Approximations of symbolic substitution systems in one dimension

Periodic approximations of quasicrystals are a powerful tool in analyzing spectra of Schr\"odinger operators arising from quasicrystals, given the known theory for periodic crystals. Namely, we seek periodic operators whose spectra approximate the spectrum of the limiting operator (of the quasicrystal). This naturally leads to study the convergence of the underlying dynamical systems. We treat dynamical systems which are based on one-dimensional substitutions. We first find natural candidates of dynamical subsystems to approximate the substitution dynamical system. Subsequently, we offer a characterization of their convergence and provide estimates for the rate of convergence. We apply the proposed theory to some guiding examples.


Introduction
In studying a Schrödinger operator coming from a quasicrystal, one often turns to study it by periodic crystals approximating the underlying dynamical structure.Examples of this approach can be seen in earlier works such as [OK85, MDO89, SB90, TFUT91, TCL93], and more recently in [SJ08, TGB + 14, EAMVD15, TDGG15, CRH19, BBDN20].This is done using finite volume approximants with either open, periodic or twisted boundary conditions, while trying to minimze the effects of the boundary conditions.In this paper, we deal with infinite approximants having periodic potential used to estimate a Schrödinger operator coming from an aperiodic configuration of atoms on the infinite lattice Z.These infinite periodic approximants are relatively well understood using Bloch-Floquet theory, which allows us to study them via finite volume operators with twisted boundary conditions.See for example [MDMPAR06] or [SV05].The Schrödinger operators we consider, which are simple cases of the tight binding model, are given by where ∆ is the discrete Laplacian and V ω is a diagonal operator whose diagonal elements come from the underlying configuration.These Recent results by Beckus, Bellissard, Cornean and Takase, [BBC19, Propostion 1.1] and [BT21, Theorem 2.1], ensure that the distance between spectra coming from two underlying configurations are at most proportional to the distance between the hulls of the configurations.For a precise version of these results see Proposition A.8 in the appendix.Given the aforementioned spectral approximation results, it is therefore natural to ask what are the restrictions on approximating aperiodic systems from periodic ones?And at which rate?We discuss in this paper such problems for one-dimensional aperiodic structures with substitution 1 symmetry, also called inflation symmetry in other sources, see [MDO89].We restrict our attention to materials coming from an aperiodic configuration of atoms on a lattice by a substitution.We encode the atoms in the configurations by a finite set of symbols, called an alphabet and denoted by A. The distance between the hulls we consider is inverse to the scale of local patterns on which they agree, and we denote this distance by d H .A precise definition of d H and its characterization are discussed in Section A.1.1 in the appendix.The local patterns in our hulls are finite words over A. Explicit details and general theory for this setting can be found in [DF22].
In this setting, and using the results in [BBC19] and [BT21], we consider an approximation scheme for substitution aperiodic structures, using only the substitution structure similar to [BBDN20].Namely, we estimate an aperiodic structure by taking a periodic configuration ω 0 and iteratively applying the substitution to ω 0 .This yields an iterative hull sequence (IHS), which we use to approximate the hull of a substitution aperiodic structure.The hull of a configuration with substitution symmetry, is called a substitution subshift and denoted Ω(S).In this paper we discuss the IHS approximation scheme and the rates at which the scheme approaches Ω(S).We restrict our attention to the class of primitive substitutions, due to their nice dynamical properties.Explicit definitions and outlines of proofs are given in the appendix.We emphasize that this paper is part of a larger collaborative work with Ram Band, Siegfried Beckus and Felix Pogorzelski.This collaboration deals with similar issues in a generalized setting, of Cayley graphs of lattices in homogeneous Lie groups, and will appear in [BBPT].The results presented here deal with the one-dimensional case, which do not fall under the framework of block substitutions used in [BHP21] and [BBPT].

