Tilings and traces

Treviño, Rodrigo

doi:10.1063/5.0015534

This paper deals with (globally) random substitutions on a finite set of prototiles. Using renormalization tools applied to objects from operator algebras, we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic $R^{d}$ action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schrödinger operators obtained from aperiodic tilings in the construction.

I. INTRODUCTION

Consider the two substitution and expansion rules defined on the half hexagons in Fig. 1, one of which is the classical half hexagon substitution rule and the other one is obtained by modifying the square (second iteration) of it. This paper is concerned about the random application of substitution and expansion rules such as these in order to construct aperiodic tilings of $R^{d}$ ⁠, the study of the statistical properties of such tilings, and an application to the study of random Schrödinger operators on quasicrystals. Figure 2 gives an example of the types of tilings one can get through random application of the substitution rules in Fig. 1.

FIG. 1.

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Two substitution and expansion rules on half hexagons.

FIG. 2.

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A patch obtained from random applications of the half hex substitution and expansion rules in Fig. 1.

Although first introduced by Godrèche and Luck (1989), interest in random substitution tilings has surged recently [e.g., Frank and Sadun (2014), Gähler and Maloney (2013), Berthé and Delecroix (2014), Rust (2016), Rust and Spindeler (2018), and Schmieding and Treviño (2021)]. Random substitutions come in two flavors: locally random constructions [e.g., Rust and Spindeler (2018)] and globally random constructions [e.g., Gähler and Maloney (2013) and Schmieding and Treviño (2021)]. The typical features of globally random tilings are repetitivity, uniform patch frequencies (equivalent to unique ergodicity, see Sec. II), and zero entropy, whereas locally random tilings typically have positive entropy and non-uniform patch frequencies. This distinction is similar to that between strictly ergodic and mixing subshifts. All of the constructions in this paper are of globally random flavor, and so, although it will not be stated repeatedly that they are of global type, the reader should assume so throughout this paper.

The present work can be seen as an extension or alternative to Schmieding and Treviño (2021). It is an extension because the class of functions for which theorems are proved here (Lipschitz functions) is much larger than the class of functions treated in that work (smooth transversally locally constant functions). It is also an alternative because the present paper develops new tools combining ideas of renormalization with objects from operator algebras. More specifically, an object called the trace cocycle is introduced, developed, and used here to obtain results on deviations of ergodic integrals for Lipschitz functions on tiling spaces coming from random substitutions. This new tool makes it possible to connect some of the invariants from AF algebras (the traces) with invariants (also traces) from certain “smooth” sub-algebras of the so-called algebras of random Schrödinger operators on aperiodic tilings while giving errors of convergence rates for the Shubin–Bellissard formula.

What both of these approaches have in common is the use of spaces of Bratteli diagrams to organize tilings, which can be constructed from applications of substitution rules defined on the same set of prototiles, and the use of subshifts as a “moduli space” of all tilings, which can be obtained from a finite set of substitution rules, whereon the shift dynamics become renormalization dynamics. In Schmieding and Treviño (2021), the topology of the resulting tiling spaces was well-studied and exploited to obtain statistical results for the tilings.

In this paper, the renormalization approach is applied to certain invariants (the traces) from operator algebras to study the properties of the random substitution tilings, although they are close in spirit to the tools used by Bufetov in his study of deviation of ergodic integrals for several classes of systems (Bufetov, 2014; 2013; and Bufetov and Solomyak, 2013). What is gained from this point of view is that there is no need to have a full understanding of the topology of the tiling spaces constructed at random, making computations easier to make, as demonstrated in Sec. IX; what is lost is the access to topological information of the tiling spaces constructed in the construction.

Here, progress is also made with the issue of boundary effects. By “boundary effects,” I mean the following: in most studies of uniquely ergodic $R^{d}$ -actions on metric spaces, when d > 1, it has been usually hard to obtain information of the error terms of ergodic integrals of functions over sets of volume ∼T^d that are smaller than T^d−1, which is the contribution of the boundary of the averaging set to the integral (Sadun, 2011; Bufetov and Solomyak, 2013; Schmieding and Treviño, 2018a; and 2021). These issues have been overcome in other settings of higher rank Abelian actions [e.g., Cosentino and Flaminio (2015)], but they have remained an obstacle in the study of tilings. In this paper, I show that given some set B, there is an arbitrarily close set B_ɛ and a set of dilations of B_ɛ such that the deviation behavior along those averaging sets are fully described by the Lyapunov spectrum of our renormalization cocycle. As the title suggests and it was suggested above, functionals from operator algebras called traces play a prominent role here, being the analog to cycles in Zorich’s theory (Zorich, 1999), currents in Forni’s theory (Forni, 2002), and finitely additive measures in Bufetov’s theory (Bufetov, 2014). [Ian Putnam recently pointed out to me that Bowen and Franks (1977), Theorem 2.1, shows that the space of traces considered here and the space of finitely additive measures that Bufetov considered are isomorphic.] Our cocycle is defined on a bundle of traces analogous to the cohomology bundle used for the Kontsevich–Zorich cocycle.

Given that aperiodic tilings serve as models for quasicrystals, the results on deviations of ergodic averages here have several applications in mathematical physics. The advantage here of using an operator algebra approach is that it makes the connection to the study of random Schrödinger operators more natural. In Schmieding and Treviño (2018b), it was shown that asymptotic properties of traces of random Schrödinger operators defined by certain self-affine aperiodic tilings are controlled by traces obtained through the behavior of ergodic integrals on the tiling space. Here, a generalization is made and the connection is made more explicit: since traces on locally finite subalgebras of AF algebras control the behavior of the ergodic integrals for randomly constructed tilings, one can obtain traces on algebras of operators that control the asymptotic properties of the integrated density of states for the so-called random Schrödinger operators.

A. Statement of results

Let Σ_N be the full N-shift, that is, the space of bi-infinite sequences of symbols from an alphabet of N symbols. Given a set of prototiles {t₁, …, t_N} and N uniformly expanding and compatible substitution rules $F = {F_{1}, \dots, F_{N}}$ on them (see the precise definition of the substitution rule in Sec. II), there is a subshift of finite type $X_{F} \subset Σ_{N}$ ⁠, which parameterizes all the tiling spaces that can be obtained by random applications of the substitution rules in $F$ ⁠: given $x \in X_{F}$ ⁠, there is a corresponding compact metric space (called a tiling space) Ω_x whose elements are tilings with the hierarchical structure dictated by the point x according to the substitution rules in $F$ ⁠. Periodic points in $X_{F}$ give rise to tiling spaces Ω_x consisting of self-similar tilings.

The tiling spaces admit a $R^{d}$ action, which is denoted by φ_t : Ω_x → Ω_x, and for many of them, this action is minimal and uniquely ergodic (this will be the scenario considered in this paper; see Proposition 2). The concept of a minimal measure is used here (see Sec. III A for the precise definition), and this roughly means that μ on $X_{F} \subset Σ_{N}$ is minimal if for μ-almost every $x \in X_{F}$ ⁠, Ω_x admits a minimal $R^{d}$ action. The shift map $σ : X_{F} \to X_{F}$ defines a homeomorphism Φ_x : Ω_x → Ω_σ(x) of tiling spaces. As such, the shift drives the renormalization dynamics.

The way of constructing Ω_x from x is through a Bratteli diagram $B_{x}$ ⁠: a point $x \in X_{F}$ establishes how a sequence of substitutions from the family $F$ are put together to obtain a tiling, and this sequence is represented by an infinite directed graph $B_{x}$ whose structure is tied to that of Ω_x. As such, any point $x \in X_{F}$ defines a *-algebra $L F (B_{x}^{+})$ ⁠, called a locally finite algebra (this is defined in Sec. V), which is dense in an approximately finite dimensional (AF) C∗-algebra $A F (B_{x}^{+})$ ⁠. The dual of $L F (B_{x}^{+}) \subset A F (B_{x}^{+})$ is the trace space of $L F (B_{x}^{+})$ ⁠, which is a finite dimensional vector space over $C$ ⁠. Here, a trace τ on a *-algebra $A$ is taken to be any linear map $τ : A \to C$ satisfying τ(ab) = τ(ba). Note that since every element of $K_{0} (A F (B_{x}^{+}))$ can be represented by an element in $L F (B_{x}^{+})$ ⁠, the space of traces $Tr (B_{x}^{+})$ can be seen as the dual to $K_{0} (A F (B_{x}^{+}))$ ⁠. The dual to $Tr (B_{x}^{+})$ as a vector space is the space of cotraces ${Tr}^{*} (B_{x}^{+})$ ⁠, and it is this space that has great importance. We define the trace bundle to be the set of pairs (x, τ′) with $τ^{'} \in {Tr}^{*} (B_{x}^{+})$ ⁠. The shift $σ : X_{F} \to X_{F}$ induces a linear map $σ_{*} : {Tr}^{*} (B_{x}^{+}) \to {Tr}^{*} (B_{σ (x)}^{+})$ ⁠, yielding a linear cocycle over the shift σ, which we call the trace cocycle. The Lyapunov spectrum of this cocycle, that is, the growth rate of cotrace vectors under the trace cocycle, is what controls the statistical properties of the tilings.

Let $L (Ω_{x})$ denote the set of Lipschitz functions on Ω_x. For any Oseledets-regular x, that is, for any x for which the conclusion of the Oseledets theorem holds (see Sec. V A), there is a map $i_{x}^{+} : L (Ω_{x}) \to L F (B_{x}^{+})$ (see Sec. VI) and we denote by $[a_{f}] = i_{x}^{+} (f)$ the image of $f \in L (Ω_{x})$ through this map. Before stating the first theorem, some notation is needed. For a set $B \subset R^{d}$ ⁠, we denote by T · B the scaling (T · Id)B. A good Lipschitz domain is defined in Sec. II A, but for now, it suffices to say that it is a set whose boundary is not too complicated.

Theorem 1.

Let $F$ be a finite family of uniformly expanding and compatible substitution rules on a finite set of prototiles {t₁, …, t_M} with $X_{F} \subset Σ_{N}$ parameterizing the possible tiling spaces and μ being a minimal, σ-invariant ergodic Borel proability measure.

There exist Lyapunov exponents

λ_{1}^{+} > λ_{2}^{+} \geq \dots \geq λ_{d_{μ}^{+}}^{+} > 0

(depending on

F

and μ) such that for μ-almost every x, there are traces

τ_{1}^{+}, \dots, τ_{d_{μ}^{+}}^{+} \in Tr (B_{x}^{+})

such that if

f \in L (Ω_{x})

satisfies

τ_{i}^{+} ([a_{f}]) = 0

for all i < j for some

j \leq d_{μ}^{+}

but

τ_{j}^{+} ([a_{f}]) \neq 0

, B a good Lipschitz domain, and

T \in Ω_{x}

⁠, then for every ɛ > 0 there exists a set B_ɛ, which is ɛ-close to B in the Hausdorff metric, a sequence T_k → ∞, and a convergent sequence of vectors τ_k such that

\underset{k \to \infty}{lim sup} \frac{\log |\int_{T_{k} (B_{ε} + τ_{k})} f ◦ φ_{t} (T) d t|}{\log T_{k}} = d \frac{λ_{j}^{+}}{λ_{1}^{+}} .

