Intermediate beta-shifts of finite type

A bstract . An aim of this article is to highlight dynamical di ﬀ erences between the greedy, and hence the lazy, β -shift (transformation) and an intermediate β -shift (transformation), for a ﬁxed β ∈ (1 , 2). Speciﬁcally, a classiﬁcation in terms of the kneading invariants of the linear maps T β,α : x (cid:55)→ β x + α mod 1 for which the corresponding intermediate β -shift is of ﬁnite type is given. This characterisation is then employed to construct a class of pairs ( β,α ) such that the intermediate β -shift associated with T β,α is a subshift of ﬁnite type and where the map T β,α is not transitive. This is in contrast to the situation for the corresponding greedy and lazy β -shifts and β -transformations, in that these two properties do not hold.


Introduction, motivation and the main results
1.1.Introduction and motivation.For a given real number β > 1 and a real number x ∈ [0, 1/(β − 1)] an infinite word (ω n ) n∈N over the alphabet {0, 1} is called a β-expansion of the point x if If β is a natural number, then the β-expansions of a point x correspond to the β-adic expansions of x.Moreover, in this case, almost all positive real numbers have a unique β-expansion.On the other hand, in [29] it has been shown that if β is not a natural number, then, for Lebesgue almost all x, the cardinality of the set of β-expansions of x is equal to the cardinality of the continuum.
The theory of β-expansions was initiated by Rényi [27] and Parry [24,25].Here an important link to symbolic dynamics is made.Indeed, through iterating the β-transformation G β : x → βx mod 1 and the β-transformation L β : x → β(x − 1) + 2 mod 1 one obtains subsets of {0, 1} N known as the greedy and (normalised) lazy β-shifts, respectively, where each point ω + of the greedy β-shift is a β-expansion, and corresponds to a unique point in [0, 1], and each point ω − of the lazy β-shift is a β-expansion, and corresponds to a unique point in [(2 − β)/(β − 1), 1/(β − 1)].(Note that in the case that (2 − β)/(β − 1) ≤ 1, if ω + and ω − are β-expansions of the same point, then ω + and ω − do not necessarily have to be equal, see [19].)Through this connection we observe one of the most appealing features of the theory of β-expansions, namely that it links symbolic dynamics to number theory.In particular, one can ask questions of the form, for what class of numbers the greedy and lazy β-shift has given properties and vice versa.In fact, although the arithmetic, Diophantine and ergodic properties of the greedy and lazy β-shifts have been extensively studied, see [6,8,28] and references therein, there are many open problems of this form.Further, applications of this theory to the efficiency of analog-to-digital conversion have also been explored in [12].Moreover, through understanding β-expansions of real numbers advances have been made in understanding Bernoulli convolutions, see [9,10,11] and reference therein.
There are many ways, other than using the greedy and lazy β-shift, to generate a β-expansion of a positive real number.For instance the intermediate β-shifts Ω β,α which The first author was supported by NSFC (no.11201155 and 11371148) and "Fundamental Research Funds for the Central Universities" SCUT (2013ZZ0085).The second author was supported by the ERC grant no.306494.The third author thanks the Universität Bremen and the South China University of Technology for their support.and where the maps T ± β,α are defined as follows.Letting p = p β,α (1 − α)/β we set T + β,α (p) 0 and T + β,α (x) βx + α mod 1, for all x ∈ [0, 1] \ {p}, see Figure 1.Similarly, we define T − β,α (p) 1 and T − β,α (x) βx + α mod 1, for all x ∈ [0, 1] \ {p}.Indeed we have that the maps T ± β,α are equal everywhere except at the point p and that 1].Observe that, when α = 0, the maps G β and T + β,α coincide, and when α = 2 − β, the maps L β and T − β,α coincide.We note that the maps defined above are sometimes also called linear Lorenz maps and arise naturally from the Poincaré maps of the geometric model of Lorenz differential equations, see for instance [13,22,30,32].Here we make the observation that for all (β, α) ∈ ∆, the symbolic space Ω β,α is always a subshift, meaning that it is invariant under the left shift map, see Corollary 2.  Here, the intermediate β-transformations and β-shifts are the main topic of study, and thus, to illustrate their importance we recall some of the known results in this area.Parry [26] proved that any topological mixing interval map with a single discontinuity is topologically conjugate to a map on the form T ± β,α where (β, α) ∈ ∆.Moreover, in [15,17] it is shown that a topologically expansive piecewise continuous map T can be described up to topological conjugacy by the kneading invariants of the points of discontinuity of T .(In the case that T = T ± β,α , the kneading invariants of the point of discontinuity p are precisely given by the points in the associated intermediate β-shift which are a β-expansion of p + α/(β − 1).)For such maps, assuming that there exists a single discontinuity, the authors of [3,17] gave a simple condition on pairs of infinite words in the alphabet {0, 1} which is satisfied if and only if that pair of sequences are the kneading invariants of the point of discontinuity of T .