The approximation scheme
Two examples for substitutions, which we use to illustrate our theory, are the Fibonacci substitution given by the rule 0 → 01 and 1 → 0, (2.1) and a counterexample substitution given by the rule 0 → 001, 1 → 200 and 2 → 102. (2.2) Given the characterization of symbolic hulls distance via their local patterns, see for example [LM95, Proposition 1.3.4],we seek to understand how these patterns change through our IHS approximations.The local patterns for a substitution subshift are the ones that occur naturally through iterations of the substitution.For example, for the Fibonacci substitution given in (2.1), one can obtain its local patterns using the following derivation chain iterating the substitution.
Local patterns occurring as a subword in this chain, are the local patterns appearing in Ω(S F ib ).They are also called legal words or patterns.One can verify that the legal 2-words in the Fiboncacci substitution case are W (S F ib ) 2 = {00, 01, 10} and the defective 2-word is 11.Relying on [Bec16, Corollary 6.2.5] and [BBDN20, Proposition 2.11], we know that the IHS Ω n (ω 0 ) coming from a starting configuration ω 0 , converge if the 2-words in ω 0 are legal.Therefore, taking the constant configuration ω 0 = 0 ∞ , yields a sequence of periodic hulls converging to Ω(S F ib ).
Building on this criterion, if the sequence of hulls eventually does not contain any defective 2-word, then the IHS converge.For example, apply the substitution given in (2.2).Using a similar derivation chain to the one in (2.3), we conclude in this case that W (S CE ) 2 = {00, 01, 02, 10, 11, 12, 20}, and the defective length 2-words are {21, 22}.Starting with the periodic configuration ω 0 = 2 ∞ and applying the substitution iteratively, one can see that the defective length 2-words appears infinitely many times in the IHS.Therefore, the hulls coming from ω 0 = 2 ∞ are bad approximants for Ω(S CE ).
To summarize, our approximation scheme is as follows.
The IHS approximation scheme • Needed: A primitive substitution S and an initial configuration ω 0 over the lattice Z.
• Iterative step: At the n-th step, consider the symbolic hull Ω n (ω 0 ) of S n (ω 0 ), obtained from applying S n-times to ω 0 .
• Result: Determine how does the distance between the IHS and Ω(S) changes.
Following this discussion, we generate a directed graph, denoted G(S), to keep track of defective 2-words occurring in the hulls along the iteration sequence.The vertices of such a G(S) graph are all the 2-words over A. We then draw edges between a defective word u to a defective word v occurring in the substitution of S(u) to pay attention to the defective 2-words in the hulls.Drawing these graphs for our examples, we obtain Figures 1 and 2, with the legal 2-words colored in gray for each substitution.Following [BBDN20, Proposition 2.11], and using the proposed graphs, we obtain the following algorithmic result to determine whether our approximation scheme succeeds.
Theorem 2.1.Let S be a primitive one-dimensional substitution and ω 0 be some initial configuration over Z.The following conditions are equivalent.
(ii) Any directed path in G(S) starting in a 2-word of ω 0 does not contain a closed subpath.
(iii) Any directed path in G(S) starting in a 2-word of ω 0 has length strictly less than |A| 2 .
The interested reader can find a proof of Theorem 2.1 in the appendix.We say that an initial configuration ω 0 and the IHS coming from it are good approximants if they satisfy the conditions in Theorem 2.1, and bad approximants otherwise.Turning back to our motivating examples, we can see that any starting configuration is good in the case of the Fibonacci substitution.More generally, any starting configuration in the case of a self-correcting substitution, see [GM13] for an explicit definition, is a good approximant and yields a good approximating IHS.The class of self-correcting substitutions constitute the majority of one-dimensional substitution examples in the literature, and include the Fibonacci, Thue-Morse, Period-doubling and the Golay-Rudin-Shapiro substitutions.See [BG13, Chapter 4] for more details.On the other hand, considering the counterexample from (2.2), we see that an initial configuration, and its derived sequence of hulls, is bad if and only if it contains a defective 2-word 21 or 22.More generally, any starting configuration containing a defective 2-word in the case of a marked substitution, see [Fri99] for a definition, will generate bad approximating IHS.