(1)

If, in addition,

d λ_{j}^{+} \geq (d - 1) λ_{1}^{+}

⁠, then

\underset{T \to \infty}{lim sup} \frac{\log |\int_{T \cdot B} f ◦ φ_{t} (T) d t|}{\log T} \leq d \frac{λ_{j}^{+}}{λ_{1}^{+}} .

(2)

Remark 1.

Some remarks are as follows:

The case of self-similar tilings, tilings that are constructed from a single substitution rule, corresponds to tiling spaces Ω_x for periodic points x under the shift σ : Σ_N → Σ_N. In other words, tiling spaces for self-similar tilings correspond to the typical points of finitely supported invariant measures on Σ_N (assuming they are minimal measures). Studies of deviations of ergodic integrals for such types of systems have been done elsewhere (Sadun, 2011; Bufetov and Solomyak, 2013; and Schmieding and Treviño, 2018a). Therefore, what is new here are the results for tiling spaces that do not come from self-similar tilings, that is, tilings that come from tiling spaces Ω_x for x a typical point of a σ-invariant, ergodic measure satisfying the hypotheses of the theorem which is not finitely supported. There is a continuum of examples in Sec. IX.
The spectral gap $λ_{1}^{+} > λ_{2}^{+}$ is a consequence of the recent general spectral gap result of Horan (Horan, 2019, Corollary 2.19).

There is a particular type of tiling space, called a solenoid, which satisfies a type of bound known as the Denjoy–Koksma inequality (the trace space is trivial for solenoids, so Theorem 1 does not yield any information). The solenoid construction here is dependent on a family of substitution rules given by a sequence of positive integers $\bar{q} = (q_{1}, q_{2}, \dots)$ ⁠, each one greater than 1. There is also a concept of function of bounded variation on the solenoid $Ω_{\bar{q}}$ ⁠, and the space of all functions of bounded variation on $Ω_{\bar{q}}$ is denoted by $B V (Ω_{\bar{q}})$ (see Sec. VII). For $\bar{q} \in N^{N}$ ⁠, denote q_(n) = q₁q₂, …, q_n, and let μ be the unique invariant measure on $Ω_{\bar{q}}$ ⁠.

Theorem 2.

Let

Ω_{\bar{q}}

be a d-dimensional solenoid. Then, for any

f \in B V (Ω_{\bar{q}})

and

p \in Ω_{\bar{q}}

,

|\int_{{[0, q_{(n)}]}^{d}} f ◦ φ_{s}^{+} (p) d s - q_{(n)}^{d} \int_{Ω_{\bar{q}}} f d μ| \leq Var (f)

for all n > 0.

The Denjoy–Koksma inequality was first proved for irrational circle rotations by Herman (Herman, 1979, Theorem VI.3.1). Theorem 2 here is the first instance of this type of inequality for higher rank systems.

Let $T$ be a repetitive tiling with finitely many prototiles. Consider the Delone set $Λ_{T} \subset R^{d}$ obtained by puncturing every prototile in its interior and forming $Λ_{T}$ as the union of all the corresponding punctures on tiles of $T$ ⁠, which correspond to punctures of the prototiles. There is a class of operators on $ℓ^{2} (Λ_{T})$ ⁠, called the Lipschitz operators of finite range, denoted by ${L A}_{x}^{fi n}$ ⁠. These operators are defined in Sec. VIII, but what is relevant here is that they contain operators of interest in mathematical physics, namely, self-adjoint operators of the form H = Δ + V, where Δ is a Laplacian-type operator and V is any potential reflecting the aperiodic and repetitive nature of all tilings in Ω_x. (A simple example to consider in one dimension is as follows. Let $T$ be a tiling of $R$ by N different tile types and $Λ_{T}$ be the collection of endpoints of tiles of $T$ ⁠. There is an obvious, order-preserving labeling of Λ by $Z$ ⁠. For i ∈ {1, …, N} and λ ≠ 0, consider the operator H_i,λ = Δ + λV_i, where Δ is the discrete Laplacian on $Z$ and the localized potential V_i is defined by V_i(p) = p if p ∈ Λ is a left endpoint of a tile of type i and otherwise V_i(p) = 0. Similar constructions can be made for tilings in higher dimensions by considering the graph $G_{T}$ given by the Delaunay triangulation of $Λ_{T}$ ⁠, considering the graph Laplacian $△_{G_{T}}$ on $G_{T}$ ⁠, and considering an operator of the form $H = △_{G_{T}} + V$ ⁠, where V is a localized potential depending only on the local pattern around $p \in Λ_{T}$ ⁠.) These types of operators sometime go under the name of random Schrödinger operators.

For $T \in Ω_{x}$ and $A \in {L A}_{x}^{fi n}$ a self-adjoint operator, we have the operator $A_{T}$ acting on $ℓ^{2} (Λ_{T})$ and this assignment is equivariant with respect to the $R^{d}$ action on Ω_x. Denote by $A_{T} |_{B}$ the restriction of $A_{T}$ to the finite dimensional subspace $ℓ^{2} (Λ_{T} \cap B) \subset ℓ^{2} (Λ_{T})$ defined by $Λ_{T} \cap B$ ⁠. For $E \in R$ and T > 0, denote

n_{T}^{A} (E) ≔ # {eigenvalues of A_{T} |_{B_{T}}, which are \leq E} .

Assuming $T$ is repetitive and has finite local complexity and uniform patch frequency, that is, $T$ corresponds to a minimal and uniquely ergodic system, the function

E \mapsto \lim_{T \to \infty} \frac{n_{T}^{A} (E)}{V o l (B_{T})}

is the distribution of a measure ρ_A (independent of $T$ in the tiling space), called the integrated density of states (Lenz and Stollmann, 2005), satisfying

ρ_{A} (φ) = τ (φ (A)) ≔ \lim_{T \to \infty} \frac{t r (φ (A |_{B_{T}}))}{V o l (B_{T})},

(3)

where $t r (φ (A |_{B_{T}}))$ is the unique (non-normalized) trace of the finite dimensional operator $φ (A |_{B_{T}})$ ⁠, and for any continuous φ. This is the Shubin–Bellissard trace formula. It should be emphasized that the fact that the limit in (3) is a trace is not trivial; see Lenz and Stollmann (2003), Lemma 3.4. For a thorough introduction to the study of spectral properties of Schrödinger operators emerging from quasicrystals, see Damanik et al. (2015), Sec. 3.

The question addressed in Schmieding and Treviño (2018b) is as follows: What can be said about the convergence in (3)? In other words, is there a λ ∈ (0, d) such that

|t r (φ (A |_{B_{T}})) - V o l (B_{T}) τ (φ (A))| \leq C T^{λ}

for some C > 0 and all T > 1? The main result of Schmieding and Treviño (2018b) showed that if the tiling or Delone set had a self-affine structure, then yes, error rates for the Shubin–Bellissard trace formula can be computed, and that they can be computed with the help of other traces.

The second main result of this paper is a generalization of the main result of Schmieding and Treviño (2018b) and answering this question in the case of random substitution tilings. Not only are the error rates for the convergence in (3) computed but also the traces responsible for them are related to the traces defined on the LF algebras $L F (B_{x}^{+})$ and the error rates are defined by the Lyapunov spectrum of the trace cocycle from Theorem 1. More precisely, in Sec. VIII, for almost every $x \in X_{F}$ ⁠, we define a map $ϒ_{x} : {L A}_{x}^{fi n} \to L F (B_{x}^{+})$ and define functionals $τ_{i}^{'} ≔ ϒ_{x}^{*} τ_{i}^{+}$ by pulling back some of the traces in $Tr (B_{x}^{+})$ ⁠. Whether or not τ_i′ is a trace on $Tr ({L A}_{x}^{fi n})$ is dependent on the Lyapunov exponent $λ_{i}^{+}$ (see Proposition 5). The following is a consequence of Theorem 1.

Theorem 3.

Let $F$ be a finite family of uniformly expanding and compatible substitution rules on a finite set of prototiles {t₁, …, t_M} with $X_{F} \subset Σ_{N}$ parameterizing the possible tiling spaces and μ being a minimal, ergodic, σ-invariant Borel ergodic proability measure.

There exist Lyapunov exponents

λ_{1}^{+} > λ_{2}^{+} \geq \dots \geq λ_{d_{μ}^{+}}^{+} > 0

(depending on

F

and μ) such that for μ-almost every x, there are traces

τ_{1}, \dots, τ_{d_{μ}^{+}} \in Tr (B_{x}^{+})

such that if

A \in {L A}_{x}^{fi n}

satisfies τ_i(ϒ_x(A)) = 0 for all i < j for some

j \leq d_{μ}^{+}

but τ_j(ϒ_x(A)) ≠ 0, for B a good Lipschitz domain, for every ɛ > 0, there exists a set B_ɛ, which is ɛ-close to B in the Hausdorff metric, a sequence T_k → ∞, and a convergent sequence of vectors τ_k such that

\underset{k \to \infty}{lim sup} \frac{\log | t r (A_{T} |_{T_{k} \cdot (B_{ε} + τ_{k})}) |}{\log T_{k}} = d \frac{λ_{r}^{+}}{λ_{1}^{+}} .

If, in addition,

d λ_{j}^{+} > (d - 1) λ_{1}^{+}

, then

τ_{j}^{'} = ϒ_{x}^{*} τ_{j}

is a trace and

\underset{T \to \infty}{lim sup} \frac{\log | t r (A_{T} |_{T \cdot B}) |}{\log T} \leq d \frac{λ_{j}^{+}}{λ_{1}^{+}} .

Remark 2.

Some remarks are as follows:

These estimates give rates of convergence for the integrated density of states in (3) for random Schrödinger operators as explained in the paragraphs following (3). For example, if under the hypotheses of the theorem, the top two Lyapunov exponents satisfy $λ_{2}^{+} > (d - 1) λ_{1}^{+} / d$ ⁠, then for any ɛ > 0,
$|t r (A |_{B_{T}}) - V o l (B_{T}) τ (A)| \leq C_{ε} T^{d \frac{λ_{2}^{+}}{λ_{1}^{+}} + ε}$
for some C_ɛ > 0 and all T > 1.
Just like many of the traces on $L F (B_{x}^{+})$ ⁠, a dense subalgebra of the C∗-algebra $A F (B_{x}^{+})$ ⁠, do not extend to the full C∗-algebra, the auxiliary traces that describe the error rates in the convergence of the integrated density of states do not extend to traces on any C∗-algebra. Thus, what is important here is not the C∗-algebra of Schrödinger operators but a dense *-subalgebra consisting of “smooth” operators, which in this case is ${L A}_{x}^{fi n}$ ⁠.
This statement has no immediate relation to any statement about gap labeling (see Kellendonk (1995) for background).
It is unclear to me what physical interpretations the traces τ_i′ in Theorem 3 have.