Our main results, Theorems 1.1 and 1.3 and Corollary 1, contribute to the ongoing efforts in determining the dynamical properties of the intermediate β-shifts and examining whether these properties also hold for the counterpart greedy and lazy β-shifts.In particular, we demonstrate that it is possible to construct pairs (β, α) ∈ ∆ such that the associated intermediate β-shift is a subshift of finite type but for which the maps T ± β,α are not transitive.In contrast to this, the corresponding greedy and lazy β-shifts are not a subshift of finite type (or even a sofic shift, namely a factor of a subshift of finite type) and, moreover, the maps G β and L β are transitive.Recall that an interval map T : [0, 1] is called transitive if and only if for all open subintervals J of [0, 1] there exists m ∈ N such that m i=1 T i (J) = (0, 1).To prove the transitivity part of our result we will use the results of Palmer [23] and Glendinning [15].Here they show that for any 1 < β < 2 the maps G β and L β are transitive.In fact, they give a complete classification of the set of points (β, α) ∈ ∆ for which the maps T ± β,α are not transitive.(For completeness we restate their classification in Section 4.) Before formally stating our main results let us emphasis the importance of subshifts of finite type.These symbolic spaces give a simple representation of dynamical systems with a finite Markov partitions.There are many applications of subshifts of finite type, for instance in coding theory, transmission and storage of data or tilings.We refer the reader to [7,18,21,30] and references therein for more on subshifts of finite type and their applications.
1.2.Main results.Recall, from for instance [8,Example 3.3.4],that the multinacci number of order n ∈ N is the real number γ n ∈ (1, 2) which is the unique positive real solution of the equation 1 The smallest multinacci number is the multinacci number of order 2 and is equal to the golden mean (1 + √ 5)/2, and γ n+1 > γ n , for all n ∈ N. Further, for n, k ∈ N 0 with n ≥ 2, we define the algebraic integers β n,k and α n,k by (β n,k ) 2 k = γ n and α n,k = 1 − β n,k /2.(Here and in the sequel, we let the symbol N 0 denote the set of non-negative integers.)Theorem 1.1.For all n, k ∈ N with n ≥ 2, we have the following.
(i) The intermediate The greedy and lazy β n,k -transformations are transitive.(iv) The greedy and lazy β n,k -shifts, Ω β n,k ,0 and Ω β n,k ,2−β n,k respectively, are not sofic, and hence not a subshift of finite type.
Part (iii) is well known and holds for every greedy, and hence lazy, β-transformations, see for instance [15,23].It is included here both for completeness and to emphasise the dynamical differences which can occur between the greedy, and hence the lazy, β-transformation (shift) and an associated intermediate β-transformation (shift).
To verify Theorem 1.1(ii) we apply Theorem 1.3 given below.This latter result is a generalisation the following result of Parry [24].Here and in the sequel, τ ± β,α (p) denotes the points in the associated intermediate β-shifts which are a β-expansion of p + α/(β − 1), where p = p β,α .Theorem 1.2 ( [24]).For β ∈ (1, 2), we have that (i) the greedy β-shift is a subshift of finite type if and only if τ − β,0 (p) is periodic and (ii) the lazy β-shift is a subshift of finite type if and only α is a subshift of finite type if and only if both τ ± β,α (p) are periodic.The following example demonstrates that it is possible to have that one of the sequences τ ± β,α (p) is periodic and that the other is not periodic.
This leads us to the final problem of studying how a greedy (or lazy) β-shift being a subshift of finite type is related to the corresponding intermediate β-shifts being a subshift of finite type.We already know that it is possible to find intermediate β-shifts of finite type in the fibres ∆(β) even though the corresponding greedy and lazy β-shifts are not subshift of finite type.Given this, a natural question to ask is, can one determine when no intermediate β-shift is a subshift of finite type, for a given fixed β.This is precisely what we address in the following result which is an almost immediate application of the characterisation provided in Theorems 1.2 and 1.3.