Approximation rates from the scheme
We now turn to estimating the rates at which the approximation scheme gives, when the IHS approximants are good.Note that the IHS Ω n (ω 0 ) are bad if and only if d H Ω n (ω 0 ), Ω(S) is bounded away from 0. To that end we need to consider how fast does the IHS agree with the substitution subshift on local patterns.Recall that a substitution S has a matrix M S , consult [Que10, Section 5.3], encoding the substitutions rule.The matrices for the substitutions given in (2.1) and (2.2) are accordingly.A substitution being primitive, implies that the associated substitution matrix M S has a leading simple eigenvalue, using Perron-Frobenius (PF) theory, consult [Mey00, Chapter 8] or [Que10, Section 5.3] for more details.This leading eigenvalue is called the PF-eigenvalue, which we denote θ S .The PF-eigenvalue for the matrices in (3.1) are 1+ √ 5 2 and 3 accordingly.The PF-theory implies that there exist constants Ĉ(S) > 0 and Č(S) > 0 such that where |u| is the length of the word u.It is well known, see [Sol98] or [Dur00], that a primitive substitution induces a linearly repetitive subshift.i.e., there exists a constant C S ≥ 1 for Ω(S), such that any legal word of length ℓ occurs inside all legal words of length greater than ℓC S .We can show, see Section A.2.2 in the appendix, that the IHS Ω n (ω 0 ) agree on ℓ-length words when n ≥ log ℓ log θ S + C1 , for some appropriate constant C1 > 0. This establishes an upper bound for the rate of convergence in (3.3).Conversely, when the initial configuration is periodic, a lower bound on the rate of the convergence follows from the number of ℓ-length words occurring in Ω(S).Using a result from Coven-Hedlund [CH73], we see that the number of ℓ-words in Ω(S) must be at least ℓ + 1.One can see that a periodic hull can have, at any fixed word length ℓ, at most as many different words as its period size.When ω 0 is a periodic configuration and Ω n (ω 0 ) agrees with Ω(S) on ℓ-length words, it follows that n ≥ log ℓ log θ S + C2 (ω 0 ; S), for an appropriate constant C2 (ω 0 ; S) > 0. Combining the bounds on n, we conclude that for a good periodic starting configuration, we get that the IHS generated from the approximation scheme satisfy where C 1 (S) > 0 and C 2 (ω 0 ; S) > 0 are appropriate constants.In particular, any periodic good approximating IHS have an asymptotically optimal distance from Ω(S) of θ −n S .Applying the estimates in (3.3) to the Fibonacci substitution gives us that any IHS converge to Ω(S F ib ) in an asymptotic rate of 1+ √ 5 2 −n .More generally, for any self-correcting substitutions with PF-eigenvalue θ S and any initial configuration ω 0 , we have that d H Ω n (ω 0 ), Ω(S) is at most proportional to θ −n S .Finally, turning back to the spectrum of a Schrödinger operator described in (1.1), the sequence of spectra arising from a good approximating IHS are at distance at most proportional to θ −n S .This shows that spectra of good approximating IHS approach the spectrum of a Schrödinger operator, with substitution symmetric potential, exponentially fast.In particular, for any self-correcting substitution and any initial configuration, the generated IHS yield an exponentially fast estimate for the spectrum of a Schrödinger operator.A precise phrasing of this statement is given in Proposition A.9 in the appendix.This can be used to prove the existence of spectral gaps similar to arguments in [HMT22].

A.1.1 Symbolic dynamical systems, dictionaries and substitutions
In this section, we collect some basic terminology and facts regarding symbolic dynamics and substitutions, which we glossed over in the main paper.We adopt terminology from [BBDN20, Section II] and [Que10, Sections 4 and 5].Other recommended references on this topic are [BG13, Chapter 4] and [Fog02, Chapter 1].Given an alphabet A, the set of possible finite words over A is given by A + := ∪ ∞ n=1 A n .We often implicitly refer to u ∈ A ℓ , for some ℓ ∈ N, as a function u : [0, ℓ) ∩ Z → A. For u 1 , u 2 ∈ A + , we write u 1 ≺ u 2 if u 1 is a subword of u 2 .A configuration over A, also called a two-sided infinite word, is any function ω : Z → A. The space of all configurations on Z is denoted by A Z .We have the natural action of Z on A Z such that, for all m ∈ Z and This action is continuous with respect to the metric between configurations given by The orbit of a configuration ω ∈ A Z , the closure of which is also called the hull of ω, is given by ⊆ Ω for all ω ∈ Ω.An invariant nonempty subset Ω ⊆ A Z is called a subshift, if it is also closed with respect to the metric in (A.2).The collections of subshifts is denoted by J .A subshift Ω ∈ J is called minimal if Ω = Orb Z (ω) for all ω ∈ Ω.A subshift Ω ∈ J is called periodic, if it is minimal and finite.We note that a periodic subshift Ω ∈ J satisfies Ω = Orb Z (ω) for all ω ∈ Ω.It follows that for any periodic Ω ∈ J , there exists a u ∈ A |Ω| and ω ′ ∈ Ω, such that ω ′ | [0,|u|) = u and ω ′ (n) = u(n − ⌊ n |u| ⌋) for all n ∈ Z.We write in this case ω = u ∞ .When Ω ∈ J is periodic, then there exists u Ω ∈ A + satisfying Ω = Orb Z (u ∞ Ω ).We now turn to discuss the notion of approximating subshifts.The space J is naturally endowed with the Hausdorff metric induced from d given in (A.2).Recall that the Hausdorff metric of two nonempty sets A, B is defined by (A. 3) The Hausdorff metric induced on J , which we denote by d H , is a complete metric.We denote by d H the standard metric on subsets of C or R. For further details and properties of the Hausdorff distance, consult [Bee93].
The alternative characterization of d H we used in the main paper relies on the notion of dictionaries.For a configuration ω ∈ A Z and a subshift Ω, their dictionaries are defined accordingly as The dictionary is comprised of the local patterns occurring in ω or Ω accordingly.This is also referred to as a language in certain literature.For every ℓ ∈ N, we denote W (Ω) ℓ := W (Ω) ∩ A ℓ and W (ω) ℓ := W (ω) ∩ A ℓ .Subshifts and dictionaries provide equivalent definitions of essentially the same objects, with the latter often considered as simpler to handle.However, to more easily obtain results from [BBC19] and [BT21], results here are formulated using Hausdorff distance.These two approaches are the equivalent by either[LM95, Proposition 1.3.4]or [BBDN20, Theorem 2.2].We require stricter quantitative estimates for the aforementioned correspondence, used implicitly when discussing the homeomorphism.To this end, we start by recalling the following Proposition, see [BG13, Lemma 4.2] or [BBDN20, Proposition 5.2], relating a dictionary of a configuration to a dictionary of its orbit.
Using Lemma A.1, we can conclude that The following useful proposition follows, shown for example in [BBC19, Lemma 2.2].