This paper is organized as follows: in Secs. II and III, we review the essential definitions related to tilings and Bratteli diagrams and how one can construct tiling spaces using Bratteli diagrams. These sections cover background material, borrowing some results from Schmieding and Treviño (2021). Section IV is an interlude that illustrates the constructions using the example of half-hexagons in Fig. 1. Section V covers locally finite subalgebras of AF algebras and their traces. It is in this section that the trace cocycle is introduced and some basic properties are derived. Section VI is devoted to the study of ergodic integrals for Lipschitz functions on tiling spaces using the trace cocycle. Section VII proves the Denjoy–Koksma inequality for general solenoids. Finally, Sec. VIII covers the application of the main theorem on deviations of ergodic averages to traces on random Schrödinger operators. Section IX shows some experimental results for the easiest non-trivial results I could come up with using half hexagons. It strongly suggests that in this case, the Lyapunov spectrum is non-singular but does have multiplicities.

II. TILINGS

This section introduces the basic concepts in the theory of tilings. For a more thorough overview, see, e.g., Baake and Grimm (2013) and Sadun (2008).

A tile t is a compact, connected subset of $R^{d}$ ⁠. Here, it will always be assumed that the boundary ∂t of a tile has finite d − 1 dimensional measure. A tiling $T$ of $R^{d}$ is a cover of $R^{d}$ by tiles, where two different tiles may only intersect along their boundaries. Here, we will consider only cases where the tilings are formed by a finite set of prototiles {t₁, …, t_M}. That is, every tile $t \in T$ is a translated copy of t_i for some i. A patch of $T$ is a finite connected union of tiles of $T$ ⁠. A tiling $T$ is called repetitive if for any patch $P$ ⁠, there exists an $R_{P} > 0$ such that any ball of radius $R_{P}$ contains a translated copy of $P$ in it. For any set $A \subset R^{d}$ ⁠, denote

O_{T}^{-} (A) = largest patch P of T completely contained in A .

A tiling has finite local complexity if for each R > 0, there exists a finite collection of patches $P_{1}, \dots, P_{N (R)}$ such that for any $x \in R^{d}$ ⁠, the patch $O_{T}^{-} (B_{R} (x))$ is a translated copy of one of the patches $P_{i}$ ⁠.

A substitution rule $F$ on a finite set of prototiles {t₁, …, t_M} is a rule that allows us to express each prototile $t_{n_{i}}$ in a subset ${t_{n_{1}}, \dots, t_{n_{F}}} \subset {t_{1}, \dots, t_{M}}$ as the finite union of scaled copies of some of the prototiles (note that this differs from the traditional definition of a substitution rule in that traditionally it is all prototiles that are subdivided, whereas here one is allowed to only consider a subset of them and ignore the rest). More precisely, suppose we identify each prototile t_i with a subset of $R^{d}$ ⁠, and we assume without loss of generality that this subset contains the origin in its interior. Then, a substitution rule consists of a collection of scaling maps (also called graph iterated function systems) $F = {f_{i, j, k} : R^{d} \to R^{d}}$ with i, j = 1, …, M, k = 1, …, r(i, j) such that

t_{n_{i}} = ⋃_{j = 1}^{M} ⋃_{k = 1}^{r (n_{i}, j)} f_{n_{i}, j, k} (t_{j}),

(4)

and if for any i, any two maps f_i,j,k and f_i,j′,k′ have f_i,j,k(t_j) ∩ f_i,j′,k′(t_j) ≠ ⊘, then the intersection happens along the boundary of the images. In other words, each $t_{n_{i}}$ can be tiled by scaled copies of the prototiles t_i. The number n(i, j) is the number of copies of a rescaled copy of the prototile t_j placed in t_i when subdividing. As such, f_i,j,1 exists only if there is a rescaled copy of t_j found when substituting the prototile t_i. The reader who has not seen a substitution rule defined as in (4) is invited to Sec. IV, where the example in Fig. 1 is illustrated from the point of view of (4).

A substitution is uniformly expanding if all maps $f_{i, j, k} \in F$ are of the form f_i,j,k(x) = rx + τ_i,j,k for some r ∈ (0, 1) and $τ_{i, j, k} \in R^{d}$ ⁠. In this case, if r is the contracting factor, $r^{- 1} \cdot t_{n_{i}}$ is the union of prototiles, and the rescaling of (4) as

r^{- 1} \cdot t_{n_{i}} = ⋃_{j = 1}^{M} ⋃_{k = 1}^{r (n_{i}, j)} r^{- 1} \cdot f_{n_{i}, j, k} (t_{j})

(5)

is a substitution and expansion rule (Fig. 1 gives an example of two such rules, one with contraction 1/2 and the other with contraction 1/4). In defining a substitution rule that is uniformly expanding, one is implicitly defining a substitution and expansion rule by (5).

We will transform tilings by two types of operations: translations and deformations. Let $T$ be a tiling of $R^{d}$ and $τ \in R^{d}$ ⁠. Then, the tiling $φ_{τ} (T) ≔ T - τ$ is the tiling of $R^{d}$ obtained by translating each tile of $T$ by the vector $τ \in R^{d}$ ⁠. This is the translation of $T$ by τ.

All the tilings that will be considered in this paper will have finite local complexity, so it will be assumed from now on. If tiling $T$ has finite local complexity, then define a metric on the set of all translates of $T$ by

d (T, φ_{t} (T)) = \min \{1, \bar{d} (T, φ_{t} (T))\},

(6)

where

\bar{d} (T, φ_{t} (T)) = \inf \{ε > 0 : O_{T}^{-} (B_{\frac{1}{ε}}) ⋈ O_{φ_{t + s} (T)}^{-} (B_{\frac{1}{ε}}) for some ‖ s ‖ \leq ε\},

(7)

where $P_{1} ⋈ P_{2}$ denotes the equivalence of patches $P_{1}$ and $P_{2}$ by a translation. In words, two tilings are close if they agree on a large ball around the origin up to a small translation. That this is a metric for tilings of finite local complexity is standard; see Baake and Grimm (2013), Sec. 5.4. The tiling space of $T$ is defined as the metric completion of all translates of $T$ with respect to the metric above,

Ω_{T} = \bar{{φ_{t} (T) : t \in R^{d}}} .

There is a natural action of $R^{d}$ on $Ω_{T}$ by translation, $φ_{t} : T^{'} \mapsto φ_{t} (T^{'})$ ⁠. The action being minimal is equivalent to $T$ being repetitive. As such, if $T$ is repetitive, then for any two $T_{1}, T_{2} \in Ω_{T}$ ⁠, we have that $Ω_{T_{1}} = Ω_{T_{2}}$ ⁠.

Suppose $T$ is a tiling of $R^{d}$ by a finite collection of prototiles. That is, there is a finite set of tiles {t₁, …, t_M} such that every tile $t \in T$ is translation equivalent to t_i for some i. For each i, pick a distinguished point in the interior of the prototile t_i and then distinguish a point in the interior of each of the tiles in $T$ by the translation equivalence between the tiles and prototiles. The canonical transversal $℧_{T} \subset Ω_{T}$ is the set

℧_{T} ≔ {T^{'} \in Ω_{T} : the distinguished point in the tile t \in T^{'} containing the origin is the origin} .

If $T$ is repetitive, then $℧_{T}$ is a true transversal for the action of $R^{d}$ on $Ω_{T}$ since it intersects every orbit.

Let $P$ be a patch of $T$ and $t \in P$ be a choice of one of the tiles in that patch. The $(P, t)$ -cylinder set is defined as

C_{P, t} = {T^{'} \in Ω_{T} : P is a patch in T^{'} and the distinguished point in t \in P is the origin},

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and note that this is a subset of $℧_{T}$ ⁠. In fact, the topology of $℧_{T}$ is generated by cylinder sets of the form $C_{P, t}$ ⁠, and it has the structure of a Cantor set whenever $T$ has finite local complexity. Note that for two tiles $t, t^{'} \in P$ (not necessarily of the same type), there exists a vector $τ = τ (P, t, t^{'})$ such that $φ_{τ} (C_{P, t}) = C_{P, t^{'}}$ ⁠.

For a patch $P$ with a distinguished point in its interior, a tile $t \in P$ ⁠, and ɛ > 0, the $(P, t, ε)$ -cylinder set is the set

C_{P, t, ε} = ⋃_{‖ t ‖ < ε} {φ_{t} (T^{'}) : T^{'} \in C_{P, t}} \subset Ω_{T} .

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For a repetitive $T$ of finite local complexity, the topology of $Ω_{T}$ is then generated by cylinder sets of the form $C_{P, t, ε}$ ⁠, with $P$ being any patch in $P$ and ɛ > 0 being arbitrarily small. This gives $Ω_{T}$ a local product structure of $B_{ε} (0) \times C$ ⁠, where $B_{ε} (0) \subset R^{d}$ is the open ball of radius ɛ and $C$ is a Cantor set.

Let $T$ be a repetitive tiling of finite local complexity. Given a patch $P \subset T$ and set $B \subset R^{d}$ ⁠, let $L_{T} (P, B)$ be the number of copies of $P$ completely contained inside of B. Then,

{f r e q}_{T} (P) = \lim_{T \to \infty} \frac{L_{T} (P, B_{T})}{V o l (B_{T})},

when it exists, is the asymptotic patch frequency of $P$ in $T$ ⁠. For the purposes of this paper, without loss of generality, it can be assumed that this limit always exists since it will be well-defined for all tilings considered here. By (8), this gives a family of Borel measures on $℧_{T}$ parameterized by $Ω_{T}$ ⁠, which are invariant under the holonomies $τ (P, t, t^{'})$ ⁠. In other words, we have a function $ν : Ω_{T} \times B (℧_{T}) \to R$ ⁠, where $B (℧_{T})$ is the Borel σ-algebra of $℧_{T}$ ⁠, with $ν (T^{'}, P) = {f r e q}_{T^{'}} (P)$ for any patch $P$ ⁠. The action of $R^{d}$ on $Ω_{T}$ is uniquely ergodic if ν does not depend on the first coordinate, that is, ${f r e q}_{T} (P)$ is independent of $T$ ⁠. This will be the typical case in this paper; see Solomyak (1997), Sec. 3 for further details about frequencies.

Given that the measures $ν_{T} ≔ ν (T, \cdot)$ are holonomy-invariant, by the local product structure of $Ω_{T}$ ⁠, they define $R^{d}$ -invariant measures on $Ω_{T}$ which are locally of the form $μ_{T} = L e b \times ν$ ⁠, where ν is defined by the restriction the frequency measure $ν_{T}$ on the Cantor set defined by the patch $P$ ⁠. Whenever $φ_{s} : Ω_{T} \to Ω_{T}$ is uniquely ergodic, we will denote by μ the unique invariant measure.