We observe that the values β n,k , which are considered in Theorem 1.1, are indeed a solution of a polynomial with coefficients in the set {−1, 0, 1} and thus do not satisfy the conditions of Corollary 1; this is verified in the proof of Theorem 1.1(iv).
1.3.Outline.In Section 2 we present basic definitions, preliminaries and auxiliary results required to prove Theorems 1.1 and 1.3 and Corollary 1.The proofs of Theorem 1.3 and Corollary 1 are presented in Section 3 and the proof of Theorem 1.1 is given in Section 4.
Theorem 2.1.A shift space Ω is a SFT if and only if there exists a finite set F of finite words in the alphabet {0, 1} such that if ξ ∈ F, then σ m (ω)| |ξ| ξ, for all ω ∈ Ω.
The set F in Theorem 2.1 is referred to as the set of forbidden words.Further, a subshift Ω is called sofic if and only if it is a factor of a SFT. For for all m ∈ N, and we write ω for all m ∈ N, and we write ω = (ω 1 , ω 2 , . . ., ω k−1 , ω k , ω k+1 , . . ., ω k+n ).
2.2.Intermediate β-shifts and expansions.We now give the formal definition of the intermediate β-shift.Throughout this section we let (β, α) ∈ ∆ be fixed and let p denote the value otherwise, for all n ∈ N. We will denote the images of the unit interval under τ ± β,α by Ω ± β,α , respectively, and write Ω β,α for the union Ω + β,α ∪ Ω − β,α .The upper and lower kneading invariants of T ± β,α are defined to be the infinite words τ ± β,α (p), respectively, and will turn out to be great importance.
Remark 1.For ease of notation let ω ± = (ω ± 1 , ω ± 2 , . . ., ) denote the infinite words τ ± β,α (p), respectively.By definition, The connection between the maps T ± β,α and the β-expansions of real numbers is given through the T ± β,α -expansions of a point and the projection map π β,α : {0, 1} N → [0, 1], which we will shortly define.The projection map π β,α is linked to the underlying (overlapping) iterated function system (IFS), namely ([0, 1]; (We refer the reader to [14] for the definition of and further details on IFSs.)The projection map π β,α is defined by An important property of the projection map is that the following diagram commutes.
(Here the symbols ≺, , and denote the lexicographic orderings on {0, 1} N .)Moreover, Remark 2. By the commutativity of the diagram in (1), we have that p) and thus, from now on we will write τ + β,α (p) for the common value Proof.This is a direct consequence of Theorem 2.2 and the commutativity of the diagram given in (1).
As one can see the code space structure of Ω ± β,α is simpler than the code space structure of Ω ± β,α .In fact, to prove Theorem 1.3, we will show that it is necessary and sufficient to show that Ω β,α is a SFT.

2.5.
Periodicity and zero/one-full words.We now give a sufficient condition for the kneading invariants of T ± β,α (and hence, by Remark 2, of T ± β,α ) to be periodic.For this, we will require the following notation and definition.For a given subset E of N, we write |E| for the cardinality of E.

Proofs of Theorem 1.3 and Corollary 1
Let us first prove an analogous result for the extended model Ω β,α .Also, throughout this section, for a given (β, α) ∈ ∆, we set p = p β,α .

Proof of Proposition 1(i).
Assume by way of contradiction that Ω β,α is a SFT, but that τ − β,α (p) is not periodic.Then by Lemma 2.4 the infinite word ν = σ(τ − β,α (p)) is zero-full, namely is not finite.Thus, there exists a non-constant monotonic sequence (n k ) k∈N of natural numbers such that This together with Corollary 2 implies that Ω β,α is not a SFT, contradicting our assumption.The statement that if Ω β,α is a SFT, then τ + β,α (p) is periodic, follows in an identical manner to above, where we use one-fullness instead of zero-fullness.
We now show in the following proposition, Proposition 2, that Ω ± β,α is a SFT if and only if Ω ± β,α is a SFT.This result together with Proposition 1 completes the proof of Theorem 1.3.In order to prove the backwards implication of Proposition 2 we will use Theorem 2.1.Namely that a subshift Ω is a SFT if and only if there exists a set of forbidden word F such that if ξ ∈ F, then σ m (ω)| |ξ| ξ, for all ω ∈ Ω. Proposition 2. For (β, α) ∈ ∆ the subshift Ω β,α is a SFT if and only if the subshift Ω β,α is a SFT.
