite sequence then α ( q ) = ( β ( q ) . . . β k − ( q ) β k ( q ) − ) ∞ where k satisfies that β j ( q ) = 0 for every j > k . We define V = V ( M ) = n q ∈ (1 , M + 1] : α ( q ) σ n ( α ( q )) α ( q ) for all n ≥ o and V = V ( M ) = { α ∈ Σ M : α σ n ( α ) α for all n ≥ } . The following result is essentially due to Parry [34]; see also [7, Theorem 2.5], [15, Proposition2.3].
Lemma 2.3.
The map
Φ : V → V given by Φ( q ) = α ( q ) is strictly increasing and biject-ive. Moreover, Φ is continuous from the left and Φ − is strictly increasing, bijective andcontinuous. Let us remind the reader that for every q ∈ (1 , M + 1] the univoque set on base q is givenby U q := { x ∈ I M,q : x has a unique q -expansion } , and U q := π − q ( U q ) is the corresponding set of q -expansions. It was shown in [14] that everysequence x ∈ U q satisfies the following lexicographic inequalities: (cid:26) σ n ( x ) ≺ α ( q ) whenever x n < M,σ n ( x ) ≻ α ( q ) whenever x n > . Also, recall that the set of univoque bases is given by U = U ( M ) := { q ∈ (1 , M + 1] : 1 has a unique q -expansion } . The set U has Lebesgue measure zero and full Hausdorff dimension—see [17, 13, 26]. More-over, the topological closure of U , U , is a Cantor set [15, Theorem 1.2]. The sets U and U were characterised symbolically in [15, Theorem 2.5, Theorem 3.9] as follows: N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 9 Lemma 2.4.
Let M ∈ N . Then: i ) U = n q ∈ (1 , M + 1) : α ( q ) ≺ σ n ( α ( q )) ≺ α ( q ) for all n ≥ o ∪ { M + 1 } ; ii ) U = n q ∈ (1 , M + 1] : α ( q ) ≺ σ n ( α ( q )) α ( q ) for all n ≥ o . Clearly U ( U ( V . The topological and symbolic properties of U , U and V aresummarised in the following theorem—see [14, 15]. Theorem 2.5.
Let M ∈ N then: i ) U \ U and V \ U are both countable; ii ) U \ U is dense in U . Moreover, if q ∈ U \ U then α ( q ) is periodic; iii ) V \ U is discrete and dense in V . Moreover, if q ∈ V \ U then α ( q ) is periodic. iv ) If q ∈ V \ U then α ( q ) is periodic. It follows from [15, Lemma 3.5] that the smallest element of V is the generalised goldenratio , denoted by q G = q G ( M ), and defined as(2.3) q G ( M ) = (cid:26) k + 1 if M = 2 k,k + 1 + √ k + 6 k + 5 / M = 2 k + 1 . Furthermore α ( q G ) = k ∞ if M = 2 k and α ( q G ) = (( k + 1) k ) ∞ if M = 2 k + 1. Thus, q G ∈ V \ U . Also, it was shown in [27] that the smallest element of U is the so-called Komornik–Loreti constant , denoted by q KL = q KL ( M ), which is defined using the classical Thue-Morse sequence ( τ i ) ∞ i =0 ; this sequence is defined as follows: τ = 0, and if τ i has alreadybeen defined for some i ≥
0, then τ i = τ i and τ i +1 = 1 − τ i . The Komornik-Loreti constantis defined explicitly by(2.4) α ( q KL ) = ( λ i ) ∞ i =1 := (cid:26) ( k + τ i − τ i − ) ∞ i =1 if M = 2 k, ( k + τ i ) ∞ i =1 if M = 2 k + 1 . Notice that the sequence ( λ i ) ∞ i =1 in (2.4) depends on M . Also, from the definition of ( τ i ) ∞ i =0 it follows that ( λ i ) satisfies the recursive equations:(2.5) λ . . . λ n +1 = λ . . . λ n λ . . . λ n + for all n ≥ . [27]. Therefore, α ( q KL ) starts with( k + 1) k ( k − k + 1) ( k − k ( k + 1) k · · · if M = 2 k, ( k + 1)( k + 1) k ( k + 1) kk ( k + 1)( k + 1) · · · if M = 2 k + 1 . Using (2.3), (2.4) and Lemma 2.3 we obtain that q G < q KL < q T . Here q T refers to the transitive base defined in (2.8).We would like to bring to mind that ( U q , σ ) is not always a subshift [14, Theorem 1.8].However we consider the subshift ( V q , σ ), where q ∈ V and V q = n x ∈ Σ M : α ( q ) σ n ( x ) α ( q ) for all n ≥ o . Accordingly, we define V q = π q ( V q ). It is easy to check that ω ∈ L ( V q ) if and only if ω = w . . . w n satisfies(2.6) α ( q ) . . . α n − i ( q ) w i +1 . . . w n α ( q ) . . . α n − i ( q ) for every 0 ≤ i ≤ n − . Also, it is clear that if ω ∈ L ( V q ) then ω ∈ L ( V q ).In [26, Proposition 2.8] Komornik et. al. showed that h top ( V q ) = h top ( U q ) for all q ∈ [ q G , M + 1] . Moreover, in [26, Theorem 1.3] the authors showed that(2.7) dim H ( U q ) = h top ( V q ) / log( q ) for all q ∈ [ q G , M + 1] . Komornik et. al. also studied the entropy function H : [ q G , M + 1] → [0 ,
1] given by H ( q ) = h top ( V q ) and the Hausdorff dimension function HD : [ q G , M + 1] → [0 ,
1] given by HD ( q ) = dim H ( V q ). In [26, Lemma 2.11] (see also, [3] [29, Theorem 2.6]) was shown that H is a devil’s staircase and, as a consequence of 2.7, it is shown in [26, Theorem 1.4] that HD is continuous, has bounded variation and has devil’s-staircase-like behaviour.In [2, Theorem 2, Theorem 3] the entropy plateaus , i.e, the maximal intervals [ p L , p R ] forwhich H ( q ) = H ( p L ) for all q ∈ [ p L , p R ] , as well as the bifurcation set B = { q ∈ (1 , M + 1] : H ( p ) = H ( q ) for any p = q } and its topological closure B = { q ∈ (1 , M + 1] : for all ε > p ∈ ( q − ε, q + ε ) such that H ( p ) = H ( q ) } were characterised. The set T = { q ∈ V : ( V q , σ ) is a transitive subshift } was also characterised (see [2, Theorem 1]).The following special classes of sequences in V were introduced in [2] (see also [1]) in orderto characterise the entropy plateaus and the sets B , B and T .Let ζ i = α ( q G ( M )) i . For every n ∈ N we define the sequence ξ ( n ) := (cid:26) ζ . . . ζ n − ( ζ . . . ζ n − + ) ∞ if M = 2 k,ζ . . . ζ n ( ζ . . . ζ n + ) ∞ if M = 2 k + 1 . Definition 2.6. i ) A sequence α ∈ V is said to be irreducible if α . . . α j ( α . . . α j + ) ∞ ≺ α whenever ( α . . . α − j ) ∞ ∈ V . ii ) A sequence α ∈ V is said to be ∗ -irreducible if there exists n ∈ N such that ξ ( n + 1) α ≺ ξ ( n ) , and α . . . α j ( α . . . α j + ) ∞ ≺ α, whenever ( α . . . α − j ) ∞ ∈ V and j > (cid:26) n if M = 2 k, n +1 if M = 2 k + 1 . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 11 We denote by I = { q ∈ V : α ( q ) is irreducible } and by I ∗ = { q ∈ V : α ( q ) is ∗ -irreducible } . We would like to mention that I and I ∗ are subsets of U [2, Lemma 4.7]. It is notdifficult to check that α ( q G ) is irreducible, and hence it is the smallest irreducible sequence.Also, the base q T = q T ( M ), called the transitive base , was introduced in [2]. The base q T isdefined implicitly as(2.8) α ( q T ) = (cid:26) ( k + 1) k ∞ if M = 2 k, ( k + 1) (( k + 1) k ) ∞ if M = 2 k + 1 . Notice that α ( q T ) ∈ V and therefore q T ∈ V . Moreover q T ∈ U and q T > q G . Thefollowing Theorem summarises [2, Theorem 1, Theorem 2, Theorem 3]. Theorem 2.7. i ) Let q ∈ V . Then ( V q , σ ) is a transitive subshift if and only if α ( q ) is irreducible, or q = q T ; ii ) The interval [ q L , q R ] ⊆ ( q KL , M +1] is an entropy plateau of H if and only if q L ∈ I ∪ I ∗ and α ( q L ) is periodic, and α ( q R ) = α ( q L ) . . . α m ( q L ) + ( α ( q L ) . . . α m ( q L )) ∞ ; iii ) The topological closure of the bifurcation set B is B = I ∪ I ∗ ⊂ U . Moreover B is a Cantor set and dim H ( B ) = 1 . The interval [ q L , q R ] described in ii ) is known as the irreducible interval generated by q L whenever q L ∈ I . We will denote an irreducible interval generated by q ∈ I with α ( q )periodic by I ( q ), whenever necessary.In [23, Remark 1.2] Kalle et. al. mentioned that B \ B is a countable set. Moreover, it isclear that if q ∈ B \ B then q is an end point of an entropy plateau. Finally in [23, Theorem2] it is shown that dim locH ( B , q ) = dim H ( U q ) = dim H ( V q ) . By [23, Theorem 2] and [26, Theorem 1.4] the following proposition holds.
Proposition 2.8.
The functions H and HD are continuous in B . Moreover the map q → dim locH ( B , q ) is continuous in B . Approximation properties of symmetric q -shifts In this section we introduce a notion of approximation of symmetric q -shifts that will be usefulfor the rest of the paper. Using this approximation we show that every transitive symmetric q -shift is coded and mixing. Definition 3.1.
Given q ∈ U we define natural approximation of q from below as the sequence { q − m } ∞ m =1 ⊂ V given by q − defined implicitly by α ( q − ) = ( α ( q ) . . . α n ( q ) − ) ∞ where n = min (cid:8) n ∈ N : ( α ( q ) . . . α n ( q ) − ) ∞ ∈ V (cid:9) ;and if q − m − is already defined then q − m is defined implicitly by(3.1) α ( q − m ) = ( α ( q ) . . . α n m ( q ) − ) ∞ where n m = min (cid:8) n ∈ N : n > n m − and ( α ( q ) . . . α n ( q ) − ) ∞ ∈ V (cid:9) . We make the following observation on Definition 3.1.
Remark . Note that Theorem 2.5 iii ) implies that for every q ∈ ( V \ U ) \{ q G } , α ( q ) is a peri-odic sequence. Set m to be the period of α ( q ). Then, since q ∈ V \ U there exists m ′ < m suchthat σ m ′ ( α ( q )) = α ( q ). We claim that, for every j > m ′ the sequence ( α ( q ) . . . α j ( q ) − ) ∞ / ∈ V .Suppose on the contrary that there exists j > m ′ such that ( α ( q ) . . . α j ( q ) − ) ∞ ∈ V . Then,( α ( q ) . . . α j ( q ) − ) ∞ σ m ′ (( α ( q ) . . . α j ( q ) − ) ∞ ) ( α ( q ) . . . α j ( q ) − ) ∞ , i.e ( α ( q ) . . . α j ( q ) − ) ∞ ( α m ′ +1 ( q ) . . . α j ( q ) − α ( q ) . . . α m ′ ( q )) ∞ ( α ( q ) . . . α j ( q ) − ) ∞ . Note that α m ′ +1 ( q ) . . . α j ( q ) − ≺ α ( q ) . . . α j − m ′ ( q ) . Then α m ′ +1 ( q ) . . . α j ( q ) − ≻ α ( q ) . . . α j − m ′ ( q ) , which is a contradiction.We have shown that if q ∈ ( V \ U ) \ { q G } there is N ∈ N such that for any m ≥ N thereis no n m > n N such that ( α ( q ) . . . α n m ( q ) − ) ∞ ∈ V . Thus, for sake of completeness, if q ∈ ( V \ U ) \ { q G } we set the natural approximation frombelow of q to be the finite sequence (cid:8) q − , . . . q − N , q − N +1 (cid:9) where q − N +1 = q . Also, since q G is thesmallest element of V we set the natural approximation from below of q G to be { q G } .In the following propositions we show that indeed, the natural approximation from belowapproximates a given q ∈ U . Proposition 3.3.
Let q ∈ U . Then, the natural approximation from below of q , { q − m } ∞ m =1 ,satisfies: i ) For every m ∈ N , q − m < q − m +1 and q − m < q . ii ) q − m ր m →∞ q . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 13 Proof.