A. Lipschitz domains

This subsection introduces Lipschitz domains, which are types of subsets of $R^{d}$ whose boundaries are well-behaved, making them useful sets over which to integrate functions. Let $H^{m}$ denote the m-dimensional Hausdorff measure.

Definition 1.

A set

E \subset R^{d}

is called m-rectifiable if there exist Lipschitz maps

f_{i} : R^{m} \to R^{d}

⁠, i = 1, 2, …, such that

H^{m} (E \ ⋃_{i \geq 0} f_{i} (R^{m})) = 0 .

Definition 2.

A Lipschitz domain

A \subset R^{d}

is an open, bounded subset of

R^{d}

for which there exist finitely many Lipschitz maps

f_{i} : R^{d - 1} \to R^{d}

⁠, i = 1, …, L, such that

H^{d - 1} (\partial A \ ⋃_{i = 1}^{L} f_{i} (R^{d - 1})) = 0 .

Lipschitz domains have d − 1-rectifiable boundaries.

Definition 3.

A subset $A \subset R^{d}$ is a good Lipschitz domain if it is a Lipschitz domain and $H^{d - 1} (\partial A) < \infty$ ⁠.

III. BRATTELI DIAGRAMS AND TILINGS

A Bratteli diagram $B = (V, E)$ is a bi-infinite directed graph partitioned such that

V = ⨆_{k \in Z} V_{k} and E = ⨆_{k \in Z \ {0}} E_{k}

with maps $r, s : E \to V$ satisfying $r (E_{k}) = V_{k} s (E_{k}) = V_{k - 1}$ if k > 0 and $r (E_{k}) = V_{k + 1} s (E_{k}) = V_{k}$ if k < 0, and with r⁻¹(v) ≠ ⊘ and s⁻¹(v) ≠ ⊘ for all $v \in V$ ⁠. We assume that $| V_{k} |$ and $| E_{k} |$ are finite for every k.

Remark 3.

The above definition is not the usual definition of a Bratteli diagrams, as usually their edge and vertex sets are indexed by $N$ ⁠. One of the reasons to index the edge set $E$ through $\bar{Z}$ instead of $Z$ is that it makes labeling choices when drawing them less awkward. The ones considered here are technically bi-infinite diagrams and the notational conventions of Lindsey and Treviño (2016) for bi-infinite Bratteli diagrams will be followed. There are two other advantages of using bi-infinite diagrams rather than the traditional diagrams indexed by $N$ ⁠; see the first paragraph of Sec. III B for details.

The positive part $B^{+}$ of $B$ is the Bratteli diagram $B^{+} = (V^{+}, E^{+})$ defined by the restriction to the non-negative indices of the data of $B$ ⁠. The negative part $B^{-}$ is similarly defined.

A path in $B$ is a finite collection of edges $\bar{e} = (e_{ℓ}, \dots, e_{m})$ such that $e_{i} \in E_{i}$ and r(e_i) = s(e_i+1) for all i ∈ {ℓ, …, m − 1}. As such, the domain of the range and source maps can be extended to all finite paths by setting $s (\bar{e}) = s (e_{ℓ})$ and $r (\bar{e}) = r (e_{m})$ ⁠. Let $E_{ℓ, m}$ be the set of all paths starting $V_{ℓ}$ to $V_{m}$ ⁠, that is, finite paths $\bar{e}$ with both $s (\bar{e}) \in V_{ℓ}$ and $r (\bar{e}) \in V_{m}$ ⁠. We can extend this to infinite paths: let $X_{B}^{+} = E_{0, \infty}$ be the set of infinite paths starting at $V_{0}$ and $X_{B}^{-} = E_{- \infty, 0}$ be the set of infinite paths ending at $V_{0}$ ⁠. The set $X_{B}^{+}$ can be topologized by cylinder sets of the form

C (\bar{e}) = {\bar{p} \in X_{B}^{+} : (p_{ℓ}, \dots, p_{m}) = (e_{ℓ}, \dots, e_{m})}

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for some finite path $\bar{e} \in E_{ℓ, m}$ with 0 ≤ ℓ < m. The set $X_{B}^{-}$ is similarly topologized, and as such, the spaces $X_{B}^{\pm}$ ⁠, when the number of vertices at every level is uniformly bounded (as they will be in the diagrams appearing in this paper), are compact metric spaces that are Cantor sets. The space of all bi-infinite paths on $B$ is then

X_{B} \subset X_{B}^{-} \times X_{B}^{+},

and it inherits the subspace topology.

Two paths $p, p^{'} \in X_{B}^{+}$ are tail-equivalent if there is an N > 0 such that $p_{i} = p_{i}^{'}$ for all i > N and this is an equivalence relation, where we denote classes by $[\bar{e}]$ ⁠. A minimal component of $X_{B}^{+}$ is a subset of the form $\bar{[\bar{e}]}$ ⁠. A Bratteli diagram $B$ is minimal if $\bar{[\bar{e}]} = X_{B}^{+}$ for all $\bar{e} \in X_{B}^{+}$ or, in other words, when there is only one minimal component. A measure μ on $X_{B}^{+}$ is invariant under the tail equivalence relation if for any N and paths $p_{1}, p_{2} \in E_{0, N}$ with r(p₁) = r(p₂), we have that μ(C(p₁)) = μ(C(p₂)).

A. Tilings from diagrams

Here, we recall the tiling construction from Schmieding and Treviño (2021). Let {t₁, …, t_M} be a set of prototiles and suppose that they admit N substitution rules $F_{1}, \dots, F_{N}$ ⁠. Given a collection $F = {F_{1}, \dots, F_{N}}$ of substitution rules, we want to parameterize all possible tilings we can obtain by different combinations of substitutions. As such, the space that organizes all of these combinations is a σ-invariant, closed subset $X_{F} \subset Σ_{N} = {1, \dots, N}^{\bar{Z}}$ of the N-shift, where $\bar{Z} ≔ Z - {0}$ ⁠, inheriting the order from $Z$ ⁠. In this section, a procedure is described for constructing from any $x \in X_{F}$ a Bratteli diagram $B_{x}^{+}$ and a construction assigning paths $\bar{e} \in X_{B}^{+}$ a tiling $T_{\bar{e}}$ ⁠. If all the substitution rules $F_{1}, \dots, F_{N}$ involve all prototiles, then $X_{F} = Σ_{N}$ ⁠. However, if one or more of the substitution rules do not involve all prototiles, then there may be restrictions as to how one can compose them, leading to a strict subset $X_{F} \subset Σ_{N}$ ⁠, which would be σ-invariant (a subshift). The reader should always keep in mind the case where all uniformly expanding substitution rules $F_{1}, \dots, F_{N}$ involve all prototiles (and so $X_{F} = Σ_{N}$ ⁠); the other cases are not usually common in the literature, but they can be handled with the machinery of this paper.

Pick $x = (x^{-}, x^{+}) = (\dots, x_{- 2}, x_{- 1}, x_{1}, x_{2}, \dots) \in X_{F}$ ⁠. We will start by defining the positive part $B_{x}^{+}$ of the Bratteli diagram $B_{x}$ ⁠. For k ≥ 0, $B_{x}^{+}$ will have $| V_{k} |$ be the number of tiles used in the substitution $F_{x_{k + 1}}$ ⁠, that is, not the number of tiles that are tiled by the rule $F_{x_{k}}$ but the number of different tiles used in that substitution rule. (If all substitution rules $F_{1}, \dots, F_{N}$ involve all prototiles, then $| V_{k} | = N$ for all k. Note that this differs from the traditional definition of a substitution rule in that traditionally it is all prototiles that are subdivided, whereas here one is allowed to only consider a subset of them and ignore the rest.) The vertices are ordered at each level so that $v_{i} \in V_{k}^{+}$ is identified with $t_{n_{i}}$ for every k. Now, starting with k = 1, consider the substitution rule $F_{x_{k}}$ ⁠. Then, for $v_{j} \in V_{k - 1}^{+}$ and $v_{i} \in V_{k}^{+}$ ⁠, there are r(i, j) edges from v_j to v_i, and we identify the corresponding map f_i,j,k with the appropriate edge $e \in E_{k}^{+}$ and denote it by f_e. Since the maps f_e are contacting, they are of the form f_e(x) = θ_ex + τ_e for some θ_e ≤ 1. This notation extends to finite paths $\bar{e} \in E_{0, k}$ by $f_{\bar{e}} = f_{e_{k}} ◦ \dots ◦ f_{e_{1}}$ ⁠.

Let $\bar{e} \in X_{B}^{+}$ ⁠, and denote by $\bar{e} |_{k}$ the truncation of $\bar{e}$ after its kth edge, that is, $\bar{e} |_{k} \in E_{0, k}^{+}$ ⁠. The kth approximant $P_{k} (\bar{e})$ is the set

P_{k} (\bar{e}) = ⋃_{\begin{array}{c} {\bar{e}}^{'} \in E_{0, k}^{+} \\ r ({\bar{e}}^{'}) = r (\bar{e} |_{k}) \end{array}} f_{\bar{e} |_{k}}^{- 1} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})})

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viewed as a tiled patch, where the tiles are the sets $f_{\bar{e} |_{k}}^{- 1} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})})$ for a path ${\bar{e}}^{'} \in E_{0, k}^{+}$ with $r ({\bar{e}}^{'}) = r (\bar{e} |_{k})$ ⁠. The hypotheses on the maps f_e guarantee that the approximants are nested, i.e., we have the inclusion of patches

{0} \subset t_{s (\bar{e})} \subset P_{1} (\bar{e}) \subset \dots \subset P_{k} (\bar{e}) \subset P_{k + 1} (\bar{e}) \subset \dots .

Patches of the form $P_{k} (\bar{e})$ are called levelk-supertiles.

Definition 4.

For

\bar{e} \in X_{B_{x}}^{+}

⁠, the tiling

T_{\bar{e}}

is the largest tiled subset of

R^{d}

such that

P_{k} (\bar{e})

is a patch of

T_{\bar{e}}

for all k and each tile of

T_{\bar{e}}

is contained in all but finitely many of the approximants

P_{k} (\bar{e})

⁠. In other words,

T_{\bar{e}} = ⋃_{k > 0} P_{k} (\bar{e}) .

Some care needs to be given in order to produce tilings that (1) cover all of $R^{d}$ and (2) have finite local complexity, as nothing guarantees that the tiling in $T_{\bar{e}}$ to have either property. The first property needed is the following:

Definition 5.

A collection $F = {F_{1}, \dots, F_{N}}$ of substitution rules is uniformly expanding if there exist numbers θ₁, …, θ_N ∈ (0, 1) such that each substitution rule $f_{i, j, k} \in F_{ℓ}$ is of the form f_i,j,k(x) = θ_ℓx + τ_i,j,k for some $τ_{i, j, k} \in R^{d}$ ⁠.

