Periodic intermediate $\beta$-expansions of Pisot numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $\beta$-shifts, namely transformations of the form $T_{\beta, \alpha} \colon x \mapsto \beta x + \alpha \bmod{1}$ acting on $[-\alpha/(\beta - 1), (1-\alpha)/(\beta - 1)]$, where $(\beta, \alpha) \in \Delta$ is fixed and where $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1,2) \; \text{and} \; 0 \leq \alpha \leq 2-\beta \}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(\beta, \alpha)$ such that $T_{\beta, \alpha}$ has the subshift of finite type property is dense in the parameter space $\Delta$. Here, they proposed the following question. Given a fixed $\beta \in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $\alpha$ in the fiber $\{\beta\} \times (0, 2- \beta)$, so that $T_{\beta, \alpha}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\alpha = 0$ to the case when $\alpha \in (0, 2 - \beta)$. That is, we examine the structure of the set of eventually periodic points of $T_{\beta, \alpha}$ when $\beta$ is a Pisot number and when $\beta$ is the $n$-th root of a Pisot number.


Introduction.
Since the pioneering work of Rényi [36] and Parry [33], β-shifts and expansions have been extensively studied and have provided practical solutions to various problems. For instance, they arise as Poincaré maps of the geometric model of Lorenz differential equations [40], and Daubechies et al. [14] proposed a new approach to analog-to-digital conversion using β-expansion. A summary of some further applications can be found in [28]. Through their study, many new phenomena have appeared, revealing a rich combinatorial and topological structure, and unexpected connections to probability theory, ergodic theory, number theory and aperiodic order [25,31,39]. Additionally, through understanding β-shifts and expansions, advances have been made in the theory of Bernoulli convolutions [1,12].
For β > 1 and x ∈ [0, 1/(β − 1)], a word (ω n ) n∈N in the alphabet {0, 1} is called a β-expansion of x if When β is a natural number, all but a countable set of real numbers have a unique β-expansion. On the other hand, in [15], it was shown that, if β is less than the golden mean, then for all x ∈ (0, 1/(β − 1)), the cardinality of the set of β-expansions of x is equal to the cardinality of the continuum. Siderov [38] extended this result and showed that if β is strictly less than two, then for Lebesgue almost all x ∈ [0, 1/(β − 1)], the cardinality of the set of β-expansions of x equals the cardinality of the continuum.
Given (β, α) ∈ ∆, the unique points in Ω + β,α and Ω − β,α corresponding to p are called the kneading invariants of Ω β,α . It is known that the kneading invariants completely determine Ω β,α , see Theorem 2.3 due to [4,19,22], and the β-shift Ω β,α is a subshift of finite type if and only if the left shift of the kneading invariants are periodic, see Theorem 2.4 due to Ito and Takahashi [23], and Parry [35], for the case α ∈ {0, 2 − β}, and Li et al. [27], for the case that α ∈ (0, 2 − β). These results immediately give us that the set of parameters in ∆ which give rise to β-shifts of finite type is countable. In a second article [28] by Li et al., it was shown that this set of parameters is in fact dense in ∆. In contrast, if one considers the dynamical property of topologically transitivity, then the structure of the set of (β, α) in ∆ such that Ω β,α is topologically transitive, with respect to the left shift map, is very different to the set of (β, α) belonging to ∆ for which Ω β,α is a subshift of finite type. It is worth noting that the former of these two sets has positive Lebesgue measure and is far from being dense in ∆, see Theorem 5.1 due to Palmer [32] and Glendinning [16].
The results of [17] and [29] in tandem with those discussed above, yield the following.
(ii) If β is not the positive n th -root of a Perron number, for some n ∈ N, then the set of α for which the β-shift Ω β,α is a subshift of finite type is empty.
Indeed, for Ω β,α to be a subshift of finite type, we require β ∈ (1, 2) to be a maximal root of a polynomial with coefficients in {−1, 0, 1} and α ∈ Q(β). This leads to the following natural question, to which we give a partial answer to in Theorem 1.1.
Question A. If β ∈ (1, 2) is a positive n th -root of a Perron number, for some n ∈ N, is the set of α for which Ω β,α is a subshift of finite type dense in (0, 2 − β)?