Let us show i ). From Lemma 2.3 it suffices to show that α ( q − m ) α ( q − m +1 ) and α ( q − m ) α ( q )for every m ∈ N . From Definition 3.1, we have for every natural number m , α ( q − m ) = ( α ( q ) . . . α n m ( q ) − ) ∞ and α ( q − m +1 ) = ( α ( q ) . . . α n m ( q ) . . . α n m +1 ( q ) − ) ∞ . Note that α n m ( q − m ) = α n m ( q ) − < α n m ( q ) = α n m ( q − m +1 ) , so α ( q − m ) ≺ α ( q − m +1 ) and q − m ≤ q − m +1 . The proof for q − m ≤ q follows from the same argument.To show ii ), let q ∈ U be fixed. Note that that d ( α ( q ) , α ( q − m )) = 1 / n m . Then, for a given ε > N ∈ N such that if m ≥ N , then d ( α ( q ) , α ( q − m )) = 1 / n m < / N < ε, that is α ( q − m ) −→ α ( q ) as m → ∞ . Then, by Lemma 2.3 we have that q − m ր m →∞ q . (cid:3) We introduced the natural approximation from below of elements of U in order to ap-proximate the subshifts ( V q , σ ) in the following way: we say that ( X, σ ) is approximated frombelow if there exists a sequence of subshifts of finite type { ( X m , σ ) } ∞ m =1 such that X m ⊂ X m +1 for every m ∈ N and X = ∞ [ m =1 X n . Lemma 3.4. If q ∈ V then the subshift ( V q , σ ) is approximated from below by the sequenceof subshifts ( V q − m , σ ) where { q − m } ∞ m =1 is the natural approximation from below of q .Proof. Firstly, let q ∈ V \ U . If q = q G then, using Remark 3.2, we have that the naturalapproximation of below of q G is { q G } . Then, the conclusion of the Lemma follows easily. Onthe other hand, if q = q G from Remark 3.2 we have that there is N ∈ N such that the naturalapproximation from below of q , { q − m } ∞ m =1 satisfies that q − m = q − N +1 = q for all m ≥ N + 1. So,it is straightforward that ( V q − m , σ ) approximates from below the subshift ( V q , σ ).Now, consider q ∈ U and let { q − m } ∞ m =1 be the natural approximation from below of q .From Proposition 3.3 we have that q − m < q − m +1 for every m ∈ N . Since { q − m } ∞ m =1 ⊂ V ,then V q − m ⊂ V q − m +1 . Since q − m < q then V q − m ⊂ V q for every m ∈ N . This implies that ∞ S m =1 V q − m ⊂ V q . Also, recall that V q is a closed subset of Σ M , then ∞ S m =1 V q − m ⊂ V q . On theother hand, consider ω ∈ L ( V q ). Since q − m ր q as m −→ ∞ , there exists n ∈ N such that ω ∈ L ( V q − m ). This implies that there exists x ∈ V q − m such that ω is a factor of x . Then ∞ S k =1 V q − m is dense in V q with respect to the metric d defined in (2.1). Thus, ∞ [ m =1 V q − m = V q . (cid:3) From Definition 3.1, if q ∈ U we have { q − m } ∞ m =1 ⊂ V . We wish to point out that theapproximation from below constructed in Lemma 3.4 is similar to the approximation ( W p L , σ )considered in [23] and [26]. One of the advantages of our approach is that it is not necessaryto compare finite words and sequences introducing a variation of the lexicographic order.Also, the constructed approximations together with Remark 3.2 allow us to always get strictinclusions when q ∈ U . That is, since q ∈ U , { q − m } ∞ m =1 ⊂ V and q − n < q − m for every n < m we have that V q − n ( V q − m for every n < m .We now show that given the natural approximation from below of q it is also possible toapproximate the language of V q by the languages of the associated subshifts of each of theelements of the natural approximation from below (compare with [23, Lemma 3.4]). Lemma 3.5.
Let q ∈ V and consider the natural approximation from below of q , { q − m } ∞ m =1 .Then, for every k ∈ N there exists J ∈ N such that if m ≥ J , then B k ( V q ) = B k ( V q − m ) = B k ( V q − J ) . Proof.
Firstly, let us assume that q ∈ V \ U . Then α ( q ) is a periodic sequence. Let J ∈ N be the period block of α ( q ). Then, from Lemma 3.4 there is N ∈ N such that q − m = q − N +1 and α ( q − m ) = α ( q ). Thus, for all k ≥ J we have B k ( V q ) = B k ( V q − m ) = B k ( V q − J ) . Suppose that q ∈ U . From Proposition 3.3 we have q − m < q − m +1 < q , for every m ∈ N and q − m ր m →∞ q . This implies V q − m ( V q − m +1 ( V q . Then, for every k ∈ N and m, J ∈ N with m ≥ J we have B k ( V q − J ) ⊂ B k ( V q − m ) ⊂ B k ( V q ) . Fix k ∈ N and let n m be the period of α ( q − m ). We claim that J = min { m ∈ N : k < n m } satisfies B k ( V q ) = B k ( V q − m ) = B k ( V q − J ) . To show this, it suffices to show that B k ( V q ) ⊂ B k ( V q − J ). Let ω ∈ B k ( V q ). Then, for every i ∈ { , . . . , k − } α ( q ) . . . α k − i ( q ) w i +1 . . . w k α ( q ) . . . α k − i ( q ) . Since n K > k then α ( q ) . . . α k ( q ) = α ( q − J ) . . . α k ( q − J ). This gives α ( q − J ) . . . α k − i ( q − J ) w i +1 . . . w k α ( q − J ) . . . α k − i ( q − J ) for every i ∈ { , . . . , k − } and the proposition follows. (cid:3) We show now the following technical, but important results. We will show that if q ∈ I then there must exist infinitely many irreducible elements in the natural approximation frombelow { q − m } ∞ m =1 . For this endeavour, we prove the following statement firstly. Lemma 3.6.
Let q ∈ I . Then, if there is j ∈ N such that q − j > q T and ( α ( q − j ) . . . α n j ( q − j )) ∞ = ( α ( q ) . . . α n j ( q ) − ) ∞ N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 15 is not irreducible then there exists a unique k < j and a unique < n k < n j such that q − k ∈ I and q − j ∈ I ( q − k ) .Proof. Let q ∈ I and q − j be such that ( α ( q − j ) . . . α n j ( q − j )) ∞ = ( α ( q ) . . . α n j ( q ) − ) ∞ with q − j > q T and q − j / ∈ I . Then, from [2, Lemma 4.9] there exists a unique irreducible interval I such that q − j ∈ I . Let q ′ ∈ I be such that I = I ( q ′ ). Since I ( q ′ ) is an irreducible intervalthen there exists a word w . . . w m such that α ( q ′ ) = ( w . . . w m ) ∞ and α ( q ) is an irreduciblesequence. From [2, Lemma 4.1] we can assume without loss of generality that m is the periodof α ( q ′ ). From the uniqueness of I we get that w . . . w m is unique. Since q − j ∈ I we have( w . . . w m ) ∞ ≺ ( α ( q − j ) . . . α n j ( q − j )) ∞ ≺ w . . . w + m ( w . . . w m ) ∞ . Observe that n j = m as if n j = m then ( w . . . w m ) ∞ is not irreducible, which is a contradic-tion. On the other hand, if m > n j then α ( q − j ) . . . α n j ( q − j ) α n j +1 ( q − j ) . . . α m ( q − j ) = α ( q − j ) . . . α n j ( q − j ) α ( q − j ) . . . α m − n j ( q j )= w . . . w n j w n j +1 . . . w m = w . . . w n j w . . . w m − n j . This implies that the period of ( w . . . w m ) ∞ is smaller than m , which is a contradiction aswell. Therefore, m < n j . Since I ⊂ U , w . . . w + m = α ( q − j ) . . . α m ( q − j ) = α ( q ) . . . α m ( q ) . This implies that w . . . w m = α ( q − j ) . . . α m ( q − j ) − = α ( q ) . . . α m ( q ) − . Then q − k = q ′ and m = n k satisfy the desired properties. (cid:3) Lemma 3.7.
Let q ∈ I and { q − m } ∞ m =1 be the natural approximation from below of q . Then,there exist infinitely many m ∈ N such that q − m ∈ I .Proof. Let q ∈ I . Let us assume on the contrary that there is N ∈ N such that q − m / ∈ I forevery m ≥ N .From Proposition 3.3, q − m ր q as m → ∞ and since q ∈ I from [2, Lemma 4.4] we knowthat q > q T . Then, there exists a minimal N ∈ N such that, q − m ∈ ( q T , M + 1) for every m ≥ N . Let N = max { N , N } . Then, from Lemma 3.6 there is a unique k < N such that q − k ∈ I and q − N ∈ I ( q − k ).We claim that for every m > N , q − m ∈ I ( q − k ). Suppose this is not true. Then, there is m > N such that q − m / ∈ I ( q − k ). Also note that Lemma 3.3 implies that q − N < q − m . Then, Lemma3.6 implies that there is a unique k ′ < m such that q − k ′ ∈ I and q − m ∈ I ( q − k ′ ). Clearly k < k ′ .From [2, Lemma 4.6] we know that I ( q − k ) ∩ I ( q − k ′ ) = ∅ . This implies that N < k ′ which is acontradiction. Therefore, q − m ∈ I ( q − k ) for every m ≥ N which gives that that q ∈ I ( q − k ), thus q / ∈ I . This establishes a contradiction. (cid:3) We show now that every transitive symmetric q -shift ( V q , σ ) is a coded system. We wantto emphasise that I ⊂ B ⊂ U —see [2, Lemma 4.7, Lemma 6.1]. Proposition 3.8.
For every q ∈ I the subshift ( V q , σ ) is coded. Proof.
Let q ∈ I . Then, from Theorem 2.7 i ) the subshift ( V q , σ ) is transitive. Then,if α ( q ) is periodic then [14, Theorem 1.7, 1.8] implies that ( V q , σ ) is coded. Similarly if α ( q ) is eventually periodic then [24, Proposition 2.14] implies that ( V q , σ ) is coded. So, let q ∈ I such that α ( q ) is neither periodic nor eventually periodic. Then, from Lemma 3.7and [2, Lemma 6.1] the natural approximation from below { q − m } ∞ m =1 contains a subsequence n q − m j o ∞ j =1 such that q − m j ∈ I , α ( q − m j ) is periodic for every j ∈ N and q − m j ր j →∞ q . Thisimplies that V q − mj ( V q − mj +1 . Then, from Lemma 3.4 it follows that the sequence of subshifts n ( V q − mj , σ ) o ∞ j =1 also approximates ( V q , σ ) from below. Moreover, since q − m j ∈ I for every j ∈ N we obtain from [2, Theorem 1] that ( V q − mj , σ ) is a transitive subshift for every j ∈ N .Finally, using [14, Theorem 1.7, 1.8] and [32, Theorem 1.3] imply that ( V q − mj , σ ) is a subshiftof finite type for every j ∈ N . Thus, ( V q , σ ) is coded. (cid:3) To finish this section, we now show that every symmetric and transitive q -shift is topolo-gically mixing. For this purpose we want to recall the usual Sharkovskiˇi order of N :3 ⊲ ⊲ ⊲ . . . ⊲ m + 1 ⊲ . . . ⊲ · ⊲ · ⊲ · ⊲ . . . ⊲ · (2 m + 1) ⊲ . . . ⊲ · ⊲ · ⊲ · ⊲ . . . ⊲ · (2 m + 1) ⊲ . . . ... ... ... ... ⊲ n · ⊲ n · ⊲ n · ⊲ . . . ⊲ n · (2 m + 1) ⊲ . . . ... ... ... ... ⊲ ∞ . . . ⊲ n ⊲ . . . ⊲ ⊲ ⊲ ⊲ . It has been shown in [6, Theorem 1.3] and [22, Theorem 1.1] that periodic points of ( V q , σ )grow with respect to the Sharkowskˇi order of N , that is, if V q contains a periodic point ofperiod m with respect to σ , then V q contain points of period n for every n ⊳ m . On theother hand it is known ([33, Proposition 4.5.10 (4)]) that a subshift of finite type ( X, σ ) istopologically mixing if and only if it is transitive and the greatest common divisor of theperiods of its periodic points is 1; that is, there exists a pair of periodic points x and y ∈ X such that gcd( m, n ) = 1, where m and n are the periods of x and y respectively. Using (2.8)it follows that if q ∈ I then ( V q , σ ) contains a periodic orbit of odd period. As a consequenceof these results we obtain the following: Proposition 3.9. If q ∈ I and α ( q ) is periodic then ( V q , σ ) is a mixing subshift of finitetype. Now we prove that every transitive symmetric q -shift is mixing. Proposition 3.10. If q ∈ I then ( V q , σ ) is a mixing subshift. N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 17 Proof.
From Proposition 3.9 we can assume that α ( q ) is not periodic. From Lemma 3.7 wehave that there is a subsequence n q − m j o ∞ j =1 of the natural approximation from below of q , { q − m } ∞ m =1 such that ( V q − mj , σ ) is a mixing subshift of finite type for every j ∈ N . Also, Lemma3.4 implies that ( V q , σ ) is approximated from below by n ( V q mj , σ ) o ∞ j =1 . Let υ, ν ∈ L ( V q ).From Lemma 3.5 there are J, J ′ ∈ N such that ( V q − J , σ ) and ( V q − J ′ , σ ) are mixing subshifts offinite type and for every m ≥ J , B | υ | ( V q ) = B | υ | ( V q − m ) = B | υ | ( V q − J )and for every m ≥ J , B | ν | ( V q ) = B | ν | ( V q − m ) = B | ν | ( V q − J ′ ) . Put J ′′ = max { J, J ′ } . Since ( V q J ′′ , σ ) is a mixing subshift then there is N ∈ N such that forevery n ≥ N there is ω ∈ B n ( V q J ′′ ) such that υων ∈ L ( V q J ′′ ). Since V q J ′′ ⊂ V q , the resultfollows. (cid:3) The specification property of ( V q , σ )In this section, we characterise the set of q ∈ V such that ( V q , σ ) has the specificationproperty. In order to do this, we introduce the notions of strongly irreducible and weaklyirreducible sequences . Definition 4.1.