Definition 6.

A collection $F = {F_{1}, \dots, F_{N}}$ of substitution rules is compatible if for any x ∈ Σ_N and ${\bar{e}}^{+} \in X_{B_{x}}^{+}$ such that $T_{\bar{e}}$ defined in Definition 4 covers all of $R^{d}$ ⁠, then $T_{\bar{e}}$ has finite local complexity.

Compatibility is automatic for d = 1. The results of Gähler et al. (2015) show that this is not asking for too much in higher dimensions. The following are standard [see Schmieding and Treviño (2021), Lemma 5]:

Lemma 1.

Let $F = {F_{1}, \dots, F_{N}}$ be a collection of compatible and uniformly expanding substitution rules defined on the same set of prototiles. For $x \in X_{F}$ ⁠, consider the Bratteli diagram $B_{x}$ where the edge set $E_{k}$ is defined by $F_{x_{k}}$ . Then, we have the following:

If $\bar{e} \sim {\bar{e}}^{*} \in X_{B_{x}}^{+}$ ⁠, then there exists $τ \in R^{d}$ such that $T_{{\bar{e}}^{*}} = T_{\bar{e}} + τ$ .
$Ω_{T_{\bar{e}}}$ only depends on the minimal component: $Ω_{T_{\bar{e}}} = Ω_{T_{{\bar{e}}^{'}}}$ for all ${\bar{e}}^{'} \in \bar{[\bar{e}]}$ .

Let ${\overset{̊}{X}}_{B}^{+} \subset X_{B}^{+}$ be the set of paths $\bar{e}$ such that $T_{\bar{e}}$ covers all of $R^{d}$ ⁠. Note that by the previous lemma, if $B_{x}^{+}$ is minimal, then $Ω_{T_{{\bar{e}}_{1}}} = Ω_{T_{{\bar{e}}_{2}}}$ for any ${\bar{e}}_{1}, {\bar{e}}_{2} \in X_{B}^{+}$ ⁠. In such cases, we denote the tiling space simply by $Ω_{B}$ ⁠, or if $B$ is defined by a parameter $x \in X_{F}$ ⁠, we write Ω_x. The following is a consequence of the previous lemma:

Corollary 1.

Let $F$ a family of N uniformly expanding and compatible substitutions, $x \in X_{F}$ , and $B_{x}$ be a Bratteli diagram such that the set $E_{k}^{+}$ in $B_{x}^{+}$ is defined by $F_{x_{k}}$ . Suppose that $B_{x}^{+}$ is minimal. Then, the assignment $\bar{e} \mapsto T_{\bar{e}}$ defines a surjective, continuous map ${\bar{Δ}}_{x} : {\overset{̊}{X}}_{B_{x}}^{+} \to ℧_{x}$ , where ℧_x is the canonical transversal of $Ω_{T_{\bar{e}}}$ , $\bar{e} \in {\overset{̊}{X}}_{B}^{+}$ .

Definition 7.

A probability measure μ on Σ_N is minimal if the set of x for which $B_{x}$ is minimal has full measure.

Proposition 2 from Schmieding and Treviño (2021) is as follows:

Proposition 1.

Let $F$ be a family of N uniformly expanding and compatible substitutions, $x \in X_{F}$ , and $B_{x}^{+}$ be a minimal Bratteli diagram such that the set $E_{k}^{+}$ in $B_{x}^{+}$ is defined by $F_{x_{k}}$ . Suppose that $μ ({\overset{̊}{X}}_{B}^{+}) = 1$ for any probability measure μ on $X_{B}^{+}$ ⁠, which is invariant under the tail equivalence relation. Then, the map ${\bar{Δ}}_{x}$ in Corollary 1 provides a bijection between measures μ on $X_{B}^{+}$ ⁠, which are invariant under the tail equivalence relation, and measures on ℧_x, which are holonomy-invariant.

Proposition 2.

Let $F$ be a family of N uniformly expanding and compatible substitutions and μ be a minimal, ergodic σ-invariant Borel probability measure on $X_{F}$ . Then, for μ-almost every $x \in X_{F}$ ⁠, we have that there is a unique probability measure μ_x on $X_{B_{x}}^{+}$ ⁠, which is invariant under the tail equivalent relation. Moreover, we have that $μ_{x} ({\overset{̊}{X}}_{B_{x}}^{+}) = 1$ and there is a unique $R^{d}$ -invariant probability measure on Ω_x.

Proof.

For

x \in X_{F}

⁠, define

λ : X_{F} \to R

by

λ_{x} = \underset{k \to \infty}{lim sup} \frac{\log | E_{0, k}^{x} |}{k}

for all

x \in X_{F}

⁠, where

| E_{0, k}^{x} |

is the number of paths from

V_{0}

to

V_{k}

on

B_{x}

⁠. Note that this is a σ-invariant function, so it is constant μ-almost everywhere. Denote by λ_μ this value and

A_{μ} \subset X_{F}

the full μ-measure set such that λ_x = λ_μ for all x ∈ A_μ.

Let x ∈ A_μ ∩ supp μ be a Poincaré recurrent point, and let

B_{x}

be the corresponding Bratteli diagram. By minimality, there exists a k∗ > 0 such that for any

v \in V_{0}^{+}

and

w \in V_{k^{*}}^{+}

⁠, there is a path

\bar{p} \in E_{0, k^{*}}^{+}

with

s (\bar{p}) = v

and

r (\bar{p}) = w

⁠. Let

U_{x} \subset X_{F}

be the cylinder set defined by

U_{x} = {y \in {\bar{Σ}}_{N} : y_{i} = x_{i} for all i = 1, \dots, k^{*}}

⁠, and note that μ(U_x) > 0. Let k_i → ∞ be the sequence of first return times to U_x for x. That is,

σ^{k_{i}} (x) \in U_{x}

for all i > 0 and σ^k(x) ∉ U_x if k ≠ k_i for some i. Note that by the definitions of U_x, k_i, and k∗, there is a positive matrix M_x such that the number of paths between

v \in V_{k_{i}}^{+}

and

w \in V_{k_{i} + k^{*}}^{+}

is given by M_x(v, w). Let λ_PF denote the Perron–Frobenius eigenvalue of M_x. Then, for any ɛ, there exists a C_ɛ such that for

v \in V_{k^{'}}^{+}

with k_i ≤ k′ < k_i+1, we have that

| E_{0, k_{i}}^{+} | \geq C_{ε} {(\frac{λ_{x}}{e^{ε}})}^{i} .

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Since μ(U_x) > 0, it follows from the estimate above that λ_μ ≥ log λ_PF − ɛ for any ɛ > 0, so λ_μ > 0. Hence, for any ɛ > 0, there is a C_ɛ so that

| E_{0, k}^{+} | \geq C_{ε} e^{(λ_{μ} - ε) k}

for all k > 0. It follows from this, minimality, and recurrence of x that for the two quantities

λ^{-} ≔ \min_{v \in {v_{1}, \dots, v_{M}}} \{\underset{k \to \infty}{lim inf} \frac{\log | E_{v}^{+} |}{k}\} and λ^{+} ≔ \max_{v \in {v_{1}, \dots, v_{M}}} \{\underset{k \to \infty}{lim sup} \frac{\log | E_{v}^{+} |}{k}\},

we have that λ⁻ = λ⁺ ≥ λ_PF > 0. That

μ ({\overset{̊}{X}}_{B_{x}}^{+}) = 1

now follows by Schmieding and Treviño (2021), Lemma 3, for any Borel probability measure μ, which is invariant under the tail-equivalence relation. That there is a unique such measure follows from the main result of Treviño (2018), so the uniqueness of an invariant measure on Ω_x follows from Proposition 1.□

B. Renormalization

There are two advantages of using bi-infinite Bratteli diagrams as opposed to the usual diagrams indexed by $N$ ⁠. The first is that the path space of a bi-infinite diagram $B_{x}$ parameterizes all tilings in a tiling space Ω_x in a continuous way (see Proposition 3 in Sec. III B) and not just the ones associated with canonical transversals, as it happens with traditional (one-sided) Bratteli diagrams. This permits one to transfer properties back and forth between the path space of the bi-infinite diagram and the corresponding tiling space (see Proposition 1). The second and more important advantage is that one can shift the labels of a diagram $B_{x}$ to obtain a diagram $B_{σ (x)}$ ⁠, and this process is equivariant with a homeomorphism of tiling spaces Φ_x : Ω_x → Ω_σ(x). Having x belong to a two-sided shift allows this operation to be invertible, which will allow for the semi-invertible Oseledets theorem to be applied in Sec. V A.

FIG. 3.

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The associated graph iterated functions systems associated with the substitution rule in Fig. 1.

FIG. 4.

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The Bratteli diagram $B_{x}$ for any x ∈ C([00.10]) looks like this around $V_{0}$ ⁠.

FIG. 5.

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Mapping a cylinder set of the positive part to a cylinder set on the canonical transversal.

FIG. 6.

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Mapping a cylinder set of the negative part to a “cylinder set” of a prototile.

FIG. 7.

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Mapping a cylinder set to a cylinder set.

FIG. 8.

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The mechanism of renormalization: the process of applying the inverse σ⁻¹ of the shift corresponds to applying the substitution and expansion rule defined by the edge set $E_{- 1}$ on $B_{x}$ ⁠. This shifts levels on $B_{x}$ to obtain $B_{σ^{- 1} (x)}$ and maps level-k supertiles to level-(k + 1), as shown with the second approximant supertile from Fig. 5. At the level of cylinder sets defined by the finite path in bold from Fig. 7, we have that $Φ_{σ^{- 1}}^{- 1} (Δ_{x} (C (\bar{e}))) = Δ_{σ^{- 1} (x)} (C (σ^{- 1} (\bar{e})))$ ⁠.

Consider a minimal measure μ and note that being minimal is a σ-invariant property: $B_{x}$ is minimal if and only if $B_{σ (x)}$ is. As such, for an σ-invariant ergodic Borel probability measure μ, the set of minimal diagrams $B_{x}$ has either full or null measure.

Let

{\overset{̊}{X}}_{B} = {(x^{-}, x^{+}) \in X_{B} : x^{+} \in {\overset{̊}{X}}_{B}^{+}} .

Proposition 3.

Let $F$ be a family of N uniformly expanding and compatible substitutions on a set of prototiles, and suppose that $B_{x}$ is minimal. Then, the map ${\bar{Δ}}_{x}$ from Corollary 1 extends to a continuous surjective map $Δ_{x} : {\overset{̊}{X}}_{B} \to Ω_{x}$ ⁠.

Proof.