Another class of subshifts which is of interest here are those which are factors of a subshift of finite type. Such subshifts are called sofic; indeed, every subshift of finite type is sofic, but not vise versa. Kalle and Steiner [24] proved that a β-shift Ω β,α is sofic if and only if its kneading invariants are eventually periodic. Combining this result with those of Li et al. [28], one obtains that the set of (β, α) ∈ ∆ for which Ω β,α is sofic is dense in ∆. This naturally leads to the study of (eventually) periodic points.
The main difficulty in proving Theorem 1.1 was in finding a way to compare the space Ω βm,α and Ω βm,α , for a fixed m and α = α . We achieved this by embedding all β m -transformations into a single (multi-valued) dynamical system and carrying out our analysis in this larger system. This result answers Question A for the class of multinacci number which belong to the wider class of algebraic numbers known as Pisôt numbers. Although many parts of our proof generalise from the class of multinacci numbers to the class of Pisot numbers, a central result (Proposition 2.7) which state that the upper kneading invariant is periodic if any only if the lower kneading invariant is periodic does not easily generalise, see Example 2.5 for an example of a point (β, α) ∈ ∆ where this is not the case. Here we would like to mention that Proposition 2.7 is closely related to the property known as matching, which has has been extensively studied [9,10].
As indicated above, Theorem 1.2 generalises the results of Schmidt [37]. Indeed, our proof is motivated by that of [37], with the following crucial difference. In the setting of [37], namely when α = 0, a key fact that is used is to any point x there exists a point y arbitrarily close to x and integers m and n, such that G n+k β (y) is arbitrarily close to zero for all k ∈ {0, 1, . . . , m}. However, this is not the case, when α > 0. To circumvent this, we appeal to the kneading theory of Milnor and Thurston discussed in Section 2.2. We also remark that a similar question to Question B was consider by Baker [3]; via different methods to ours, and also Schmidt's, Theorem 1.2 (i) maybe concluded from the work of Baker and Theorem 1.2 (ii) can be seen as a strengthening of Baker's results.
In addition to this, combining the results of Palmer [32] and Glendinning [16] as well as Parry [32,34] with Theorems 1.1 and 1.2, we may (i) determine a set of α which lie dense in a subset of positive Lebesgue measure of the fibre ∆(β 1/n m ), for all integers m and n ≥ 2, and (ii) classify the set Preper(β, α), in the case that β is the n-th root of a Pisot number and T β,α is non-transitive.
In order to state these results we require a few preliminaries.
Let n and k ∈ N with k < n and gcd(n, k) = 1 be given, and let s ∈ {0, 1, . . . , k − 1} be such that , and where, for 2 ≤ j ≤ s, see Figure 5.1 for a sketch of the intervals I n,k (β). If β = 2 1/n , then I n,k (β) is a single point and, if β ∈ (0, 2 1/n ), then I n,k (β) is an interval of positive Lebesgue measure. Further, for a fixed β ∈ (1, 2), in [16], it was shown that the Lebesgue measure of remains bounded away from zero as l ∈ N tends to infinity.
Corollary 1.4. Let m and n ≥ 2 denote two natural numbers, and let k ∈ N be such that k < n and gcd(n, k) = 1. There exists a dense set of α in I n,k ( n √ β m ) with Ω n √ βm,α a subshift of finite type. Moreover, if β is a Pisot number, then there exists a dense set of α in I n,k ( n √ β) with Ω n √ β,α sofic.

Preliminaries
We divide this section into three parts: Sections 2.1 and 2.2 in which we discuss aspects of symbolic dynamics and β-shifts; and Section 2.3 where we review results concerning a related class of interval maps, namely uniform Lorenz maps, which are in essence scaled versions of β-transformations.
2.1. Subshifts. We equip the set {0, 1} N of infinite words with the topology induced by the ultra metric where |ω ∧ ν| := min{i ∈ N : ω i = ν i }, for ω = (ω 1 , ω 2 , . . . ) and ν = (ν 1 , ν 2 , . . . ). This topology coincides with the product topology on {0, 1} N , where {0, 1} is endowed with the discrete topology. For n ∈ N and ω ∈ {0, 1} N , we set ω| n = (ω 1 , . . . , ω n ) and call n the length of ω| n denoted by |ω| n |. We define the (left) Given a subshift Ω and a natural number n, we set and denote by Ω * := n∈N Ω| n the collection of all finite words. A subshift Ω is said to be of finite type if there exists a finite set F of finite words such that (i) ν| n ∈ F for all ν ∈ Ω and n ∈ N; The set F is often referred to as the set of forbidden words of Ω. If Ω ⊆ {0, 1} N is a factor of a subshift of finite type, then it is called sofic.