We say that an irreducible sequence α = α ( q ) ∈ V is strongly irreducible ifthere exists N ∈ N such that for all m ≥ N one has q − m ∈ I . We also say that an irreduciblesequence is weakly irreducible if there are infinitely many m ∈ N such that q − m / ∈ I .In a similar fashion we introduce the notion of strongly irreducible number . Definition 4.2.
A number q ∈ I is called strongly irreducible if α ( q ) is strongly irreducible,similarly q ∈ I weakly irreducible if α ( q ) is weakly irreducible.We will use the notations S I = { q ∈ U : q is strongly irreducible } and W I = { q ∈ U : q is weakly irreducible } . Clearly W I = I \ S I and I , W I ( B ∩ [ q T , M + 1] . In this section we will show some properties of the set of strongly and weakly irreduciblesequences. Firstly, we mention that there are three different kinds of strongly irreduciblesequences.
Definition 4.3.
We say that a strongly irreducible sequence α ( q ) is: i ) of Type 1 if for every m ∈ N , q − m ∈ I ; ii ) of Type 2 if there is N ∈ N such that q − m ∈ I for every m ≥ N and q − k < q T for every k < N ; iii ) of Type 3 if there exists an N ∈ N and m < N such that q − m / ∈ I with q − m > q T and q − k ∈ I for all k ≥ N .Let us illustrate Definition 4.3 with some examples. Example 4.4.
Consider M = 1, then:(1) Let n ≥
3. The sequence (1 n ∞ is strongly irreducible of type . Note that here q − = q G (1), so by [2, Lemma 3.1] α ( q − ) is an irreducible sequence;(2) The sequence (11010) ∞ is strongly irreducible of type . Here N = 4 and the corres-ponding n N = 7. The last non-irreducible sequence of the approximation from belowis q − where α ( q − ) = (110100) ∞ α ( q T (1));(3) The sequence (1110010010) ∞ is strongly irreducible of type . Here N = 5 and thelast non-irreducible sequence of the approximation from below is q − where α ( q − ) = (111001000) ∞ < α ( q T (1)) . Consider M = 2, then:(1) Let n ≥
2. The sequence (2 n ∞ is strongly irreducible of type 1.(2) The sequence (211211121111) ∞ is strongly irreducible of type 2. Here N = 4 and thelast non-irreducible sequence of the approximation from below is q − where α ( q − ) =(210) ∞ α ( q T (2));(3) The sequence (22010101) ∞ is strongly irreducible of type 3. Here N = 6 and the lastnon-irreducible sequence of the approximation from below is q − where α ( q − ) = (22010100) ∞ < α ( q T (2)) . We can distinguish strongly irreducible numbers of types
1, 2 and defining q implicitly,as was done in Definition 4.2.To start our investigation, we want to show the reader why our intuition is that stronglyirreducible sequences are, loosely speaking, the “right ones” to look for the specificationproperty; i.e. a symmetric q -shift with the specification property is parametrised by a stronglyirreducible sequence and vice versa. Proposition 4.5.
Set
Per( I ) = { q ∈ I : α ( q ) is periodic } . Then,
Per( I ) ⊂ S I .Proof. Let q ∈ Per( I ). Suppose that α ( q ) = ( α ( q ) , . . . α k ( q )) ∞ has period k . Let { q − m } ∞ m =1 the natural approximation from below of q and set(4.1) α ( q − m ) = ( α ( q ) . . . α n m ( q ) − ) ∞ = ( α ( q − m ) . . . α n m ( q − m )) ∞ the quasi-greedy expansion of q − m for every m ∈ N .Since q ∈ I then for every m ∈ N we have(4.2) α ( q ) . . . α n m ( q )( α ( q ) . . . α n m ( q ) + ) ∞ = α ( q − m ) . . . α n m ( q − m ) + ( α ( q − m ) . . . α n m ( q − m )) ∞ ≺ α ( q ) . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 19 Also, since q ∈ Per( I ) ⊂ I then Lemma 3.7 implies that there are infinitely many m ∈ N such that q − m ∈ I .Let N = min { m ∈ N : q − m ∈ I and n m > k } , i.e. N is the first irreducible element of the natural approximation from below such that α ( q − m ) has larger period than the period of α ( q ). Observe that Lemma 3.7 implies that N iswell-defined.We claim that for every m ≥ N , q − m ∈ I , i.e. α ( q − m ) = ( α ( q ) . . . α n m ( q ) − ) ∞ is anirreducible sequence. We will proceed by induction. From the definition of N we have that q − N ∈ I . Suppose that for every N ≤ j ≤ m we have that q − j ∈ I . We show now that q − m +1 ∈ I . Clearly, n m < n m +1 . We show in the following two cases that for every i ∈ N such that ( α ( q − m +1 ) . . . α i ( q − m +1 ) − ) ∞ ∈ V then(4.3) α ( q − m +1 ) . . . α i ( q − m +1 )( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ≺ α ( q − m +1 ) . • Assume firstly that 1 ≤ i < n m . From the induction hypothesis q − m ∈ I then using(4.2) we have α ( q − m +1 ) . . . α i ( q − m +1 )( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ≺ α ( q − m ) . . . α i ( q − m )( α ( q − m ) . . . α i ( q − m ) + ) ∞ ≺ α ( q − m ) ≺ α ( q − m +1 ) . So, (4.3) holds in this case. • Consider now i ≥ n m . Observe that α ( q ) . . . α i ( q ) = α ( q − m +1 ) . . . α n m ( q − m +1 ) . . . α i ( q − m +1 ) ∈ L ( V q − m +1 ) , and(4.4) ( α ( q − m +1 ) . . . α i ( q − m +1 ) − ) ∞ and ( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ∈ V q − m +1 . We claim that for every ℓ ∈ N (4.5) α ( q − m +1 ) ≺ σ ℓ (( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ) ≺ α ( q − m +1 ) . Let us show (4.5). From (4.4) we have α ( q − m +1 ) σ ℓ (( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ) α ( q − m +1 ) . for every ℓ ∈ N . Now, clearly ( α ( q − m +1 ) . . . α i ( q − m +1 ) − ) ∞ ≺ α q − m +1 . Since( α ( q − m +1 ) . . . α i ( q − m +1 ) − ) ∞ ∈ V , then Lemma 2.3 implies there is a unique p ∈ V such that α ( p ) = ( α ( q − m +1 ) . . . α i ( q − m +1 ) − ) ∞ and p ≺ q − m +1 . Moreover V p ( V q . Therefore, for every ℓ ∈ N α ( q − m +1 ) ≺ ( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ σ ℓ (( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ) ( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ ≺ α ( q − m +1 ) . So (4.5) holds. Then (4.5) implies( α ( q − m +1 ) . . . α i ( q − m +1 ) + ) ∞ = ( α ( q ) . . . α n m ( q ) . . . α i ( q ) + ) ∞ ≺ σ i ( α ( q − m +1 ))which implies that (4.3) holds in this case. (cid:3) We now show that transitive sofic symmetric q -shifts are parametrised by strongly irredu-cible numbers. Proposition 4.6.
Let q ∈ I . Then, if α ( q ) is eventually periodic then q ∈ S I .Proof. Let q ∈ I . Suppose that α ( q ) = α ( q ) . . . α r ( q )( α r +1 ( q ) . . . α n ( q )) ∞ . Set k to be theperiod of σ r ( α ( q )). Since q ∈ I then for every j ∈ N such that ( α ( q ) . . . α j ( q ) − ) ∞ ∈ V then(4.6) α ( q ) . . . α j ( q )( α ( q ) . . . α j ( q ) + ) ∞ ≺ α ( q ) . Let { q − m } ∞ m =1 the natural approximation from below of q . We will define N in a similar wayas in Proposition 4.5. Let N = min (cid:8) m ∈ N : q − m ∈ I and the period of α ( q − m ) > r + k (cid:9) . Using Lemma 3.7 we get that N is well defined. The argument to show that for every j > N , q − j ∈ I is exactly the same as in Proposition 4.5. (cid:3) Existence of non periodic strongly irreducible sequences and weakly irredu-cible sequences.
We now show that, for a fixed M ∈ N , I contains non-periodic andnon-eventually periodic sequences, which can be either strongly or weakly irreducible.We start our investigation by recalling that in [26, Theorem 1.6] the following class ofsubsets was constructed. Let M ∈ N and fix N ≥ V that satisfy(4.7) 0 N ≺ α ( r · N )+1 ( q ) . . . α ( r +1) · N ( q ) ≺ M N , for all r ≥ , namely(4.8) I N = { q ∈ V : α ( q ) . . . α N ( q ) = M N − } In [2, Lemma 6.2] it is shown that I N ⊂ I for every N ≥
2. Moreover I N ⊂ U . Then,Lemma 2.4 i ) implies that for every q ∈ I N , α ( q ) is not periodic. Furthermore, in [2, Lemma6.4] it is shown that dim H ( I N ) > N ≥
2, so I N is uncountable. Proposition 4.7.
Let M ∈ N . Then, for every N ≥ , I N ⊂ S I . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 21 Proof.
Let q ∈ I N . Since q ∈ I then from Lemma 3.7 we have that(4.9) K = min (cid:8) k ∈ N : ( α ( q ) . . . α n k ( q ) − ) ∞ ∈ I and n k ≥ N (cid:9) is well-defined. We claim that for every j ≥ K the sequence ( α ( q ) . . . α n j ( q ) − ) ∞ is irreducible.Fix j ≥ K and let q − j be the corresponding element of the natural approximation from below.Then α ( q − j ) = ( α ( q − j ) . . . α n j ( q − j )) ∞ = ( α ( q ) . . . α n j ( q ) − ) ∞ . Since N ≥ α ( q ) satisfies (4.8) then the word 0 N − is not a factor of α ( q − j ).We split the proof in two cases: • Suppose that i ∈ N satisfies ( α ( q − j ) . . . α n i ( q − j ) − ) ∞ ∈ V and 1 ≤ n i ≤ N . Then α ( q − j ) . . . α n i ( q − j )( α ( q − j ) . . . α n i ( q − j ) + ) ∞ = M n i (0 n i − ∞ . Since 1 ≤ n i ≤ N then M n i (0 n i − ∞ ≺ ( α ( q − j ) . . . α n j ( q − j )) ∞ . • Suppose that i ∈ N satisfies that ( α ( q − j ) . . . α n i ( q − j ) − ) ∞ ∈ V and 2 N ≤ n i . Then α ( q − j ) . . . α n i ( q − j )( α ( q − j ) . . . α n i ( q − j ) + ) ∞ = α ( q − j ) . . . α n i ( q − j )(0 N − M α N +1 ( q ) . . . α n i ( q − j ) + ) ∞ . Since 0 N − is not a factor of α ( q − j ) we obtain α ( q − j ) . . . α n i ( q − j )(0 N − M α N +1 ( q ) . . . α n i ( q − j ) + ) ∞ ≺ α ( q − j )and the result follows. (cid:3) Construction of strongly irreducible sequences.
We describe a construction toobtain non-periodic strongly irreducible sequences. This construction will allow us to findstrongly irreducible sequences in I \ S N ≥ I N .Recently, Allaart in [5, Definition 2.1] introduced the notion of fundamental word . Given M ∈ N , a word ω = w . . . w n ∈ L (Σ M ) with n > fundamental word if(4.10) w . . . w n − i w i +1 . . . w n ≺ w . . . w n − i for every 1 ≤ i ≤ n − . It is clear that if α = α . . . α m is a fundamental word then ( α . . . α m ) ∞ ∈ V [5, p.6510]. Lemma 4.8.
Let q ∈ U . Then, there exists a strictly increasing sequence { m j } ∞ j =1 ∈ N suchthat α ( q ) . . . α m j ( q ) is a fundamental word.Remark . We want to make clear that the statement of Lemma 4.8 remains valid for q ∈ V making the following modifications. From Theorem 2.5 iv ) we have that for every q ∈ V \ U , α ( q ) is a periodic sequence. If α ( q ) = s ∞ with s ∈ { k + 1 , . . . , M } given that M = 2 k + 1 or s ∈ { k, . . . , M } if M = 2 k , then no factor of α ( q ) is a fundamental word, however s ∞ ∈ U and ( α ( q ) . . . α m ( q )) ∞ = s ∞ for every m ∈ N . Also, if q ∈ V \ U with α ( q ) = s ∞ as in theformer case then α ( q ) = ( α ( q ) . . . α m ( q )) ∞ with m = 1. It is clear that the periodic block α ( q ) . . . α m ( q ) is a fundamental word. Then, the sequence m j = m · j for each j ∈ N satisfiesthe consequence of Lemma 4.8. Proof of Lemma 4.8.