Proof.We first prove the forward direction: if the shift space Ω β,α is a SFT, then Ω β,α is a SFT.We proceed by way of contradiction.Suppose that Ω β,α is a SFT, but Ω β,α is not a SFT.Proposition 1 then implies that τ ± β,α (p) are both periodic and we may assume, without loss of generality, that they have the same period length m.Moreover, by Corollary 2, since, by assumption, Ω β,α is not a SFT, there exist ω In particular, as Ω β,α is a SFT, Theorem 2.2 implies that there exists an l ∈ {0, 1, . . ., n − 1} such that either We will now show the converse: if the shift space Ω β,α is a SFT, then Ω β,α is a SFT.Let F denote the set of forbidden words of Ω β,α , and assume that F is the smallest such set.Then for each ξ ∈ F either is a finite set of forbidden words for Ω β,α .For if not, this would contradict Theorem 2.2.
Proof of Theorem 1.3.This is an immediate consequence of Propositions 1 and 2.
Let us conclude this section with the proof of Corollary 1.
Proof of Corollary 1.By construction π β,α • τ ± β,α is the identity function on [0, 1] and so β is a root of the polynomial . By the definition of the projection map π β,α , this latter polynomial is independent of α and all of its coefficients belong to the set {−1, 0, 1}.An application of Theorem 1.2 and 1.3 completes the proof.

Proof of Theorem 1.1
Having proved Theorem 1.3, we are now equipped to prove Theorem 1.1.Before setting out to prove Theorem 1.1, let us recall the result of Palmer and Glendinning on the classification of the point (β, α) ∈ ∆ such that T + β,α , and hence T − β,α , is transitive.Definition 4.1.Suppose that 1 < k < n are natural numbers such that gcd(k, n) = 1.Let s, m ∈ N be such that 0 ≤ s < k and n = mk + s.For 1 ≤ j ≤ s define V j and r j by jk = V j s + r j , where r j , V j ∈ N and 0 ≤ r j < s.Also, define h j inductively via the formula We define the set D k,n to be the set of points (β, α) ∈ ∆, such that 1 < β n ≤ 2 and Here, for 2 ≤ j ≤ s.Further, for each natural number n > 1, we define the set D 1,n to be the set of points (β, α) ∈ ∆ such that 1 < β n ≤ 2 and (See Figure 3 for an illustration of the regions D k,n .) , and hence T + β,α , is not transitive.In order to prove Theorem 1.1 we will also use the following notation.For n ∈ N set ).Further, for n, k ∈ N, we set ξ We also make the observation that if k ≥ 2 and if l ∈ {2, 3, . . ., k}, then we have the following equalities.
This observation will become important in the proof of Theorem 1.1(ii).
and that The later inequalities give in ( 6) and ( 7) both hold true since 1 . Therefore, we have that (β n,k , α n,k ) ∈ D 1,2 and so the results follows from an application of Theorem 4.2.
Proof of Theorem 1.1(ii).Let n, k ∈ N 0 with n ≥ 2 be fixed.By combining Theorems 1.3 and 2.5 together with the results of [3] it suffices to show the following, for all n, k ∈ N 0 with n ≥ 2.
(i) The pair (ξ − n,k , ξ + n,k ) are admissible, namely that, for all m ∈ N, where ξ ± n,k,m denotes the m-th letter of the word ξ ± n,k , respectively.(iii) The following equalities hold: However, (i) is precisely the result given in Proposition 3, (ii) is the result presented in Proposition 4 and (iii) is the result given in Proposition 5; these propositions, together with their proofs, are presented directly below.
Proposition 3.For n, k ∈ N 0 with n ≥ 2, we have that (ξ − n,k , ξ + n,k ) is admissible.Proof.We first prove the statement for the cases k = 0, k = 1 and k = 2.We then show the result for a given k > 2 using an inductive argument.
Using the observations that ξ − n,2 = (0, 1, 1, 0, 1, 0, 0, 1 , . . ., 1, 0, 0, 1 it is easy to verify, using the repeating structure of the words ξ ± n,k , that the pairs (ξ − n,0 , ξ + n,0 ), (ξ − n,1 , ξ Hence, all that remains is to prove, for k > 2 and for m = 4p, where p ∈ N, that Thus by the equality given in (5) it follows that Further, by a symmetrical argument we obtain that n,k otherwise.Now assume that for some l ∈ {1, 2, . . ., k − 2} and for all p ∈ N that and that To complete the proof we will show, for all p ∈ N with p = 1 mod 2, that To this end observe that by construction, the words ξ − n,k and ξ + n,k are made up of the finite words ξ . Hence, we have one the following cases.