An important property of τ ± β,α and π β,α is that the following diagram commutes.
Next, we recall a result which shows that Ω ± β,α is completely determined its kneading invariants.
A necessary and sufficient condition on the kneading invariants of an intermediate β-shift for determining when it is a subshift of finite type is as follows. With the above at hand, it is natural to ask if β ∈ (1, 2) and α ∈ (0, 2 − β), then is true that τ + β,α (p) is periodic if and only if τ − β,α (p) is periodic and vice versa? In Proposition 2.7 we show that this is indeed the case when β is a multinacci number. However, there exist values of β ∈ (1, 2) for which this does not hold, as the following counterexample demonstrates. Thus, it would be interesting to investigate if there exists other values of β ∈ (1, 2), for which τ + β,α (p) is periodic if and only if τ − β,α (p). In fact this idea is very closely linked to the concept of matching which has recently attracted much attention.
Let S 0 (x) := β m x + α and S 1 (x) := β m x + α − 1. It suffices to show β(T − β,α ) j+1 (p) + α = β(S 1 j • S 0 (p)) + α is strictly greater than 1. To this end, observe that Let us now describe the relation between uniform Lorenz maps and β-transformations. For this we require the following concept, which determines when two dynamical systems are 'the same'. Let X and Y denote two topological spaces and let f : X and g : Y . We say that f and g are topologically conjugate if there exists a homeomorphism h : X → Y such that h • f = g • h. The maps f and g are called topologically semi-conjugate if h is a continuous surjection.
As in the setting of Section 2.2 we have that the following diagram commutes.
The main benefit of using uniform Lorenz maps stems from the idea that every β-trnasformation has a realisation as a uniform Lorenz map, as discussed above, and that every uniform Lorenz map is defined on [0, 1] and has the same fixed points. Thus, it allows one to easily compare the kneading invariants of systems with the same expansion rate, namely β, but with different translates, namely α.

Fiber Denseness of intermediate β-shifts of finite type -Proof of Theorem 1.1 -
The aim of this section is to prove Theorem 1.1. We divide the proof into two parts. We show that the sets Per ± (β) := {α ∈ ∆(β) : τ ± β,α (p) is periodic}, for a given β ∈ (1, 2), are dense in ∆(β) with respect to the Euclidean norm, and with the help of Proposition 2.7, we have that τ + β k ,α is periodic if and only if τ − β k ,α is periodic. Theorem 1.1 follows by combining these two results together with Theorem 2.4.

Periodic expansions of Pisot and Salem numbers -Proof of Theorem 1.2 -
Throughout this section, let β ∈ (1, 2) denote an algebraic integer with minimal polynomial where z ∈ C, d ∈ N and a 1 , a 2 , . . . , a d ∈ Z. In which case, x ∈ Q(β) ∩ J β,α can be written in the form where p 1 , p 2 , . . . , p d−1 ∈ Z and q ∈ N. We assume that the integer q in (4.1) is as small as possible yielding a unique representation for x. Let p 1 , p 2 , . . . , p d−1 ∈ Z and and q ∈ N denote the corresponding terms for α ∈ Q(β): Fix α ∈ Q(β) ∩ (0, 2 − β) and x ∈ Q(β) ∩ J β,α with the forms given in (4.1) and (4.2). For i ∈ N, let ω ± i (x) respectively denote the i-th letter of τ ± β,α (x). From the commutative diagram given in (2.1), we have, for n a non-negative integer, that Lemma 4.1. For x ∈ Q(β) ∩ J β,α and n ∈ N 0 , there exists a unique vector (r Proof. By (4.1), (4.2) and (4.3) we have that The result for n = 1 follows from the fact that q and q are fixed and that B := {β, β 2 , . . . , β d } is a basis for Q(β). An inductive argument yields the general result.
With the above two lemmas at hand we are ready to prove Theorem 1.2 (i).
From this we obtain the following chain of inequalities.
This yields a contradiction, and concludes the proof.
For the proof of Theorem 1.2 (ii) we require an additional lemma. The following are equivalent.
A main ingredient in the proof of this result is to show that for given n, k ∈ N with 1 ≤ k < n and gcd(n, k) = 1, there exists a one-to-one correspondence between points in ∆ and points in D n,k . More precisely, on the one hand, given (β, α) ∈ ∆, there exists a unique a ∈ I n,k ( n √ β), namely a = α n,k (β, α), see otherwise.