Let q ∈ U . Then α ( q ) ≺ σ n ( α ( q )) ≺ α ( q ) for every n ∈ N . Let m = min { m ∈ N : α m ( q ) < α ( q ) } . Since q ∈ U then α ( q ) is not a periodic sequence, so m is well defined. Note that α ( q ) . . . α m − i ( q ) α i +1 ( q ) . . . α m ( q ) ≺ α ( q ) . . . α m − i ( q )since α ( q ) . . . α m − i ( q ) = α ( q ) m − i for every i ∈ { , . . . , m − } .Clearly, ( α ( q ) . . . α m ( q )) ∞ ≻ α ( q ) . Let us set ( α ( q ) . . . α m ( q )) ∞ = α ( q +1 ) where q +1 isdefined implicitly. Then, there exists a minimal integer ℓ > m such that α ℓ ( q ) < α ( q ) = α ( q +1 ) ,α ( q +1 ) . . . α ℓ − ( q +1 ) ≺ α ( q ) . . . α ℓ − ( q ) = α ( q +1 ) . . . α ℓ − ( q +1 )and α ℓ ( q +1 ) ≤ α ℓ ( q ) < α ℓ ( q +1 ) . So, let m = ℓ . Observe that α ( q ) . . . α m − i ( q ) α i +1 ( q ) . . . α m ( q ) α ( q ) . . . α m − i ( q )holds for every 1 ≤ i ≤ m −
1; this, the definition of m and the minimality of m give that α ( q ) . . . α m − i ( q ) α i +1 ( q ) . . . α m ( q ) ≺ α ( q ) . . . α m − i ( q )for every 1 ≤ i ≤ m −
1. Thus, the word α ( q ) . . . α m ( q ) is fundamental and α ( q +1 ) ≻ ( α ( q ) . . . α m ( q )) ∞ ≻ α ( q ) . Let us call α ( q +2 ) = ( α ( q ) . . . α m ( q )) ∞ .Now we proceed by induction. Suppose that n ∈ N and that m . . . m n and q +1 . . . q + n havebeen already defined. Then we define m n +1 to be the smallest integer ℓ > m n such that α ℓ ( q ) < α ( q ), α ( q + n ) . . . α ℓ − ( q + n ) ≺ α ( q ) . . . α ℓ − ( q ) = α ( q + n ) . . . α ℓ − ( q + n )and α ℓ ( q + n ) ≤ α ℓ ( q ) < α ℓ ( q + n ) . We claim that α ( q ) . . . α m n +1 ( q ) is a fundamental word. From the definition of m n +1 we onlyneed to show that for every 1 ≤ i ≤ m n +1 − α ( q ) . . . α m n +1 − i ( q ) α i ( q ) . . . α m n +1 ( q ) ≺ α ( q ) . . . α m n +1 − i ( q ) . From the induction hypothesis (4.11) is valid for 1 ≤ i < m n − m n +1 we obtain that (4.11) holds for m n ≤ i ≤ m n +1 − (cid:3) In comparison with Definition 3.1, in Lemma 4.8 we constructed another “natural” approx-imation of q ∈ V . It is possible to show that the sequence { q + n } ∞ n =1 is strictly decreasing and q + n ց n →∞ q . We call this the natural approximation from above of q . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 23 Lemma 4.10. If q ∈ B then for every j ∈ N such that α ( q ) . . . α m j ( q ) is a fundamentalword, the sequence ( α ( q ) . . . α m j ( q )) ∞ is irreducible or ∗ -irreducible.Proof. Let q ∈ B . Let us assume that q ≥ q T . The proof for q < q T follows from asimilar argument. In [23, Lemma 2.6] (see also [2, Theorem 3]) it was shown that B ( U . Also, from [2, Lemma 4.10] we have that if q ∈ B satisfies that α ( q ) is periodic then α ( q ) ∈ U \ U . Then, the sequence given in Remark 4.9 satisfies that ( α ( q ) . . . α m j ( q )) ∞ isirreducible for every j ∈ N . So, let us assume that α ( q ) is not periodic. Fix j ∈ N such that( α ( q ) . . . α m j ( q )) ∞ ∈ V and set α ( q + j ) = ( α ( q ) . . . α m j ( q )) ∞ . We want to show that if i ∈ N is such that ( α ( q ) . . . α i ( q + j ) − ) ∞ ∈ V then(4.12) α ( q ) . . . α i ( q + j )( α ( q ) . . . α i ( q + j ) + ) ∞ ≺ α ( q + j ) . Suppose that there exists i ∈ N such that α ( q ) . . . α i ( q + j )( α ( q ) . . . α i ( q + j ) + ) ∞ < α ( q + j ) . Without loss of generality we assume that such i is minimal. Then, from [2, Lemma 4.9 (1)]there exists a unique k < i such that ( α ( q + j ) . . . α k ( q + j ) − ) ∞ is irreducible and( α ( q + j ) . . . α k ( q + j ) − ) ∞ ≺ α ( q + j ) . . . α i ( q + j )( α ( q ) . . . α i ( q + j ) + ) ∞ (4.13) ≺ α ( q + j ) . . . α k ( q + j )( α ( q ) . . . α k ( q + j ) + ) ∞ . Then (4.13) and Lemma 4.8 imply that, whenever n ≥ j , the sequence( α ( q + j ) . . . α k ( q + j ) − ) ∞ ≺ α ( q + n ) ≺ α ( q ) . . . α k ( q + j )( α ( q ) . . . α k ( q + j ) + ) ∞ . Thus, q + n ∈ I ( q + j ) for every n ≥ j , which is a contradiction to q ∈ B . (cid:3) We now construct non-periodic strongly irreducible sequences. Fix q ∈ Per( I ) with α ( q ) = ( α ( q ) . . . α m ( q )) ∞ . Then q parametrises an irreducible interval I ( q ). Let p bethe right end point of I ( q ). Then, p ∈ B ∩ U and the quasi-greedy expansion of p is givenby α ( p ) = α ( q ) . . . α m ( q ) + ( α ( q ) . . . α m ( q )) ∞ . Since p ∈ B then from Lemmas 4.8 and 4.10 there are infinitely many m ∈ N such that α ( p ) . . . α m ( p ) is a fundamental word and ( α ( p ) . . . α m ( p )) ∞ is an irreducible sequence.Taking m ′ = m + 1 it is clear that α ( p ) . . . α m ′ ( p ) is a fundamental word, so there existsat least one m ∈ { m + 1 , . . . , · m } where α ( p ) . . . α m ( p ) is a fundamental word.Let m ∈ { m + 1 , . . . , · m } such that α ( p ) . . . α m ( p ) is a fundamental word. Let q be defined implicitly to be such that α ( q ) = ( α ( p ) . . . α m ( p )) ∞ . Thus, q generates an irreducible interval I ( q ) with right end point p .Set m ∈ { m + 1 . . . · m } such that α ( p ) . . . α m ( p ) is a fundamental word and m = m . This generates q . So, let us assume that q , . . . , q n and p , . . . , p n has been already defined. Let m n +1 ∈ { m n +1 . . . · m n } such that α ( p n ) . . . α m n +1 ( p n ) is a fundamental wordand m n +1 = m n , so we define q n +1 implicitly to have quasi-greedy expansion α ( q n +1 ) = ( α ( p n ) . . . α m n +1 ( p n )) ∞ . Then { q n } ∞ n =1 is a strictly increasing sequence of elements of I . Moreover, q n < p +1 where p +1 is given by the proof Lemma 4.8 applied to p . Thus q n converges to a number q ∈ ( q , p +1 ).We claim that q is strongly irreducible.Let α ( q ) be the quasi-greedy expansion of q . From the above construction, it is clear that q ∈ B . To show that α ( q ) is an irreducible sequence, note that whenever i ∈ N satisfies that( α ( q ) . . . α i ( q ) − ) ∞ ∈ V then there exists K ∈ N such that for every k ≥ K , α ( q ) . . . α i ( q )( α ( q ) . . . α i ( q ) + ) ∞ ≺ α ( q k ) ≺ α ( q ) . This implies the irreducibility of α ( q ). Let us show that α ( q ) is strongly irreducible. Weclaim that for every i ≥ m + 1 such that(4.14) ( α ( q ) . . . α i ( q ) − ) ∞ ∈ V ( α ( q ) . . . α i ( q ) − ) ∞ is an irreducible sequence.Observe that for every n ∈ N there is no i , m n < i < m n +1 , satisfying ( α ( q ) . . . α i ( q ) − ) ∞ ∈ V . This holds since α m n +1 ( q ) . . . α i ( q ) = α ( q + n − ) . . . α i − m n − ( q + n − ) for every m n < i 2. We have then proved: Proposition 4.11. The set S I \ S N ≥ I N is uncountable. Construction of weakly irreducible sequences. Let us now construct a weaklyirreducible sequence. The idea behind this construction is similar to the one for strongly irre-ducible sequences. The difference between both constructions is that for strongly irreduciblesequences, in each step n we find an irreducible sequence “relatively far” from each entropyplateau, whereas for the case of weakly irreducible sequences we will strive to be “relativelyclose” to entropy plateaus at each step.Fix q ∈ I with α ( q ) = ( α ( q ) . . . α m ( q )) ∞ . Then, q parametrises an irreducibleinterval I ( q ). Let p the right end point of I ( q ). Then, p ∈ B ∩ U and the quasi-greedyexpansion of p is given by α ( p ) = α ( q ) . . . α m ( q ) + ( α ( q ) . . . α m ( q )) ∞ . Fix N ∈ N . Then Lemmas 4.8 and 4.10 imply that there exist infinitely many m ∈ N such that α ( p ) . . . α m ( p ) is a fundamental word and ( α ( p ) . . . α m ( p )) ∞ is an irreduciblesequence. In particular, there must exists at least one m ∈ { ( N + 1) · m , . . . , ( N + 2) · m } N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 25 where α ( p ) . . . α m ( p ) is a fundamental word. Let m ∈ { ( N + 1) · m , . . . , ( N + 2) · m } be such that α ( p ) . . . α m ( p ) is a fundamental word. Let q be defined implicitly to be suchthat α ( q ) = ( α ( p ) . . . α m ( p )) ∞ . Notice that for every i = k · m with 1 ≤ k ≤ N .(4.15) α ( p ) . . . α i ( p ) = α ( q ) . . . α m ( q ) + ( α ( q ) . . . α m ( q )) k − α ( q ) . . . α m ( q ) − ≺ α ( p ) . Then, the cardinality of the set of integers i such that ( α ( q ) . . . α i ( q ) − ) ∞ ∈ V and( α ( q ) . . . α i ( q ) − ) ∞ is not irreducible is at least N .Note that q < q . Now, recall that q generates an irreducible interval I ( q ) with rightend point p . Fix N ∈ N . Then there exists m ∈ { ( N + 1) · m , . . . ( N + 2) · m } suchthat α ( p ) . . . α m ( p ) is a fundamental word. We define q implicitly to have quasi-greedyexpansion ( α ( p ) . . . α m ( p )) ∞ . Now, note that for every i = k · m with 1 ≤ k ≤ N (4.16) α ( p ) . . . α i ( p ) = α ( q ) . . . α m ( q ) + ( α ( q ) . . . α m ( q )) k − α ( q ) . . . α m ( q ) − ≺ α ( p ) . This implies that (cid:8) i ∈ { , . . . , m } : ( α ( q ) . . . α i ( q ) − ) ∞ ∈ V and ( α ( q ) . . . α i ( q ) − ) ∞ is not irreducible (cid:9) has cardinality at least N + N .So, let us assume that q , . . . , q n , p , . . . , p n and N , . . . , N n have been already defined. Fix N n +1 and let m n +1 ∈ { ( N n + 1) · m n . . . ( N n + 2) · m n } such that α ( p n ) . . . α m n +1 ( p n ) is afundamental word, so we define q n +1 implicitly to have quasi-greedy expansion α ( q n +1 ) = ( α ( p n ) . . . α m n +1 ( p n )) ∞ . Then { q n } ∞ n =1 is a strictly increasing sequence of elements of I . Moreover, q n < p +1 as inLemma 4.8 applied to p . Thus q n converges to a number q ∈ ( q , p +1 ). Observe that q ∈ I since for every i ∈ N with ( α ( q ) . . . α i ( q ) − ) ∞ there is k ∈ N such that α ( q ) . . . α i ( q )( α ( q ) . . . α i ( q ) + ) ∞ ≺ α ( q k ) ≺ α ( q ) . Note that, for each n ≥ q n forces the set (cid:8) i ∈ { , . . . , m } : ( α ( q n ) . . . α i ( q n ) − ) ∞ ∈ V and ( α ( q n ) . . . α i ( q n ) − ) ∞ is not irreducible (cid:9) to have cardinality at least P n − j =1 N j . So q ∈ W I . This finishes our construction of weaklyirreducible subsequences.We want to remark once more that our construction now depends entirely on the choice of m n and N n at each step n . Note that at each step n the number of choices of m n +1 is strictlygreater to the number of choices we have at the step m n since the period of α ( q n ) is strictly smaller than the period of α ( q n +1 ). Moreover, the set of sequences { N n } ∞ n =1 with N n ∈ N isuncountable. Then, the following proposition holds. Proposition 4.12. The set W I is uncountable. From Propositions 