Let $\bar{e} = (e^{-}, e^{+}) \in {\overset{̊}{X}}_{B_{x}}$ ⁠. The discussion leading to Corollary 1 shows how $e^{+} \in {\overset{̊}{X}}_{B_{x}}^{+}$ determines a point in the canonical transversal $℧_{x} \subset Ω_{x} = Ω_{T_{e^{+}}}$ ⁠. It is left to show what role e⁻ plays.

What e⁻ determines is a vector $τ_{e^{-}}$ so that $Δ (\bar{e}) = φ_{τ_{e^{-}}} (T_{e^{+}})$ ⁠, and this is done as follows (there is a concrete example worked out in Sec. IV, in case the reader would find that helpful as they read the construction). Consider the tile t containing the origin in $T_{e^{+}}$ ⁠. The assumptions about the substitution rules imply that the origin is in the interior of this tile, and it can be subdivided according to the substitution rule $F_{x_{- 1}}$ into |r⁻¹(v_t)| ≥ 1 tiles, where $v_{t} \in V_{0}$ is the vertex identified with the tile t containing the origin. The edge e₋₁ corresponds to a choice of one of the smaller tiles that make up t. Now, $F_{x_{- 2}}$ gives a rule for subdividing this tile into |r⁻¹(s(e₋₁))| ≥ 1 smaller tiles, and the edge e₋₂ corresponds to choosing one of the smaller tiles in this subdivision. Carrying on recursively, after ending up with a small connected subset at level −k, the substitution rule $F_{x_{- k - 1}}$ yields a collection of smaller pieces that make up this connected subset and the edge e_−k−1 of $\bar{e}$ determines a choice of one of the smaller pieces. Since $S^{-} (\bar{e}) \leq c$ ⁠, on average, the pieces are contracting at a rate of e^−ck. Thus, performing this procedure infinitely many times yields a unique point $p_{e^{-}} \in t$ ⁠. The vector $τ_{e^{-}}$ is now defined to be the unique vector that takes $p_{e^{-}} \in t \in T_{e^{+}}$ to the origin. That is, the point $φ_{τ_{e^{-}}} (p_{e}) = 0$ ⁠. This assignment can readily be seen to be continuous.□

Let $B_{x}$ be a Bratteli diagram determined by a family of substitution rules $F_{1}, \dots, F_{N}$ and a point $x \in X_{F}$ ⁠. There is a natural homeomorphism $h_{x} : X_{B_{x}} \to X_{B_{σ (x)}}$ defined by the shifting of indices in $X_{B_{x}}$ by 1. This yields a homeomorphism of tiling spaces, which is proved by Schmieding and Treviño (2021) (Proposition 6).

Proposition 4.

Let $F = {F_{1}, \dots, F_{N}}$ be a family of uniformly expanding and compatible substitution rules, and suppose that $B_{x}$ is minimal. The shift $σ : X_{F} \to X_{F}$ induces a homeomorphism of tiling spaces Φ_x : Ω_x → Ω_σ(x) satisfying Φ_x◦Δ_x = Δ_σ(x)◦h_x. In addition, level-k supertiles on $T_{\bar{e}} \in Ω_{x}$ are mapped to level-k − 1 supertiles on $Φ_{x} (T_{\bar{e}}) = T_{σ (\bar{e})} \in Ω_{σ (x)}$ .

IV. INTERLUDE: AN EXAMPLE

Before proceeding to the second, more technical part of this paper, I will take the time to relate the example in Fig. 1 to the constructions of tilings from Bratteli diagrams in Sec. III and to the renormalization procedure in Sec. III B.

Figure 1 illustrates part of two different substitution rules on six different prototiles, which are rotated copies of the half-hexagon prototile illustrated in Fig. 1 by 2πk/6, k = 1, …, 5. The substitution rules for the rest of the prototiles in these cases are then defined by looking at the substitution for the prototile in Fig. 1 and rotating them by 2πk/6, k = 1, …, 6. The graphs for the corresponding graph iterated function systems that define the substitution rules as in (4) are illustrated in Fig. 3.

To connect this more concretely with the substitution rule expressed in (4), we take each prototile as a subset of $R^{2}$ with its barycenter at the origin (note that this choice will define the canonical transversal). Each edge in the graphs of Fig. 3 corresponds to a contracting linear map from (4), which places a scaled copy of a prototile inside another prototile, and so we can express the subset of $R^{2}$ corresponding to a prototile as the union of images of contracting linear maps, i.e., as in (4).

Given that these two substitution rules are primitive, for any point x ∈ Σ₂, we obtain a minimal Bratteli diagram $B_{x}$ where the edge information $E_{k}$ is given by the graph on the left in Fig. 3 if x_k = 0 and otherwise by the graph on the right. Consider now a point x ∈ Σ₂ where x = (…, x₋₂, x₋₁.x₁, x₂, …) = (…, 0, 0.1, 0, …) ∈ C([00.10]), where the dot (.) denotes the break between the negative and positive parts and C([w]) ⊂ Σ₂ denotes the obvious cylinder set in Σ₂ defined by the word w defined for a specific set of indices. Figure 4 illustrates the common part of a Bratteli diagram $B_{x}$ for any x ∈ C([00.10]).

Consider now the positive part $B_{x}^{+}$ of the Bratteli diagram $B_{x}$ for x ∈ C([00.10]) and its associated path space $X_{B_{x}}^{+}$ ⁠, and consider a path ${\bar{e}}^{+} = (e_{1}, e_{2}, \dots) \in X_{B_{x}}^{+}$ ⁠, the first two edges of which are outlined in bold blue in the left part of Fig. 5. This path defines both a cylinder set $C ((e_{1}, e_{2})) \subset X_{B_{x}}^{+}$ as in (10), as well as a second approximant $P_{2} (\bar{e})$ as in (11), which is denoted on the right part of Fig. 5. As such, it also denotes cylinder sets $C_{P_{2} ({\bar{e}}^{+}), b}$ ⁠, where b is the distinguished point corresponding to the barycenter of the tile. In fact, using the continuous map ${\bar{Δ}}_{x}$ from Corollary 1, it follows that ${\bar{Δ}}_{x} (C ((e_{1}, e_{2}))) = C_{P_{2} ({\bar{e}}^{+}), b}$ ⁠.

Now consider the negative part $B_{x}^{-}$ ⁠. A finite path ${\bar{e}}^{-} = (e_{- ℓ}, \dots, e_{- 1})$ on $B_{x}^{-}$ defines a cylinder set $C ({\bar{e}}^{-}) \subset X_{B_{x}}^{-}$ ⁠, as well as a measurable subset of the prototile associated with the vertex $r ({\bar{e}}^{-}) \in V_{0}$ ⁠. This subset is precisely $f_{e_{- 1}} ◦ \dots ◦ f_{e_{- ℓ}} (t_{s ({\bar{e}}^{-})}) \subset t_{r ({\bar{e}}^{-})}$ ⁠, where f_−i is the contracting map associated with the edge $e_{- i} \in E_{- i}$ and t_r(e) is the prototile corresponding to the vertex $r (e) \in V_{0}$ ⁠. As such, the blue path ${\bar{e}}^{-} = (e_{- 1}, e_{- 2})$ denoted in bold blue in Fig. 6 defines a cylinder set $C ({\bar{e}}^{-}) \subset X_{B_{x}}^{-}$ and, on the right, the associated subset denoted in blue on the tile.

Putting Figs. 5 and 6 together, one obtains a path ${\bar{e}}^{'} = (e_{- 2}, e_{- 1}, e_{1}, e_{2})$ that defines a cylinder set $C ({\bar{e}}^{'}) \subset X_{B_{x}} \subset X_{B_{x}}^{-} \times X_{B_{x}}^{+}$ ⁠. The image of this cylinder set under the map $Δ_{x} : {\overset{̊}{X}}_{B_{x}} \to Ω_{x}$ from Proposition 3 is a cylinder set in Ω_x, although not of the canonical form as in (9). In any case, the cylinder set $Δ_{x} (C ({\bar{e}}^{'}))$ is described as all the tilings in Ω_x having a patch around the origin, which is a translation copy of $P_{2} ({\bar{e}}^{+})$ in Fig. 5, and where the origin is somewhere in the blue region of Fig. 6. This is illustrated in Fig. 7.

It remains to illustrate how renormalization works in this example. As described in Sec. III B, renormalization is driven by the shift σ : Σ₂ → Σ₂. Figure 8 illustrates what a step of renormalization does to the cylinder set in Fig. 7. The illustration uses the inverse of the shift, as it shows the relationship between renormalization and the substitutions encoded in $B_{x}$ ⁠.

V. LF ALGEBRAS AND TRACES

A multimatrix algebra is a *-algebra of the form

M = M_{ℓ_{1}} \oplus \dots \oplus M_{ℓ_{n}},

where M_ℓ denotes the algebra of ℓ × ℓ matrices over $C$ ⁠. Let $M_{1} = M_{ℓ_{1, 1}} \oplus \dots \oplus M_{ℓ_{n, 1}}$ and $M_{2} = M_{ℓ_{1, 2}} \oplus \dots \oplus M_{ℓ_{n, 2}}$ be multi-matrix algebras, and suppose $ϕ : M_{1} \to M_{2}$ is a unital homomorphism of $M_{1}$ into $M_{2}$ ⁠. Then, ϕ is determined up to unitary equivalence in $M_{2}$ by a ℓ_n,2 × ℓ_n,1 non-negative integer matrix A_ϕ [Davidson (1996), Sec. III 2]. It follows that the inclusion of a multi-matrix algebra $M_{0}$ into a larger multimatrix algebra $M_{1}$ is determined up to unitary equivalence by a matrix A₀, which roughly states how many copies of a particular subalgebra of $M_{0}$ goes into a particular subalgebra of $M_{1}$ ⁠.

Let $B$ be a Bratteli diagram, and let $A_{k}^{+}$ ⁠, $k \in N$ ⁠, be the connectivity matrix at level k. In other words, A_k(i,j)⁺ is the number of edges going from $v_{j} \in V_{k - 1}$ to $v_{i} \in V_{k}$ ⁠. An analogous matrix $A_{k}^{-}$ can be defined for k < 0. Starting with $M_{0} = C^{| V_{0} |}$ ⁠, the matrices $A_{k}^{\pm}$ define two families of inclusions $i_{| k |}^{\pm} : M_{| k | - 1}^{\pm} \to M_{| k |}^{\pm}$ (up to unitary equivalence), one for + and one for −, where each $M_{k}^{\pm}$ is a multimatrix algebra. More explicitly, if

M_{k}^{\pm} = M_{n_{1}^{\pm}} \oplus \dots \oplus M_{n_{k}^{\pm}},

then starting with the vector $h^{0} = {(1, \dots, 1)}^{T} \in C^{| V_{0} |}$ and defining $h^{k, +} = A_{k}^{+} h^{k - 1, +} = A_{k}^{+} \dots A_{1}^{+} {(h^{0})}^{T}$ for k ≥ 0 and $h^{k, -} = h^{k - 1, -} A_{k}^{-} = h^{0} A_{1}^{-} \dots A_{k}^{-}$ for k ≤ 0, we have that

M_{k}^{+} = M_{h_{1}^{k, +}} \oplus \dots \oplus M_{h_{n_{k}}^{k, +}} and M_{k}^{-} = M_{h_{1}^{k, -}} \oplus \dots \oplus M_{h_{n_{k}}^{k, -}},

and the inclusions $i_{| k |}^{\pm} : M_{| k | - 1}^{\pm} \to M_{| k |}^{\pm}$ are defined up to unitary equivalence by the matrices $A_{k}^{\pm}$ ⁠. With these systems of inclusions, one can define the inductive limits

L F (B^{+}) ≔ ⋃_{k} M_{k}^{+} = \lim_{\to} (M_{k}^{+}, i_{k}^{+}), L F (B^{-}) ≔ ⋃_{k} M_{k}^{-} = \lim_{\to} (M_{k}^{-}, i_{k}^{-}),

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which are *-algebras called the locally finite (LF) algebras defined by $B$ ⁠. Their C∗-completions

A F (B^{+}) ≔ \bar{L F (B^{+})}, A F (B^{-}) ≔ \bar{L F (B^{-})}

are the approximately finite-dimensional (AF) algebras defined by $B$ ⁠.