Proof.Observe, by the recursive definition of the infinite words ξ This latter term is equal to zero if and only if either However, x 2 k > 0 for all x > 1, and, by the definition of a multinacci number, the maximal real solution of for all x > 1.Given this claim, which we will shortly prove, the result follows.
To prove the claim we proceed by induction on k.
Suppose that the statement is true for some k ∈ N 0 , then Since x > 1 the value of the term 1 − x −2 k is positive and finite and by our inductive hypothesis This completes the proof of the claim.
We claim that for all n, k ∈ N. We will shortly prove the claim, then using (9) we prove via an inductive argument the statement of the proposition.
Fixed n ∈ N. We now prove (9) by induction on k.By definition we have that This competes the proof of the base case of the induction.So suppose that (9) holds true for all integers m < k, for some integer k > 1.Then, by construction of ξ n,k , definition of κ and the inductive hypothesis, we have that κ This completes the induction and hence the proof of the claim.
For a fixed integer n ≥ 2, we now prove, via an inductive argument on k, the statement of the proposition.Recall that α n,k = 1 − β n,k /2 and observe, for k = 0, by definition of the projection map π β n,0 ,α n,0 , that where the last equality follows by using the fact that β n,0 is the multinaccinumber of order n and so 1 = β n,0 −1 + β n,0 −2 + • • • + β n,0 −n .Further, we note that m = π β n,0 ,α n,0 (ξ + n,0 ) This completes the proof of the base case of the induction.So suppose the statement of the proposition holds true for all integers m < k, for some k ∈ N. By the definition of the projection map π β n,k ,α n,k , that β n,k 2 = β n,k−1 and ( 9), we have that This completes the induction.
Proof of Theorem 1.1(iii).This is a direct consequence of Theorem 4.2 For the proof of Theorem 1.1(iv) we will require the following additional preliminaries.(i) A Pisôt number is a positive real number β whose Galois conjugates all have modulus strictly less than 1.A Perron number is a positive real number β whose Galois conjugates all have modulus strictly less than β.(ii) For n ≥ 2, the n-th multinacci number is a Pisôt number, see [8,Example 3.3.4].(iii) If the shift space Ω β,0 is sofic, then β is a Perron number, see [28,Theorem 2.2(3)]; here the result is attributed to [5,20,24].
Proof of Theorem 1.1(iv).Since, by definition, any SFT is sofic, it is sufficient to show that β n,k is not a Perron number, for all n, k ∈ N with n ≥ 2. To this end let P n,k denote the polynomial given by Suppose, by way of contradiction, that β n,k was a Perron number.By the definition of β n,k and P n,k , we have that P n,k (β n,k ) = 0, and so the minimal polynomial Q n,k of β n,k divides P n,k .Further, if β was a root of Q n,k not equal to β n,k , then it would also be a root of P n,k .Moreover, |β| < 1.Since if not, then P n,k (β) = 0, since β is a root of Q n,k , and, by our assumptions, 1 < |β 2 k | < γ n .Hence, β 2 k would be a Galois conjugate of γ n with modules greater than 1, contradicting the fact that γ n is a Pisôt number.Thus, all of the roots of Q n,k with the exception of β n,k would be of absolute value strictly less than 1, and hence β n,k would be a Pisôt number.However, it is well known that there are only two Pisôt numbers θ 0 and θ 1 less than 2 1/2 , and moreover, θ 1 > θ 0 > 2 1/3 .Thus, since γ n ∈ (1, 2), if k ≥ 2 we obtain a contradiction to our assumption that β n,k is a Perron number.Furthermore, by numerical calculations we know that 1.8 > θ 0 2 , 1.925 > θ 1 2 , P 4,0 (1.8) < 0 and P 4,0 (1.925) < 0. Therefore, for all n ≥ 4, we have that θ 0 nor θ 1 is a root of P n,1 .Furthermore, for n ∈ {2, 3}, it is a simple calculation to show that the minimal polynomials of θ 0 and θ 1 , namely x → x 3 − x − 1 and x → x 4 − x 3 − 1 respectively, do not divide P n,1 , for n ∈ {2, 3}.Hence, θ 0 , θ 1 {β 2,1 , β 3,1 }.This yields a contradiction to the assumption that β n,1 is a Perron number.

Figure 1 .
Figure 1.Plot of T + β,α for β = ( √ 5 + 1)/2 and α = 1 − 0.474β, and the corresponding lazy and greedy β-transformations.(The height of the filled in circle determines the value of the map at the point of discontinuity.) Plot of T − β,α .