3.3 and 4.5, and Lemma 3.7 we obtain the following result. Proposition 4.13. The set of irreducible sequences I is dense in B ∩ [ q T , M + 1] . Moreover, S I is dense in B ∩ [ q T , M + 1] . We now show that W I is also a dense subset relative to B ∩ [ q T , M + 1]. Proposition 4.14. The set W I is dense in B ∩ [ q T , M + 1] Proof. Let q ∈ ( B ∩ [ q T , M + 1]) \ W I with quasi-greedy expansion α ( q ). Fix ε > 0. Then,from Lemma 2.3 there is δ > d ( α ( p ) , α ( q )) < δ then | p − q |≤ ε . Let n ∈ N besufficiently large to satisfy 1 / n ≤ δ/ 2. From Proposition 4.13 and Lemma 3.7 there exists q ′ ∈ I such that α ( q ′ ) = ( α ( q ′ ) . . . α m ( q ′ )) is periodic of period m , irreducible, α ( q ′ ) ≺ α ( q )and 0 < d ( α ( q ′ ) , α ( q )) < / n . Since α ( q ) is irreducible and α ( q ′ ) ≺ α ( q ) we have α ( q ′ ) . . . α m ( q ′ ) + ( α ( q ′ ) . . . α m ( q ′ )) ∞ ≺ α ( q ) . Then, from Lemma 3.7 and Lemma 4.10 there exist N ∈ N and i ∈ { , . . . , m } such that α ( q ′ ) . . . α m ( q ′ ) + ( α ( q ′ ) . . . α m ( q ′ )) N α ( q ′ ) . . . α i ( q ′ )is a fundamental word and α ( q ′ ) . . . α m ( q ′ ) + ( α ( q ′ ) . . . α m ( q ′ )) N α ( q ′ ) . . . α i ( q ′ ) ≺ α ( q ) . Let q be defined implicitly by α ( q ) = ( α ( q ′ ) . . . α m ( q ′ ) + ( α ( q ′ ) . . . α m ( q ′ )) N α ( q ′ ) . . . α i ( q ′ )) ∞ . Then, appliying 4.3 to q we obtain a weakly irreducible sequence p such that0 < d ( α ( p ) , α ( q )) < / n < δ/ . So, d ( p, q ) < ε . This shows that W I is dense in B ∩ [ q T , M + 1]. (cid:3) Characterisation of the specification property for ( V q , σ ) . We proceed now tocharacterise the set of q ∈ V such that ( V q , σ ) has the specification property. For thisendeavour we would like to recall the definition of the specification property: ( V q , σ ) has the specification property if there exists S ∈ N such that for any υ, ν ∈ L ( V q ) there exists ω ∈ B S ( V q ) such that υων ∈ L ( V q ). Recall that, given q ∈ V with q ≥ q T the sequence s n ( V q ) is given by s n = s n ( V q ) = inf { k ∈ N : for every υ, ν ∈ B n ( V q ) there exists ω ∈ B k ( V q )such that υων ∈ L ( V q ) } . (4.17)Also, ( V q , σ ) has the specification property if and only if lim n →∞ s n < ∞ .We now study the properties of s n ( V q ). Firstly, let us prove some technical lemmas. Letus recall that { q − j } ∞ j =1 stands for the natural approximation from below of q and let α ( q − j ) beas in Definition 3.1. Lemma 3.6 will allow us to show that weakly irreducible sequences do N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 27 not have specification. To prove that assertion, we find a lower bound for s n , for large valuesof n . Lemma 4.15. Let q ∈ I be such that there is j ∈ N such that q − j > q T and ( α ( q − ) . . . α n j ( q − j )) ∞ = ( α ( q ) . . . α n j ( q ) − ) ∞ is not irreducible. Let k and n k be given by Lemma 3.6. Then there exists N ∈ N such that s n ( V q ) > N · n k − for every n ≥ n k .Proof. Let q ∈ I . Since there is j ∈ N such that q − j > q T and ( α ( q ) . . . α n j ( q ) − ) ∞ is notirreducible, then there exists N ∈ N such that α ( q ) . . . α n k · ( N +1) − ( q ) = α ( q − j ) . . . α n k · ( N +1) − ( q − j )= α ( q − k ) . . . α n k ( q − k ) + ( α ( q − k ) . . . α n k ( q − k )) N − α ( q − k ) . . . α n k − ( q − k ) , (4.18)where k and m k are given by Lemma 3.6. Let N be the maximal integer satisfiying (4.18).Note that α ( q j ) is periodic and q − j parametrises the subshift of finite type ( V q − j , σ ) . Moreover V q − j ⊂ V q . This shows that N is well defined.Let n ≥ n k and consider υ = u . . . u n , and let ν = v . . . v n ∈ L ( V q ) be such that u n k +1 . . . u n = α ( q ) . . . α n k ( q )and α ( q ) . . . α n k ( q ) + ≺ v . . . v n k ≺ α ( q ) . . . α n k ( q ) − . Since q ∈ I then ( V q , σ ) is a transitive subshift. Then, there exists ω ∈ L ( V q ) such that υων ∈ L ( V q ). Also, from the choice of υ and ν we have u n k +1 . . . u n and v . . . v n ∈ L ( V q − j ) . Moreover, since ( V q − j , σ ) is not a transitive subshift, then ω ∈ L ( V q ) \ L ( V q − j ). Note thatthe period of α ( q − j ) = N · n k . From Lemma 3.5 we have that | ω |≥ N · n k − (cid:3) Proposition 4.16. If the subshift ( V q , σ ) has the specification property then q ∈ S I .Proof. We will show that ( V q , σ ) does not have the specification property if q / ∈ S I . Firstly,note that if q ∈ V ∩ (( q G , M ] \ I ), α ( q ) is not an irreducible sequence. Then, from [2, Theorem1] ( V q , σ ) is not transitive, thus ( V q , σ ) cannot have the specification property.Recall that if ( V q , σ ) is a transitive subshift then q ∈ I . Moreover, if q / ∈ S I then q ∈ W I , that is, there are infinitely many j ∈ N such that q − j satisfies that α ( q − j ) is periodicbut not irreducible. On the other hand, [2, Proposition 6.1] and Proposition 3.8 imply thatthere are also infinitely many m ∈ N such that α ( q − m ) is periodic and irreducible. This impliesthat there exist infinitely many j ∈ N such that q − j and such that α ( q − j ) is not irreducibleand q − j ≥ q T . Let m ∈ N be such that q − m is irreducible and q − m − ≥ q T and q − m − is notirreducible. Let m be the period of α ( q − m ). Let k and n k be given by Lemma 3.6. Then,from Lemma 4.15 we have that for every n ≥ m , s n ( V q ) ≥ N · n k for some N ∈ N . Since q is weakly irreducible, then there exists m > m + 1 such that q − m is irreducible and q − m − is not irreducible. Since { q − j } is an increasing sequence then q T < q m − < q m < q m − < q m .Then again, Lemma 3.6 and Lemma 4.15 imply that there are N ∈ N and n k ∈ N such thatfor every n ≥ m , s m ( V q ) ≥ N · n k − N ∈ N . Note that n k ≥ n m ≥ N · n k − s m ( V q ) > s m ( V q ). Then for every r ∈ N , it is clear that s n ( V q ) > s m r ( V q ) ≥ N r · n k r − . Also, n k r −→ ∞ as n → ∞ . Then lim n →∞ s n ( V q ) is not bounded from above, which shows that( V q , σ ) does not have the specification property. (cid:3) Showing that strongly irreducible sequences parametrise symmetric subshifts with the spe-cification property is more complicated since it is necessary to find an upper bound for s n .The following technical lemma will allow us to obtain such an upper bound. Lemma 4.17. Let q ∈ S I . Then, there exists j ∈ N such that either ω = α ( q ) . . . α j ( q ) − or ω = α ( q ) . . . α j ( q ) + satisfy ων or ων are in L ( V q ) , for any ν ∈ L ( V q ) .Proof. Let ν = v . . . v r ∈ L ( V q ). We consider three cases: Case 1 : Suppose that α ( q ) is strongly irreducible of type 1. Let us assume that α ( q ) Let q ∈ V . Then, for any word υ ∈ L ( V q ) and any m > | υ | there exists η ∈ L ( V q ) such that υη ∈ L ( V q ) and α ( q ) . . . α m ( q ) or α ( q ) . . . α m ( q ) is a suffix of υη .Proof. Let υ = u . . . u n ∈ L ( V q ) and let m > n be fixed. Then,(4.22) α ( q ) . . . α n − i ( q ) u i +1 . . . u n α ( q ) . . . α n − i ( q ) for every i ∈ { , , . . . , n − } . If(4.23) α ( q ) . . . α n − i ( q ) ≺ u i +1 . . . u n ≺ α ( q ) . . . α n − i ( q ) for every i ∈ { , , . . . , n − } , then it is clear that the words η = α ( q ) . . . α m ( q ) and η ′ = α ( q ) . . . α m ( q ) satisfy theconclusion of the lemma.Suppose now that(4.24) α ( q ) . . . α n − i ( q ) ≺ u i +1 . . . u n α ( q ) . . . α n − i ( q ) for every i ∈ { , , . . . , n − } . Then, we define(4.25) s + = s + ( υ ) = min { s ∈ { , , . . . , n − } : u s +1 . . . u n = α ( q ) . . . α n − s ( q ) } . Then, the minimality of s + and (4.22) imply that α ( q ) . . . α n − i ( q ) ≺ u i +1 . . . u n ≺ α ( q ) . . . α n − i ( q ) for all 0 ≤ i < s + . Then, for every j ∈ N the word η = α n − s + +1 ( q ) . . . α n − s + + j ( q ) satisfies the conclusion of thelemma.Now, let us assume that(4.26) α ( q ) . . . α n − i ( q ) u i +1 . . . u n ≺ α ( q ) . . . α n − i ( q ) for every i ∈ { , , . . . , n − } , In a similar way to (4.25) we define(4.27) s − = s − ( υ ) = min n s ∈ { , , . . . , n − } : u s +1 . . . u n = α ( q ) . . . α n − s ( q ) o . Here, the minimality of s − and (4.22) imply that α ( q ) . . . α n − i ( q ) ≺ u i +1 . . . u n ≺ α ( q ) . . . α n − i ( q ) for all 0 ≤ i < s − . So, for every j ∈ N the word η = α n − s − +1 ( q ) . . . α n − s − + j ( q ) satisfies the conclusion of thelemma.Finally, suppose that(4.28) α ( q ) . . . α n − i ( q ) u i +1 . . . u n α ( q ) . . . α n − i ( q ) for every i ∈ { , , . . . , n − } , Using symmetry, we may assume without losing generality that s + < s − , i.e u s + +1 . . . u n = α ( q ) . . . α n − s + ( q ) . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 31 If s + = 0, then υ = α ( q ) . . . α n ( q ). Therefore, for every j ∈ N the word η = α n +1 ( q ) . . . α n + j ( q )satisfies the conclusion of the lemma. Let us assume now that s + = 0. Then, the minimalityof s + , as well as (4.22) and the inequality s + < s − imply α ( q ) . . . α n − i ( q ) ≺ u i +1 . . . u n ≺ α ( q ) . . . α n − i ( q ) for all 0 ≤ i < s + . Then, for every j ∈ N the word η = α n − s + +1 ( q ) . . . α n − s + + j ( q ) satisfies the conclusion oflemma. (cid:3) Proposition 4.19. If q ∈ S I and there exists K ∈ N such that d ( σ k ( α ( q )) , α ( q )) ≥ / K for every k ∈ N then ( V q , σ ) has the specification property.Proof. Let q ∈ S I . Then, there exists N ∈ N such that for every m ≥ N , q − m ∈ I . Let usassume that there is K ∈ N such that d ( σ k ( α ( q )) , α ( q )) ≥ / K for every k ∈ N . Clearly, d ( σ k ( α ( q )) , α ( q )) ≥ / K for every k ∈ N . Set α ( q − m ) = ( α ( q − m ) . . . α n m ( q − m )) ∞ = ( α ( q ) . . . α n m ( q ) − ) ∞ for every m ∈ N . Let(4.29) J = min n m ∈ N : α ( q ) . . . α K ( q ) , α ( q ) . . . α K ( q ) ∈ L ( V q − m ) and q − m ∈ I o . It is easy to check that n J > K . Claim A : For every m ≥ J we have that n m +1 − n m ≤ K + 1 . To show this, suppose that there is m ≥ J such that n m +1 − n m > K + 1. Then, for every n m + 1 ≤ j ≤ n m +1 , ( α ( q ) . . . α j ( q ) − ) ∞ / ∈ V . This, combined with the assumption n m +1 − n m > K + 1, implies that α n m +1 ( q ) . . . α n m + K +1 ( q ) = α n m +1 ( q ) . . . α n m +1 − i ( q ) = α ( q ) . . . α K +1 ( q )for some i ≥ . Then, d ( σ n m ( α ( q )) , α ( q )) ≤ / K +1 which is a contradiction. Therefore, our claim is true.Let υ = u . . . u ℓ and ν = v . . . v ℓ ∈ B ℓ ( V q ). From Lemma 3.5 there exists L ∈ N suchthat