Definition 8.

A trace on a *-algebra $A$ is a linear functional $τ : A \to C$ that satisfies τ(ab) = τ(ba) for all $a, b \in A$ ⁠. [There is no assumption that traces are poitive, that is, τ(aa*) > 0.] The set of all traces of $A$ forms a vector space over $C$ ⁠, and it is denoted by $Tr (A)$ ⁠. A cotrace τ′ is an element of the dual vector space ${Tr}^{*} (A) ≔ Tr {(A)}^{*}$ ⁠.

For M_ℓ, the algebra of ℓ × ℓ matrices, Tr(M_ℓ) is one-dimensional and generated by the trace $τ_{ℓ} : a \mapsto \sum_{i = 1}^{ℓ} a_{i i}$ ⁠. For a multimatrix algebra $M = M_{ℓ_{1}} \oplus \dots \oplus M_{ℓ_{n}}$ ⁠, the dimension of $Tr (M)$ is n and is generated by the traces $τ_{ℓ_{i}} \in Tr (M_{ℓ_{i}})$ for i = 1, …, n.

Let $i_{k}^{+} : M_{k - 1}^{+} \to M_{k}^{+}$ be the family of inclusions defined by the positive part of a Bratteli diagram $B^{+}$ ⁠. Then, there is a dual family of inclusions $i_{k}^{*} : Tr (M_{k}^{+}) \to Tr (M_{k - 1}^{+})$ [and an analogous family $i_{k}^{*} : Tr (M_{k}^{-}) \to Tr (M_{k - 1}^{-})$ ]. The trace spaces of the LF algebras defined by a Bratteli diagram $B$ are then the inverse limits

\begin{aligned} Tr (B^{+}) & ≔ Tr (L F (B^{+})) = \lim_{\leftarrow} (i_{k}^{*}, Tr (M_{k}^{+})), \\ Tr (B^{-}) & ≔ Tr (L F (B^{-})) = \lim_{\leftarrow} (i_{k}^{*}, Tr (M_{k}^{-})), \end{aligned}

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which are vector spaces. The respective spaces of cotraces are then

{Tr}^{*} (B^{+}) = \lim_{\to} ({(i_{k}^{*})}^{*}, {Tr}^{*} (M_{k}^{+})) and {Tr}^{*} (B^{-}) = \lim_{\to} ({(i_{k}^{*})}^{*}, {Tr}^{*} (M_{k}^{-})) .

Remark 4.

Note that since every class [p] of the dimension group $K_{0} (A F (B^{+}))$ can be represented by an element $p \in L F (B^{+})$ ⁠, the set $Tr (B^{+})$ also defines the dual space $Tr (K_{0} (A F (B^{+}))) ≔ K_{0} {(A F (B^{+}))}^{'}$ ⁠. As such, the trace spaces that will be used can be thought of as the dual of the invariant $K_{0} (A F (B^{+}))$ ⁠.

Let ${B_{x}}$ be a family of Bratteli diagrams parameterized by x ∈ X ⊂ Σ_N, where X is a closed, σ-invariant subset of Σ_N (an example of this is $X_{F}$ ⁠, where $F$ is a family of substitutions on N tiles, as described in Sec. III B). In what follows, we will focus on the invariants defined by the positive part of $B_{x}$ ⁠, so we will drop the + superscripts used earlier. The shift induces a *-homomorphism $σ_{*} : M_{0}^{x} \to M_{0}^{σ (x)}$ as follows: For $a = (a_{1}, \dots, a_{| V_{0}^{+} |}) \in M_{0}^{x}$ ⁠, consider its image $i_{1}^{x} a = ({(i_{1}^{x} a)}_{1}, \dots, {(i_{1}^{x} a)}_{| V_{1}^{+} |}) \in M_{1}^{x}$ ⁠. Composing this with the evaluation by $T_{1}^{x}$ ⁠, which takes $a = (a_{1}, \dots, a_{| V_{1} |}) \in M_{1}^{x}$ to $T_{1}^{x} (a) = (τ_{1} (a_{1}), \dots, τ_{| V_{1}^{+} |} (a_{| V_{1}^{+} |})) \in M_{0}^{σ (x)}$ ⁠, we obtain the map

\begin{aligned} σ_{*} = T_{1}^{x} ◦ i_{1}^{x} : a \mapsto (τ_{1} (i_{1}^{x} a), \dots, τ_{| V_{1}^{+} |} (i_{1}^{x} a)) & = (\sum_{j} a_{j} A_{1} (1, j), \dots, \sum_{j} a_{j} A_{1} (| V_{1} |, j)) \\ = A_{1} {(a_{1}, \dots, a_{| V_{0} |})}^{T} \in M_{0}^{σ (x)} = C^{| V_{1} |} . \end{aligned}

As such, the map $σ_{*} : M_{0}^{x} = C^{| V_{0} |} \to C^{| V_{1} |} = M_{0}^{σ (x)}$ coincides with the linear map $A_{1} : C^{| V_{0} |} \to C^{| V_{1} |}$ defined by the first matrix of the Bratteli diagram. As such, there is a dual map $σ^{*} : Tr (M_{0}^{σ (x)}) \to Tr (M_{0}^{x})$ ⁠, and so we have the isomorphisms

Tr (B_{x}^{+}) ≅ \lim_{\leftarrow} (Tr (M_{0}^{σ^{k} (x)}), σ^{*}) and {Tr}^{*} (B_{x}^{+}) ≅ \lim_{\to} ({Tr}^{*} (M_{0}^{σ^{k} (x)}), σ_{*}) .

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Now consider the composition $σ_{*} ◦ σ_{*} = T_{1}^{σ (x)} ◦ i_{x}^{σ (x)} ◦ T_{1}^{x} ◦ i_{1}^{x} : M_{0}^{x} \to M_{0}^{σ^{2} (x)}$ ⁠. Since both $Tr (M_{1}^{σ (x)})$ and $Tr (M_{2}^{x})$ are isomorphic to $C^{| V_{2}^{+} |}$ and there is a canonical correspondence between their bases ${τ_{1}, \dots, τ_{| V_{2}^{+} |}}$ and ${τ_{1}^{'}, \dots, τ_{| V_{2}^{+} |}^{'}}$ ⁠, respectively, we have that

τ_{ℓ} (i_{1}^{σ (x)} (T_{1}^{x} ◦ i_{1}^{x} (a))) = τ_{ℓ}^{'} (i_{2}^{x} i_{1}^{x} (a))

for all $ℓ \in {1, \dots, | V_{2}^{+} |}$ ⁠. Hence, we can now write the composition in detail as follows:

\begin{aligned} T_{1}^{σ (x)} ◦ i_{x}^{σ (x)} ◦ T_{1}^{x} ◦ i_{1}^{x} (a) & = (τ_{1} (i_{1}^{σ (x)} (T_{1}^{x} ◦ i_{1}^{x} (a))), \dots, τ_{| V_{2}^{+} |} (i_{1}^{σ (x)} (T_{1}^{x} ◦ i_{1}^{x} (a)))) \\ = (τ_{1} (i_{2}^{x} i_{1}^{x} (a)), \dots, τ_{| V_{2}^{+} |} (i_{2}^{x} i_{1}^{x} (a))) \\ = ({(i_{2}^{x} i_{1}^{x})}^{*} τ_{1} (a), \dots, {(i_{2}^{x} i_{1}^{x})}^{*} τ_{| V_{2}^{+} |} (a)) \\ = ({(A_{2} A_{1})}^{*} τ_{1} (a), \dots, {(A_{2} A_{1})}^{*} τ_{| V_{2}^{+} |} (a)), \end{aligned}

where we have abused notation slightly in using τ_ℓ to denote both the ℓth canonical trace in $Tr (M_{2}^{x})$ and the one in $Tr (M_{1}^{σ (x)})$ ⁠. This immediately generalizes to

\begin{aligned} σ_{*}^{(k)} ≔ σ_{*} ◦ \dots ◦ σ_{*} & : M_{0}^{x} \to M_{0}^{σ^{k} (x)} defined by \\ a \mapsto & ({(A_{k} \dots A_{1})}^{*} τ_{1} (a), \dots, {(A_{k} \dots A_{1})}^{*} τ_{| V_{k}^{+} |} (a)) . \end{aligned}

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A. The trace cocycle

Let $F$ be a family of substitution rules on the set of prototiles t₁, …, t_M, and let $X_{F}$ be the subshift that it defines.

Definition 9.

The trace bundle $p : Tr (F) \to X_{F}$ is the bundle over $X_{F}$ ⁠, where $p^{- 1} (x) = Tr (M_{0}^{x})$ for all $x \in X_{F}$ ⁠. The cotrace bundle $q : {Tr}^{*} (F) \to X_{F}$ is the dual of the trace bundle, where $q^{- 1} (x) = {Tr}^{*} (M_{0}^{x})$ for all $x \in X_{F}$ ⁠.

Definition 10.

The trace cocycle is the bundle map $Θ : {Tr}^{*} (F) \to {Tr}^{*} (F)$ defined by Θ_x : (x, τ′) ↦ (σ(x), σ_*(τ′)) for all $x \in X_{F}$ ⁠, $τ^{'} \in {Tr}^{*} (M_{0}^{x})$ ⁠.

Since ${Tr}^{*} (M_{0}^{x})$ is a finite dimensional vector space, we endow it with a norm ‖·‖. Note that for all $y \in X_{F}$ close enough to x, we will have ${Tr}^{*} (M_{0}^{x}) = {Tr}^{*} (M_{0}^{y})$ ⁠, and thus, all these spaces inherit the same norm. With a norm in every space $Tr (M_{0}^{x})$ ⁠, we now appeal to Oseledets theorem. Let ‖·‖_op be the operator norm. Since the maps σ_* can be singular but the base transformation $σ : X_{F} \to X_{F}$ is invertible, we can appeal to the semi-invertible Oseledets theorem (Froyland et al., 2013) and obtain a decomposition of the trace spaces, which is invariant under the dynamics.