Remark. This proposition helps shed some light on the different behaviour of distance between subshifts and distance between configurations. Heuristically, the distance between two configurations corresponds to how much they agree around the origin. On the other hand, the distance between two subshifts corresponds to the lengths of words on which their dictionaries agree.
We now recall relevant notions for substitutions from [Que10, Section 5] and [BBDN20, Section II].A substitution rule, S 0 : A → A + , is extended to maps on A + and A Z by concatenation.The substitution matrix of S is the matrix M (S) ∈ R s×s , such that [M (S)] i,j is the number of times the letter a i occurs in S(a j ), where A = {a 1 , ..., a s }.The matrix M (S) is called the substitution matrix of S. Recall that S is called primitive, if there exist a p ∈ N, such that b ≺ S p (a) for all a, b ∈ A. The collection of legal words is called the dictionary of S and denoted W (S). Equivalently, For general primitive substitutions, the definition of W (S) yields a unique subshift Ω(S) ∈ J , called the substitution subshift satisfying W (Ω(S)) = Ω(S).Using the introduced terminology and notations, the IHS approximants are rigorously defined as Ω n (ω 0 ) := Orb Z S n (ω 0 ) .Another classic property of Ω(S), that we rely on for the proof of Theorem A.5, is linear repetitivity of Ω(S).
Recall that a configuration ω ∈ A Z is called linear repetitive if there exists some C ≥ 1 such that, for all u, w ∈ W (S), if |w| ≥ C|u| then u ≺ w.In this case, we say that the aforementioned C is a linear repetitivity constant of ω.We choose some linear repetitivity constant of some ω ∈ Ω(S) and denote it by C S ≥ 1.This C S is well defined by [Sol98, Lemma 2.3] or [Dur00, Proposition 6].

A.1.2 Directed graphs
To clarify the terms used in Theorem 2.1, we recall several relevant graph notions in this subsection.A finite set V , over which a graph is defined, is called a vertex set.A directed graph G over Theorem A.5.Let S be a primitive one-dimensional substitution over an alphabet A, with PFeigenvalue θ S .Given an initial configuration ω 0 ∈ A Z , consider the generated IHS Ω n (ω 0 ).Then there exists a constant C 1 = C 1 (S) > 0 such that if Ω n (ω 0 ) are good approximating IHS, then If furthermore, ω 0 is a periodic configuration, then there exists a constant We once again only outline the proof for Theorem A.5, due to similarities with arguments in [BBPT].Theorem A.5 is proved using the following lemmas.
Lemma A.6.Let S be a primitive one-dimensional substitution over an alphabet A, with linear repetitive constant C S and PF-eigenvalue θ S .Let r ∈ N, ω 0 ∈ A Z and consider the generated IHS Ω n (ω 0 ).

A.2.3 Specrtal convergence rates
For a Schrödinger operator H ω , as in (1.1), we denote the arising spectrum by σ(H ω ).A precise version of the results in [BBC19] and [BT21], is given in the following proposition.where ω S ∈ A Z is a configuration with substitution symmetry.
This result follows directly by combining [BBC19, Proposition 1.1] and Theorem A.5, so we leave its proof to the reader.

A.3 Further examples of substitutions
In this subsection we consider some more examples of substitutions and review their resulting G(S) graphs.One can see from the lack edges in the following figures, that all these substitutions are indeed self-correcting.
Its G(S P D ) graph is the same as that for the Fibonacci substitution and is given in Figure 1.

Figure 1 :
Figure 1: G(S F ib ) for the Fibonacci and the period doubling substitutions.Figure 2: G(S CE ) for the substitution coming from the rule in (2.2).

Figure 2 :
Figure 1: G(S F ib ) for the Fibonacci and the period doubling substitutions.Figure 2: G(S CE ) for the substitution coming from the rule in (2.2).
d H (A, B) := max sup a∈A d(a, B), sup b∈B d(b, A) , with d(a, B) := inf b∈B d(a, b).

Figure 3 :
Figure 3: G(S T M ) for the Thue-Morse substitution.