υ, ν ∈ L ( V q − m ) for every m ≥ L . Consider(4.30) J ′ = min n m > max { L, N, J } : υ, ν ∈ L ( V q − m ) o . Note that Proposition 3.9 ( V q − J , σ ) and ( V q − J ′ , σ ) are mixing subshifts of finite type. Also,(4.31) J ≤ N and K < n J ≤ n N ≤ n J ′ . Moreover, if ℓ ≤ m N then J ′ = N + 1. Since υ ∈ B ℓ ( V q − J ′ ), then(4.32) α ( q − J ′ ) . . . α ℓ − i ( q − J ′ ) u i +1 . . . u ℓ α ( q − J ′ ) . . . α ℓ − i ( q − J ′ ) . We now split the proof in three cases: Case 1 : Strict inequalities hold in (4.32).Then from Lemma 4.18, we have that the words α ( q − J ′ ) . . . α t ( q − J ′ ) and α ( q − J ′ ) . . . α t ( q − J ′ )satisfy that υα ( q − J ′ ) . . . α t ( q − J ′ ) , υα ( q − J ′ ) . . . α t ( q − J ′ ) ∈ L ( V q − J ′ )for every t ∈ N . Since J < J ′ , Lemma 3.5 implies that υα ( q ) . . . α n J ( q ) − = υα ( q − J ′ ) . . . α n M ( q − J ′ ) − and υα ( q ) . . . α n J ( q ) + = υα ( q − J ′ ) . . . α n J ( q − J ′ ) + ∈ L ( V q − J ′ ) . From Lemma 4.17 there is j ∈ N such that α ( q ) . . . α j ( q ) − ν or α ( q ) . . . α j ( q ) + ν ∈ L ( V q J ′ ) . Then, either ω = α ( q − J ′ ) . . . α m M ( q − J ′ ) − α ( q ) . . . α j ( q ) − or ω = α ( q − J ′ ) . . . α m J ( q − J ′ ) + α ( q ) . . . α j ( q ) + ∈ L ( V q − J ′ )satisfy that υων ∈ L ( V q − J ′ ), so υων ∈ L ( V q ). Since J < J ′ we obtain that | ω | = n J + j . Case 2 : Let s + , s − given by Lemma 4.18 and let s = min { s + , s − } . We prove the case when s = s + since the proof for other case is analogous. Case 2 a ): Suppose that s = 0. Then, there is N ∈ N such that either(4.33) υ = ( α ( q − J ′ ) . . . α n M ′ ( q − J ′ )) N or(4.34) υ = ( α ( q − J ′ ) . . . α n J ′ ( q )) N α ( q − J ′ ) . . . α l ( q − J ′ ) for l ∈ { , . . . , n J ′ − } . Suppose that (4.33) holds. Again, as a consequence of Lemma 4.18, we obtain that α ( q − J ′ ) . . . α t ( q − J ′ ) and α ( q − J ′ ) . . . α t ( q − J ′ )satisfy υα ( q − J ′ ) . . . α t ( q − J ′ ) , υα ( q − J ′ ) . . . α t ( q − J ′ ) ∈ L ( V q − J ′ )for every t ∈ N . Using a similar argument as in Case 1), we have that either ω = α ( q − J ′ ) . . . α n J ( q − J ′ ) − α ( q ) . . . α j ( q ) − or ω = α ( q − J ′ ) . . . α n J ( q − J ′ ) + α ( q ) . . . α j ( q ) + ∈ L ( V q − J ′ )satisfy υων ∈ L ( V q − J ′ ), and υων ∈ L ( V q ). In this case, we also have | ω | = n J + j . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 33 Suppose now that (4.34) holds. Then l ∈ { , . . . n J ′ − } . If l ∈ { , . . . , N − } then υα ( q ) l +1 . . . α n N ( q ) − ∈ L ( V q − J ′ ). Then either ω = α ( q ) l +1 . . . α n N ( q ) − α ( q ) . . . α j ( q ) − or ω = α ( q ) l +1 . . . α n N ( q ) − α ( q ) . . . α j ( q ) + satisfy that υων ∈ L ( V q J ′ ) ⊂ L ( V q ). Note that | ω |≤ n N − j . If l ∈ { m N . . . n J ′ − } ,Claim A together with (4.31) imply that there is l ′ ∈ { m N . . . m J ′ − } such that l < l ′ ,( α ( q ) . . . α l ( q ) − ) ∞ is an irreducible sequence and l ′ ≤ l ≤ K + 1. Then, using Lemma 4.17 wehave that ω = α n N +1 ( q ) . . . α l ′ ( q ) α ( q ) . . . α j ( q ) − or ω = α n N +1 ( q ) . . . α l ′ ( q ) α ( q ) . . . α j ( q ) + satisfies that υων ∈ L ( V q N ′′ ) ⊂ L ( V q ) . Note that | ω | = l ′ − l + j ≤ K + 1 + j . Case 2 b ): Finally, let us assume that s = 0. Then(4.35) u s +1 . . . u ℓ = α ( q − J ′ ) . . . α ℓ − s ( q − J ′ ) . Then, there exists N ∈ N such that(4.36) u s +1 . . . u ℓ = ( α ( q − J ′ ) . . . α n J ′ ( q − J ′ )) N or(4.37) u s +1 . . . u ℓ = ( α ( q − J ′ ) . . . α n J ′ ( q )) N α ( q − J ′ ) . . . α l ( q − J ′ )for l ∈ { , . . . , m J ′ − } . Then, we can proceed as in Case a ).Combining cases and and Proposition 3.10 we get that for every υ, ν ∈ B ℓ ( V q ) there is ω such that υων ∈ L ( V q ) and | ω | = S = max { n J + j, K + 1 + j } . Observe that | ω | does notdepend on ℓ . This gives that s ℓ ( V q ) = S for every ℓ ∈ N . Then we conclude that s V q = S and that ( V q , σ ) has the specification property. (cid:3) Corollary 4.20. Let N ≥ . Then, if q ∈ I N then ( V q , σ ) has specification.Proof. Fix M ∈ N , N ≥ q ∈ I N . Then, Proposition 4.7 implies that q ∈ I . Then α ( q ) satisfies that for any r ≥ N ≺ α ( rN )+1 ( q ) . . . α ( N +1) r ( q ) ≺ M N . This implies that d ( σ n ( α ( q )) , α ( q )) ≥ / N . Then the result follows directly from Proposition4.19. (cid:3) Note that Proposition 4.16 shows the necessity for q ∈ S I in order to get the specificationproperty. As we show in the Proposition 4.21 the strong irreducibility of q is not sufficientto get the specification property. In those cases d ( σ n ( α ( q )) , α ( q )) is very small for infinitelymany n ∈ N . Proposition 4.21. There exists q ∈ S I such that ( V q , σ ) has no specification.Proof. We construct now q ∈ S I such that for any K ∈ N there is n ∈ N such that d ( σ n ( α ( q )) , α ( q )) ≤ / K . Fix p ∈ Per( I ) with quasi-greedy expansion α ( p ) = ( α ( p ) . . . α m ( p )) ∞ . Let I ( p ) the irreducible interval generated by p as [2, Theorem 2] and define q implicitlyby α ( q ) = α ( p ) . . . α m ( p ) + ( α ( p ) . . . α m ( p )) ∞ . From Lemma 4.8 and Lemma 4.10 there exists infinitely many n ∈ N such that α ( q ) . . . α n ( q )is a fundamental word and ( α ( q ) . . . α n ( q )) ∞ is an irreducible sequence. Let m = max (cid:26) m ∈ { m + 1 , . . . · m } : α ( q ) . . . α m ( q ) is a fundamental word and( α ( q ) . . . α m ( q )) ∞ is irreducible (cid:27) . Let p be defined implicitly as α ( p ) = ( α ( q ) . . . α m ( q )) ∞ . Set q to have quasi-greedy expansion α ( q ) = α ( p ) . . . α m ( p ) + ( α ( p ) . . . α m ( p )) ∞ . Again, applying [2, Theorem 2] we have that I ( p ) is an entropy plateau. Also note that p < q < p < q . As before, applying Lemma 4.8 and Lemma 4.10 we obtain m = max (cid:26) m ∈ { m + 1 , . . . · m } : α ( q ) . . . α m ( q ) is a fundamental word and( α ( q ) . . . α m ( q )) ∞ is irreducible (cid:27) . Then, we can define implicitly p to be α ( p ) = ( α ( q ) . . . α m ( q )) ∞ . Assuming that I ( p n ) is already defined, we define p n +1 to satisfy α ( p n +1 ) = ( α ( q n ) . . . α m n +1 ( q n )) ∞ and m n +1 = max (cid:26) m ∈ { m n + 1 , . . . · m n } : α ( q n ) . . . α m ( q n ) is a fundamental word and( α ( q n ) . . . α m ( q n )) ∞ is irreducible (cid:27) . Then, p n < q n < p n +1 for every n ∈ N . Note that, m n < m n +1 for every n ∈ N . Furthermore,using 4.2 we obtain that p = lim n →∞ p n exists and p ∈ S I . As we constructed p we have that d ( σ m n ( α ( p )) , α ( p )) ≤ / m n . Since m n < m n +1 we obtain the desired conclusion. (cid:3) Remark . The procedure described in Subsection 4.2 can also be performed to get a base p ∈ S I with the specification property. In Proposition 4.21 the sequence { m n } ∞ n =1 is notbounded. If a bounded sequence { m n } ∞ n =1 ⊂ N is considered then the resulting limit point p will satisfy that the subshift ( V p , σ ) has specification. Here K = max { m n } ∞ n =1 + 1 willsatisfy the hypothesis of Proposition 4.19.On the other hand, it is not difficult to show thatPer( I ∗ ) = { q ∈ I ∗ : α ( q ) is periodic } N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 35 will satisfy that there is N ∈ N with d ( σ n ( α ( q )) , α ( q )) ≥ / N for every n ∈ N . However, from[2, Lemma 3.3], the subshift ( V q , σ ) is not transitive, thus, it can not have the specificationproperty.We obtain the following corollary as a direct consequence of [2, Lemma 6.2] Proposition4.12, and Corollary 4.20. Corollary 4.23. The class C ′ is uncountable and ( B ∩ [ q T , M + 1]) \ C ′ is an uncountableset. Synchronised q -subshifts In this section we characterise the set of q ∈ V such that ( V q , σ ) is synchronised. Recall thata word ω ∈ L ( X ) for a transitive subshift ( X, σ ) is intrinsically synchronising (colloquially ω is a magic word ) if whenever υω and ων ∈ L ( X ) we have υων ∈ L ( X ) . We call a transitivesubshift ( X, σ ) to be synchronised if there exists an intrinsically synchronising word ω ∈ L ( X ).Following this, let us observe that C ′ = ∅ since for every q ∈ V ∩ ( q G , q T ), ( V q , σ ) cannot bea synchronised subshift. Also, notice that C ′ ⊂ I . We will show the existence of an intrinsically synchronising word whenever α ( q ) is irredu-cible and the orbit of α ( q ) is not dense in V q ; that is, there is a word ω ∈ L ( V q ) such that ω is not a factor of α ( q ). Our intuition is based on Propositions 4.16 and 4.21. Lemma 5.1. If α ( q ) is an irreducible sequence and the orbit of α ( q ) under σ is not dense in V q then there exists an intrinsically synchronising word ω ∈ L ( V q ) .Proof. We claim that any word ω ∈ L ( V q ) such that ω is neither a factor of α ( q ) nor a factorof α ( q ) is an intrinsically synchronising word. Let υ = u . . . u ℓ and ν = v . . . v n ∈ L ( V q )such that υω ∈ L ( V q ) and ων ∈ L ( V q ). Suppose that | ω | = m . Since, υω ∈ L ( V q ) we havethat(5.1) α ( q ) . . . α ℓ + m − i ( q ) u i +1 . . . u ℓ w . . . w m α ( q ) . . . α ℓ + m − i ( q ) . for every i ∈ { , . . . ℓ − } . Moreover, since ων ∈ L ( V q ) we obtain that(5.2) α ( q ) . . . α m + n − j ( q ) w j +1 . . . w m v . . . v n α ( q ) . . . α m + n − j ( q )for every j ∈ { , . . . m − } . Suppose that υων / ∈ L ( V q ). From (5.1) and (5.2) there exists i ∈ { , . . . ℓ − } such that either α ( q ) . . . α ℓ + m + n − i ( q ) u i +1 . . . u ℓ w . . . w m v . . . v n or α ( q ) . . . α ℓ + m + n − i ( q ) < u i +1 . . . u ℓ w . . . w m v . . . v n . Suppose that α ( q ) . . . α ℓ + m + n − i ( q ) u i +1 . . . u ℓ w . . . w m v . . . v n . Then, (5.1) implies the following: either(5.3) u i +1 . . . u ℓ w . . . w m = α ( q ) . . . α ℓ + m − i ( q ) and v > α ℓ + m − i +1 ( q ) or there is j ′ ∈ { , . . . n } such that(5.4) u i +1 . . . u ℓ w . . . w m v . . . v j ′ − = α ( q ) . . . α ℓ + m + j ′ − − i ( q ) and v j > v ℓ + m + j ′ − i . Both (5.3) and (5.4) contradict that ω is not a factor of α ( q ). If α ( q ) . . . α ℓ + m + n − i ( q ) < u i +1 . . . u ℓ w . . . w m v . . . v n the proof follows from a similar argument. (cid:3) The following proposition gives a sufficient condition on α ( q ) to guarantee that there exists q ∈ I such that ( V q , σ ) is transitive and non-synchronised. Proposition 5.2. If q ∈ I and α ( q ) has dense orbit under σ then no word ω ∈ L ( V q ) isintrinsically synchronising.Proof. Let ω ∈ L ( V q ) and let us assume that | ω | = n . Since the orbit under σ of α ( q ) is densein V q , it is clear that the orbit of α ( q ) is also dense in V q . Recall that the follower set of ω is given by F V q ( ω ) = { ν ∈ L ( V q ) : ων ∈ L ( V q ) } . Fix m ∈ N and let F m V q ( ω ) = (cid:8) ν ∈ F V q ( ω ) : | ν | = m (cid:9) . Since α ( q ) and α ( q ) have dense orbitsin V q then for every ν ∈ F m V q ( ω ) there exist k, k ′ ∈ N such that ων is a prefix of σ k ( α ( q )) and ων is a prefix of σ k ′ ( α ( q )). Fix γ, ν ∈ F m V q ( ω ) with γ ≺ ν . Then, there exist k and k ′ ∈ N such that σ k ( α ( q )) = ωγ and σ k ′ ( α ( q )) = ων . Clearly ωγ ≺ ων . Let υ = α ( q ) . . . α k ( α ( q )).Then υω ∈ L ( V q ). However υων / ∈ L ( V q ) since υων ≻ α ( q ) . . . α k + n + m . Thus, ω is not asynchronising word. (cid:3) As a direct consequence of (2.2) and Corollary 4.23, we obtain the following corollary. Corollary 5.3. The class C ′ is uncountable. Existence non synchronised and transitive symmetric q -subshifts. We will shownow that there are bases q ∈ I with α ( q ) dense in V q . We will perform a constructioninspired by the one given by Schmeling in [36, Proof of Theorem B]. Unfortunately, thepresented construction is algorithmically complicated. Nonetheless, it is not our objective togive an optimal construction for such bases.Firstly, fix q ∈ Per( I ). So, α ( q ) = ( α ( q ) . . . α m ( q )) ∞ where m is the period of α ( q ).Consider B m ( V q ) ordered in a decreasing way with respect to the lexicographical order. Notethat the largest element of B m ( V q ) is α ( q ) . . . α m ( q ) and the smallest is α ( q ) . . . α m ( q ). Nowwe recall [2, Lemma 3.9]. Lemma 5.4. Let q ∈ [ q T , M + 1] ∩ V and m ∈ N . Then for any ν ∈ L ( V q ) there exists η ∈ L ( V q ) such that k m is a prefix of ην ∈ L ( V q ) if M = 2 k , or (( k + 1) k ) m is a prefix of ην ∈ L ( V q ) if M = 2 k + 1 . Then, for every ω ∈ B n ( V q ) there is a word η ω ∈ L ( V q ) such that ( k + 1) k is a prefixof η ω ω ∈ L ( V q ) if M = 2 k + 1, or k is a prefix of η ω ω ∈ L ( V q ) if M = 2 k . On the otherhand, Lemma 4.18 assures us that for every j > | η ω ω | , there exists a word γ η ω ω ∈ L ( V q ) N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 37 such that η ω ωγ η ω ω ∈ L ( V q ) and the word α ( q ) . . . α m ( q ) is a suffix of η ω ωγ η ω ω . For each i ∈ { , . . . , B m ( V q ) − } and for each word ω i ∈ B m ( V q ) we call the word(5.5) ω ′ i = η ω i ω i γ η ωi ω i an extended word of ω i . For ω = α ( q ) . . . α m ( q ) we set ω ′ = ( α ( q ) . . . α m ( q )) t for a fixed t ∈ N with t ≥ and for ω B m ( V q ) = α ( q ) . . . α m ( q ) we set ω ′ B m ( V q ) = ω B m ( V q ) .From [2, Theorem 1] there exists { υ i } B m ( V q ) − i =1 ⊂ L ( V q ) such that(5.6) δ ( q ) = ω ′ υ ω ′ υ . . . ω ′ B m ( V q ) − υ B m ( V q ) − ω ′ B m ( V q ) ∈ L ( V q ) . Clearly, the word δ ( q ) defined in 5.6 satisfies that ν is a factor of δ ( q ) for every ν ∈ m S k =0 B k ( V q ).We recall now the notion of primitive word introduced in [2, Definition 3.10]. Given a finiteword ω = w . . . w n we say that ω is primitive if(5.7) w . . . w n − i ≺ w i +1 . . . w n w . . . w n − i for every i ∈ { , . . . m − } . Note that (5.7) is similar to (4.10). Also, it is clear that(5.8) d ( δ ( q ) , α ( q ) . . . α | δ ( q ) | ) ≤ / m · t . Here, we are inducing the distance d for Σ M as a distance in B | δ ( q ) | ( V q ).The main idea of our construction is to show that for any q ∈ Per( I ) it is possible to finda set { υ i } B m ( V q ) − i =1 ⊂ L ( V q ) satisfying that δ ( q ) is a prefix of a fundamental word θ ( q ).For this purpose, we need to recall the notion of reflection recurrence word introduced in [2,Definition 3.11] and some related results.Given a primitive word ω = w . . . w n , the reflection recurrence word of ω is the truncatedword R ( ω ) = w . . . w s where s = min (cid:8) s ∈ { , . . . , n − } : w s +1 . . . w − n = w . . . w n − s (cid:9) . In the case that s = 0, we have that R ( ω ) = ǫ . We summarise the results proven in [2,Lemmas 3.13, 3.14. 3.15 and 3.16] in the following lemma. Lemma 5.5. Suppose that ω is a primitive word with | ω | = m . i ) If m ≥ then m/ ≤ | R ( ω ) |≤ m ; ii ) R ( ω ) is primitive; iii ) For n ∈ N set R n ( ω ) = R ( R n − ( ω )) and R ( ω ) = R ( ω ) . If m ≥ then there exists j ∈ { , . . . , m } such that either R j +1 ( ω ) = R j ( ω ) with | R j ( ω ) |≤ or R j ( ω ) = w w . This technical condition will help us to prove that the word generated in Lemma 5.7 will parameterise atransitive symmetric subshift. iv ) Let q ∈ I . There exists infinitely many m ∈ N such that α ( q ) . . . α m ( q ) is a primitiveword and for each of such m ∈ N there exists N = N ( m ) such that α ( q − N ) ≺ σ n ( α ( q ) . . . α m ( q )( α ( q ) . . . α m ( q ) + ) ∞ ) ≺ α ( q − N );(5.9) α ( q − N ) ≺ σ n ( α ( q ) . . . α m ( q )( α ( q ) . . . α m ( q ) − ) ∞ ) ≺ α ( q − N );(5.10) and for every r ∈ N α ( q − N ) ≺ σ n ( α ( q ) . . . α m ( q ) − ( R ( α ( q ) . . . α m ( q )) − ) ∞ ) ≺ α ( q − N );(5.11) α ( q − N ) ≺ σ n ( α ( q ) . . . α m ( q ) + ( R ( α ( q ) . . . α m ( q )) + ) ∞ ) ≺ α ( q − N ) . (5.12)We now show the desired properties for δ ( q ). We want to remark that the proof of thefollowing lemma is strongly based on the argument used to prove [2, Propositon 3.17]. Lemma 5.6. Let q ∈ Per( I ) . Then there exists { υ i } B m ( V q ) − i =1 ⊂ L ( V q ) such that δ ( q ) is aprefix of a fundamental word θ ( q ) .Proof. Let α ( q ) = ( α ( q ) . . . α m ( q )) ∞ and consider B m ( V q ) ordered in a decreasing way withrespect to the lexicographical order. For each ω i ∈ B m ( V q ) let ω ′ i be a extended word of ω i given in 5.5. From Lemma 5.5 there is a word γ such that(5.13) ω γ ∈ L ( V q ) and ω γ is primitive.We can consider γ to have minimal length and satisfy (5.13). From [2, Proposition 3.17]there exists ν ∈ L ( V q ) such that ω ′ γ ν ω ′ ∈ L ( V q ) and ν = ( R ( ω ′ γ ) + ) N . . . R ℓ ( ω ′ γ ) + ) N , where ℓ ∈ { , . . . , | ω ′ γ |} and N is given by (5.12) of Lemma 5.5. Let us set ω ′ γν ω ′ = b . . . b K with K = | ω ′ γν ω ′ | . Since ω ≻ ω we have that for any j ∈ { , . . . , K } ,b . . . b K − j σ j ( b . . . b K ) b . . . b K − j . Since ω ′ is the suffix of b . . . b K we have that α ( q ) . . . α m ( q ) is the suffix of length m of b . . . b K . Then, from Lemma 5.5 there is γ such that α ( q ) . . . α m ( q ) γ ∈ L ( V q ) is primitive.We can consider again γ to have minimal lenght. Then [2, Proposition 3.17] implies thatthere is ν ∈ L ( V q ) such that ω ′ γ ν ω ′ γ ν ω ′ ∈ L ( V q ) and ν = ( R ( α ( q ) . . . α m ( q ) γ ) + ) N . . . R ℓ ( α ( q ) . . . α m ( q ) γ ) + ) N , with ℓ ∈ { , . . . , m + | γ |} and N ∈ N . Set K = | ω ′ γ ν ω ′ γ ν ω ′ | and ω ′ γ ν ω ′ γ ν ω ′ = b . . . b K . Similarly, since ω ≻ ω then for any j ∈ { , . . . , K } we obtain that b . . . b K − j σ j ( b . . . b K ) b . . . b K − j . N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 39 Iterating this procedure we obtain that the set { υ i } B m ( V q ) i =1 with υ i = γ i ν i for every i ∈{ , . . . B m ( V q ) − } satisfies that δ ( q ) = b . . . b K = ω ′ υ ω ′ . . . υ B m ( V q ) − ω ′ B m ( V q ) ∈ L ( V q )and b . . . b K − j σ j ( δ ( q )) b . . . b K − j for every j ∈ { , . . . , K − } . Observe that for j = K − m we have that σ j ( δ ( q )) = α ( q ) . . . α m ( q ). Consider θ ( q ) = δ ( q ) α ( q ). Note that σ j ( θ ( q )) = α ( q ) . . . α m ( q ) α ( q ). Since α ( q ) . . . α m ( q ) is a fundamental word and satisfies α ( q ) . . . α m ( q ) α ( q ) ≺ α ( q ) . . . α m ( q ) α ( q ) , we have that for every j ∈ { , . . . m } , α ( q ) . . . α m +1 − i ( q ) α i +1 ( q ) . . . α m ( q ) α ( q ) ≺ α ( q ) . . . α m +1 − i ( q ) . This implies that θ ( q ) is a fundamental word. (cid:3) As a consequence of Lemma 5.6 we obtain the following result. Lemma 5.7. Let q ∈ Per( I ) with α ( q ) = ( α ( q ) . . . α m ( q )) ∞ and let θ ( q ) = θ ( q, t ) given byLemma 5.6. Then: i ) θ ( q ) is a fundamental word; ii ) For any ν ∈ m S k =0 B k ( V q ) , ν is a factor of θ ( q ) . iii ) ( θ ( q )) ∞ ≺ α ( q ) iv ) ( θ ( q )) ∞ is irreducible.Proof. Note that i ) and ii ) are direct consequences of Lemma 5.6. Moreover, since ω ≻ ω weobtain iii ) directly from the definition of θ ( q ). It remains to show that ( θ ( q )) ∞ is irreducible.Let j ∈ N such that ( θ ( q ) . . . θ j ( q ) − ) ∞ ∈ V . Then if j ∈ { , . . . , t · m } then(5.14) θ ( q ) . . . θ j ( q )( θ ( q ) . . . θ j ( q ) + ) ∞ ≺ ( θ ( q )) ∞ holds from the irreducibility of α ( q ). On the other hand for t · m ≥ j note that since α ( q ) . . . α m ( q ) ≺ ω ′ i for every i ∈ { , . . . , B m ( α ( q )) − } and α ( q ) . . . α m ( q ) t is a factor of α ( q ) . . . α j ( q ) we have that (5.14) holds. Thus θ ( q ) is irreducible. (cid:3) Proposition 5.8. There exists q ∈ I such that { σ n ( α ( q )) } ∞ n =0 is a dense subset of V q .Proof. Let q ∈ Per( I ) with quasi-greedy expansion α ( q ) = ( α ( q ) . . . α m ( q )) ∞ . Fix t ∈ N with t ≥ 2. From Lemma 5.6 and Lemma 5.7 there is a fundamental word θ ( q ) suchthat: i ) ( θ ( q )) ∞ is irreducible; ii ) ( θ ( q )) ∞ ≺ α ( q ); and iii ) ν is a factor of θ ( q ) for any ν ∈ m S k =0 B k ( V q ) . Since ( θ ( q )) ∞ is irreducible then α ( q T ) ≺ ( θ ( q )) ∞ . Set q be defined explicitly such that α ( q ) = ( θ ( q )) ∞ = ( α ( q ) . . . α m ( q )) ∞ . Clearly m < m . Also, (5.8) implies that d ( α ( q ) , α ( q )) < / m · t . Moreover, from Lemma 2.3 we have that q ≥ q . This combined with [33, 1.5.10] imply that B m ( V q ) = B m ( V q ) . Then, iii) implies that ν is a factor of α ( q ) for every ν ∈ m S k =0 B k ( V q ) . Consider t n ∈ N with t n ≥ n ≥ 2. Suppose that q n is already defined. We define q n +1 implicitly as α ( q n +1 ) = ( θ ( q n )) ∞ = ( θ ( q n , t n )) ∞ .Note that from i) we have that α ( q n ) is irreducible, so q n ∈ I for every n ∈ N . Also, Lemma2.3 and ii imply that { q n } ∞ n =1 is a decreasing sequence. Furthermore, { q n } is bounded frombelow by q T . Thus q n ց q ∈ U ∩ [ q T , M + 1] as n → ∞ . This implies V q ⊂ ∞ \ n =1 V q n . Let us consider the quasi-greedy expansion of q , α ( q ). If x = ( x i ) ∈ ∞ T n =1 V q n then α ( q n ) σ m (( x i )) α ( q n ) for every n, m ∈ N . Therefore, α ( q n ) ≺ σ m (( x i )) ≺ α ( q n ) for every n, m ∈ N , which gives that α ( q ) σ m (( x i )) α ( q ), i.e. x ∈ V q . Thus,(5.15) V q = ∞ \ n =1 V q n . Since, m n < m n +1 , it is clear that α ( q ) is not a periodic sequence. Note that iii) implies thatfor every n ∈ N and for every ν ∈ m n S k =0 B k ( V q n ), ν is a factor of α ( q n +1 ). Also, observe thatfor every n ∈ N , α ( q n ) . . . α m n ( q n ) is a prefix of α ( q ). Then, for any ν ∈ ∞ [ n =1 m n [ k =0 B k ( V q n ) ,ν is a factor of α ( q ). Fix ω = w . . . w k ∈ L ( V q ). Then α ( q ) . . . α k − i ( q ) w i +1 . . . w k α ( q ) . . . α k − i ( q ) for every i ∈ { , . . . , k − } . Then, from (5.15) and since q n ց q , [33, 1.5.10] implies that there is N ∈ N such that forevery n ≥ N , ω ∈ B k ( V q n ) = B k ( V q N ) and B k ( V q n ) = B k ( V q ). Thus, ω is a factor of α ( q n )for every n ≥ N which gives that ω is a factor of α ( q ). Then, since each word in L ( V q ) hasto appear in α ( q ) infinitely often we obtain that { σ n ( α ( q )) } ∞ n =0 is a dense subset of V q andthat V q is a transitive subshift. Therefore, [2, Theorem 1] gives that q ∈ I . (cid:3) N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 41 Corollary 5.9. For any q ∈ B ∩ [ q T , M +1] there is p such that { σ n ( α ( p )) } ∞ n =0 is dense in V p and p is arbitrarily close to q , i.e. { p ∈ I : α ( p ) is dense in V p } is dense in B ∩ [ q T , M + 1] .Proof. Let q ∈ B ∩ [ q T , M + 1] with quasi-greedy expansion α ( q ). We note here that if q ∈ I ( p ′ ) for some p ′ ∈ Per( I ) then, from [2, Proposition 4.11] Lemma 3.7 we obtain that q = p ′ , or q satisfies α ( q ) = α ( p ′ ) . . . α m ( p ′ ) ( p ′ ) + ( α ( p ′ ) . . . α m ( p ′ ) ( p ′ )) ∞ , where m ( p ′ ) is the period of α ( p ′ ). Then, we have to consider three cases: Case 1 : Suppose that q ∈ I ( p ′ ) and q = p ′ . Let α ( q ) = ( α ( q ) . . . α m ( q )) ∞ the quasi-greedyexpansion of q . Then, since Per( I ) is dense in B ∩ [ q T , M + 1] then, for every N ∈ N thereexists p N ∈ Per( I ), α ( p N ) = ( α ( p N ) . . . α m N ( p N )) ∞ such that d ( α ( p N ) , α ( q )) ≤ / N +1 . Inparticular, for any ε > N, M ∈ N such that d ( α ( p N ) , α ( q )) ≤ / ( M · m )+1 < / N < ε/ . So, fix t ∈ N with t ≥ θ ( p N ) given by Lemma 5.7 and set p N definedimplicitly by α ( p N ) = ( θ ( p N )) ∞ . Since t ≥ 2, we get d ( α ( p N , α ( p N )) ≤ / · m N . Then, d ( α ( p N ) , q ) ≤ / · m N + 1 / ( M · m )+1 < / ( M · m )+1 ≤ ε. Consider { θ ( p N i ) } ∞ i =1 , the sequence generated in the proof of Proposition 5.8 and the associ-ated sequence { α ( p N i ) } ∞ i =1 with limit p . Note that for every i ≥ α ( p N i ) . . . α t · m N ( p N i ) = α ( p N ) . . . α t · m N ( p N ) , where m N is the period of α ( p N ). Then, d ( α ( p N i ) , α ( q )) < ε for every i ∈ N , thus d ( α ( p ) , α ( q )) < ε. As a consequence of Lemma 2.3 we obtain the result for this case. Case 2 : Suppose that q ∈ I ( p ′ ) and α ( q ) = α ( p ′ ) . . . α m ( p ′ ) ( p ′ ) + ( α ( p ′ ) . . . α m ( p ′ ) ( p ′ )) ∞ . Let ε > 0. From Lemmas 4.8 and 4.10 there exists N ∈ N such that for every n ≥ N , i ) α ( q ) . . . α m n ( q ) is a fundamental word; ii ) ( α ( q ) . . . α m n ( q )) ∞ is an irreducible sequence; and iii ) d (( α ( q ) . . . α m n ( q )) ∞ , α ( q )) < / N · m n < ε. Fix n ∈ N such that i), ii) iii) holds for for ε and let q ′ with α ( q ′ ) = ( α ( q ) . . . α m n ( q )) ∞ .Fix t ∈ N with t ≥ 2. Let θ ( p ) given by Lemma 5.7 and set p defined implicitly by α ( p ) = ( θ ( q ′ )) ∞ . Fix a sequence { t n } ∞ n =2 ⊂ N with t n ≥ n ≥ 2. Then, forevery n ≥ { θ ( p n ) } ∞ i =2 the sequence generated in the proof of Proposition 5.8and the associated sequence { α ( p n ) } ∞ i =1 . From a similar argument as in Case 1, the quasi-greedy expansion of p , α ( p ), where p is the limit of the sequence { p n } ∞ n =1 satisfies the desiredconclusion. Case 3 : Suppose that q ∈ ( B ∩ [ q T , M + 1]) \ S p ′ ∈ Per( I ) I ( p ′ ). In such case, note that for any ε > 0, there are p + , p − ∈ Per( I ) such that d ( α ( p − ) , α ( q )) < ε/ α ( p − ) ≺ α ( q ); and d ( α ( p + ) , α ( q )) < ε/ α ( p + ) ≻ α ( q ) . Then, applying the construction exposed on Case 1 to α ( p − ) = ( α ( p − ) . . . α m p − ( p − )) ∞ andthe construction exposed in Case 2 to α ( p + ) = ( α ( p + ) . . . α m p + ( p + )) ∞ we can construct thedesired base. (cid:3) As in the previous constructions 4.2 and 4.3, the construction of a base q with dense quasi-greedy expansion α ( q ) in V q depends on a sequence { t n } ∞ n =1 . Then, the following corollaryholds. Corollary 5.10. The subclass C ′ ∩ I is uncountable. Hausdorff dimension of the classes C ′ , C ′ , C ′ We will investigate the Hausdorff dimension of the classes C ′ , C ′ , C ′ . Let us summarise theresults concerning of this classes so far.(A) C ′ = n q ∈ I : q ∈ S I and there is K ∈ N with d ( σ n ( α ( q )) , α ( q )) ≥ / K for n ∈ N o ;(B) C ′ = { q ∈ I : σ n ( α ( q )) is not dense in V q } ;(C) C ′ = { q ∈ I : σ n ( α ( q )) is dense in V q } ;Observe that (A) is a consequence of Proposition 4.16, Proposition 4.19 and Proposition4.21. Also, (B) and (C) are consequences of Lemma 5.1 and Proposition 5.2. Finally, Corollary4.23, Corollary 5.3, Proposition 5.9 and Corollary 5.10 imply C ′ , C ′ , and C ′ are uncountableand dense subsets of B ∩ [ q T , M + 1].Let us establish the necessary results to complete the proof of Theorem A. From Theorem2.7 we have that dim H ( B ) = 1. Furthermore, from [2, Lemma 4.10] we have B ⊂ U ⊂ V . Then, combining [2, Theorem 3], [23, Lemma 2.6] it holds thatdim H ( B ) = dim H ( U ) = dim H ( V ) = 1 . Now, from [23, Theorem 2] we know that for any q ∈ B we have that dim locH ( B q ) =dim H ( V q ).We recall now [2, Lemma 6.4]. Lemma 6.1. Let N ≥ . Then dim H ( I N ) ≥ log(( M + 1) N − /N log( M + 1) . Then, from Lemma 6.1 we obtain directly the following: Proposition 6.2. dim H ( C ′ ) = 1 . Moreover, dim H ( C ′ ) = 1 . Proof. As in the proof of [2, Theorem 3], we note that I N ⊂ C ′ for every N ≥ 2. FromLemma 6.1 the statement holds by letting N → ∞ . Finally, (2.2) gives us dim H ( C ′ ) = 1 . (cid:3) N THE SPECIFICATION PROPERTY AND SYNCHRONISATION OF UNIQUE q -EXPANSIONS 43 Now, as a consequence of Lemma 2.2, Proposition 2.8 and Proposition 4.14 we havedim H ( W I ) = dim H (([ q T , M + 1] ∩ V ) \ C ′ ) = 1 , thus dim H ( V \ C ′ ) = 1 . Finally, Lemma 2.2, Proposition 2.8 and Corollary 5.9 imply the following statement. Proposition 6.3. dim H ( C ′ ) = 1 . Proof of Theorem A . Theorem A follows from Propositions 4.16, 4.19, 4.21. 6.2, 5.2, 6.3and Lemma 5.1. (cid:3) Final comments and open questions We notice that Definition 4.1 can be generalised in the following way. We say that a sequence α = ( α i ) ∈ V is strongly ∗ -irreducible if α is ∗ -irreducible and there exists N ∈ N such that forevery n ≥ N with ( α . . . α − n ) ∞ ∈ V , then ( α . . . α − n ) ∞ is ∗ -irreducible. Similarly, a sequence( α i ) ∞ ∈ V is weakly ∗ -irreducible if α is ∗ -irreducible and there exist infinitely many n ∈ N such that ( α . . . α − n ) ∞ ∈ V and ( α . . . α − n ) ∞ is not ∗ -irreducible. We can also define stronglyirreducible numbers and weakly irreducible numbers as in Definition 4.2.Using the constructions performed in Subsections 4.2 and 4.3 with small modifications itis not difficult to see that there exist uncountably many strongly ∗ -irreducible sequences anduncountably many weakly ∗ -irreducible sequences. Also, it is not hard to check that stronglyirreducible numbers and weakly irreducible numbers are dense in ( q KL , q T ) ∩ V where q KL is defined in (2.4). Then, it is natural to investigate the Hausdorff dimension of the set ofstrongly ∗ -irreducible numbers and ∗ -weakly irreducible numbers. Question 7.1. What is the Hausdorff dimension of strongly irreducible and weakly irreduciblenumbers?We would like to make some comments about the symbolic dynamics of ( V q , σ ) when q ∈ ( q KL , q T ). In [2, Proposition 5.10] it was shown that if q is a ∗ -irreducible base q suchthat α ( q ) is periodic then for every p ∈ I ∗ ( q ), V p contains a unique subshift of finite type( X, σ ) such that h top ( X ) = h top ( V q ) = h top ( V p ). To the best of the knowledge of the authorit is not known that ( X, σ ) is a mixing subshift. Also, if q ∈ V ∩ [ q KL , q T ) it is known that( V q , σ ) is not a transitive subshift [2, Lemma 3.3]. However, we do not know much about thesymbolic dynamics of the transitive components of ( V q , σ ). Question 7.2. Consider q ∈ B ∩ ( q KL , q T ). i ) Is it true that if ( V q , σ ) is sofic, then there exists a unique transitive subshift ( X, σ ) suchthat h top ( X ) = h top ( V q ) and ( X, σ ) is a sofic subshift? ii ) Is it true that if q is a strongly ∗ -irreducible number then ( V q , σ ) contains a unique trans-itive subshift ( X, σ ) such that h top ( X ) = h top ( V q ) and ( X, σ ) is a has the specificationproperty? iii ) What are the conditions for the quasi-greedy expansion of q to ensure that ( V q , σ )contains a unique transitive subshift ( X, σ ) such that h top ( X ) = h top ( V q ) and ( X, σ ) issynchronised?It would be also interesting to know if for every q ∈ V the subshift ( V q , σ ) is balanced or boundedly supermultiplicative as in [9, Definition 3.1, Definition 3.2]. In particular, thisquestion is interesting from two different perspectives: firstly, it is not clear if there is anexample of a non-transitive balanced shift or a non-transitive boundedly supermultiplicativesubshift. Secondly, in [9, p. 639] the authors ask if there is an example of a balanced subshiftwithout the almost specification property. It would be interesting to find such example in C ′ or in B ∩ [ q T , M + 1] \ C ′ .It would be interesting to classify the set of q ∈ ( B ∩ [ q T , M + 1]) \ C ′ satisfying theweaker forms of specification defined in [31] and calculate their size. Finally, to the best ofthe knowledge of the author, it is not known if every transitive symmetric q -shift is entropyminimal , i.e. that every subshift ( X, σ ) of ( V q , σ ) satisfies h top ( X ) < h top ( V q )— see[21]. Acknowledgements The author is indebted with Felipe Garc´ıa-Ramos and with Edgardo Ugalde for all theirsupport during the development of this research. 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