Theorem 4

[semi-invertible Oseledets theorem (Froyland et al., 2013)]. Let $F$ be a family of substitution rules on t₁, …, t_M tiles and μ be a minimal and σ-invariant Borel ergodic probability measure on $X_{F}$ . Suppose that $\log^{+} ‖ σ_{*} ‖_{o p} \in L_{μ}^{1}$ . Then, there exist numbers $λ_{1}^{\pm} \geq λ_{2}^{\pm} \geq \dots \geq λ_{r^{\pm}}^{\pm}$ , where $λ_{i}^{+} > 0$ and $λ_{i}^{-} \leq 0$ , such that for μ-almost every x, there is a measurable, σ_*-invariant family of subspaces $V_{j}^{\pm} (x), V^{\infty} (x) \subset {Tr}^{*} (M_{0}^{x})$ :

We have ${Tr}^{*} (B_{x}^{+}) = E_{x}^{+} \oplus E_{x}^{-}$ ⁠, where
$E_{x}^{\pm} = \oplus_{i = 1}^{r^{\pm}} V_{i}^{\pm} (x) a n d {Tr}^{*} (M_{0}^{x}) = {Tr}^{*} (B_{x}^{+}) \oplus V^{\infty} (x) .$
$σ_{*} V_{j}^{\pm} (x) = V_{j}^{\pm} (σ (x))$ and σ_*V^∞(x) ⊂ V^∞(σ(x)).
For any $v^{\pm} \in V_{i}^{\pm} (x)$ and v₀ ∈ V^∞(x), we have that
$\lim_{n \to \infty} \frac{\log ‖ σ_{*}^{(n)} v^{\pm} ‖}{n} = λ_{i}^{\pm} and \lim_{n \to \infty} \frac{\log ‖ σ_{*}^{(n)} v_{0} ‖}{n} = - \infty .$

The collection of numbers $λ_{i}^{\pm}$ associated with the measure μ are the Lyapunov exponents of μ. The set of all exponents is the Lyapunov spectrum of μ. Given an invariant measure μ satisfying the hypotheses of Oseledets theorem, an Oseledets-generic or Oseledets-typical point is a point x for which the conclusions of the theorem hold.

In (i) of the above theorem, we have made the identification of the cotrace space ${Tr}^{*} (B_{x}^{+})$ with subspace of ${Tr}^{*} (M_{0}^{x})$ ⁠, which consists of vectors that are not in the kernel of $σ_{*}^{(k)}$ for all k > 0. This is justified by (15). Thus, the restriction of σ_* to ${Tr}^{*} (B_{x}^{+})$ is the linear map on the cotrace space induced by the shift σ. There is an analogous, dual, invariant decomposition of $Tr (B_{x}^{+})$ as $Tr (B_{x}^{+}) = T_{x}^{+} \oplus T_{x}^{-}$ ⁠, where

T_{x}^{\pm} = ⨁_{i = 1}^{r^{\pm}} T_{i}^{\pm} (x), a n d Tr (M_{0}^{x}) = Tr (B_{x}^{+}) \oplus T^{\infty} (x) .

The rest of this section is devoted to defining, for Oseledets-typical points x ∈ Σ_N, a map $j_{x}^{+} : {Tr}^{*} (B_{x}^{+}) \to L F (B_{x}^{+})$ and deducing its equivariant properties with respect to the renormalization dynamics, that is, with respect to the shift map σ : Σ_N → Σ_N, which is given by (20). These properties will be used in Sec. VI in the study of ergodic integrals.

Denote by ${τ_{1}, \dots, τ_{| V_{0}^{+} |}}$ the standard basis of $Tr (M_{0}^{x})$ and by ${δ_{1}, \dots, δ_{| V_{0}^{+} |}}$ the dual basis for ${Tr}^{*} (M_{0}^{x})$ ⁠. Oseledets theorem above gives a canonical identification of ${Tr}^{*} (B_{x}^{+})$ with a subspace of ${Tr}^{*} (M_{0}^{x})$ ⁠, so any cotrace in ${Tr}^{*} (B_{x}^{+})$ can be written as

τ^{*} = \sum_{i = 1}^{| V_{0}^{+} |} β_{i} (τ^{*}) δ_{i} \in {Tr}^{*} (B_{x}^{+}) \subset {Tr}^{*} (M_{0}^{x}) .

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We now define a map

j_{x}^{+} : {Tr}^{*} (B_{x}^{+}) \to L F (B_{x}^{+}) \subset A F (B_{x}^{+})

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as follows: For $τ^{*} \in {Tr}^{*} (B_{x}^{+})$ ⁠, the image $[a_{τ^{*}}] = [j_{x}^{+} (τ^{*})]$ is defined through its representative in $M_{0}^{x}$ ⁠,

j_{x}^{+} (τ^{*}) = (β_{1} (τ^{*}), \dots, β_{| V_{0}^{+} |} (τ^{*})) \in ⨁_{i = 1}^{| V_{0}^{+} |} C = M_{0}^{x},

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which is well-defined by expression (17). We denote by $[a_{τ^{*}}] = [j_{x}^{+} (τ^{*})]$ its class in $L F (B_{x}^{+})$ ⁠. Note that by (16), we have that

j_{σ (x)}^{+} (σ_{*} v) = (τ_{1} (i_{1}^{x} j_{x}^{+} v), \dots, τ_{| V_{1}^{+} |} (i_{1}^{x} j_{x}^{+} v)) \in C^{| V_{1}^{+} |} = M_{0}^{σ (x)},

where τ_ℓ is the canonical generator for $Tr (M_{n_{ℓ}})$ ⁠, the trace space for the ℓth summand of the multimatrix algebra $M_{1}^{x}$ ⁠. In general, (16) gives

\begin{aligned} j_{σ^{k} (x)}^{+} (σ_{*}^{(k)} v) & = (τ_{1} (i_{k}^{x} \dots i_{1}^{x} j_{x}^{+} (v)), \dots, τ_{| V_{k}^{+} |} (i_{k}^{x} \dots i_{1}^{x} j_{x}^{+} (v))) \\ = (σ_{(k)}^{*} τ_{1} (j_{x}^{+} (v)), \dots, σ_{(k)}^{*} τ_{| V_{k}^{+} |} (j_{x}^{+} (v))) \in C^{| V_{k}^{+} |} = M_{0}^{σ^{k} (x)}, \end{aligned}

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where $σ_{(k)}^{*}$ is the dual to $σ_{*}^{(k)}$ ⁠.

VI. ERGODIC INTEGRALS

This section is devoted to the proof of the main result of this paper (Theorem 1). First, some necessary notions are introduced and some estimates derived. Then, in Sec. VI A, a proof of the upper bound (2) in Theorem 1 is derived. This is followed by the construction of special averaging sets in Sec. VI B and a proof of (1) in Theorem 1.

Throughout this section, we assume that we are working with a minimal, ergodic σ-invariant Borel probability measure on $X_{F}$ and that the collection $F$ of substitutions are uniformly expanding and compatible. Throughout this section, we also assume that $x \in X_{F}$ is an Oseledets typical, Poincaré recurrent point. Let

{\overset{̊}{X}}_{B_{x}}^{0} ≔ \{\bar{e} \in {\overset{̊}{X}}_{B} : Δ_{x} (\bar{e}) \in ℧_{x}\} .

Definition 11.

Let $F$ be a family of substitution tilings on the tiles t₁, …, t_M, and let $Ω_{x} = Δ_{x} ({\overset{̊}{X}}_{B_{x}})$ be the tiling space given by the minimal Bratteli diagram $B_{x}$ ⁠. A spanning system of patches for Ω_x is a collection $Γ = {Γ_{k}}_{k \geq 0}$ of sets of patches $Γ_{k} = {P_{v}}_{v \in V_{k}^{+}}$ with the following properties: for each $v \in V_{k}^{+}$ ⁠, there is a path ${\bar{e}}_{v} = ({\bar{e}}_{v}^{-}, {\bar{e}}_{v}^{+}) = (\dots, e_{- 2}, e_{- 1}, e_{1}, e_{2}, \dots) \in {\overset{̊}{X}}_{B_{x}}^{0}$ with $r ({\bar{e}}_{v}^{+} |_{k}) = v$ ⁠, and in that case, $P_{v} = P_{k} ({\bar{e}}_{v}^{+})$ ⁠.

A spanning system of patches gives a catalog of all the supertiles in a given space. Along with this catalog, we can find a subset of the tiling space itself, which corresponds to each of the patches in this catalog. More specifically, given a spanning system of patches Γ, there is a corresponding system of plaques. For each patch $P_{v}$ given by the system Γ, the corresponding plaque in Ω_x is

P_{v}^{'} ≔ ⋃_{t \in P_{v}} φ_{t} (T_{{\bar{e}}_{v}}) \subset Ω_{x} .

We will denote by $X_{B}^{Γ} \subset {\overset{̊}{X}}_{B}^{0}$ the set of paths parameterized by $V^{+}$ ⁠, which give the spanning system of patches Γ.

Let $L (Ω_{x})$ be the set of Lipschitz functions on Ω_x, and for each $f \in L (Ω_{x})$ ⁠, denote by L_f the Lipschitz constant. Given a spanning system of patches Γ, we define for $f \in L (Ω_{x})$ and each $k \in N$ ⁠, the vector

\begin{aligned} V_{Γ}^{k} (f) & = (\int_{P_{v_{1}}^{'}} f d \bar{s}, \dots, \int_{P^{'} v_{| V_{k}^{+} |}} f d \bar{s}) \\ = (\int_{P_{v_{1}}} f ◦ φ_{s} (T_{e_{v_{1}}^{+}}) d s, \dots, \int_{P_{v_{| V_{k}^{+} |}}} f ◦ φ_{s} (T_{e_{v_{| V_{k}^{+} |}}^{+}}) d s) \in C^{| V_{k}^{+} |}, \end{aligned}

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where $d \bar{s}$ is the natural, leafwise volume form on Ω_x. In words, the vectors are obtained by integrating the function f along level-k super tiles of all possible types, and we use the plaques $P_{v_{i}}^{'}$ given by the spanning system of patches. This will allow us to know how the function integrates along bigger and bigger orbits.

Since $\dim {Tr}^{*} (M_{0}^{σ^{k} (x)}) = | V_{k}^{+} |$ ⁠, there is a canonical isomorphism between ${Tr}^{*} (M_{0}^{σ^{k} (x)})$ and $C^{| V_{k}^{+} |}$ ⁠, taking the dual of the generator $τ_{ℓ_{i}} \in Tr (M_{ℓ_{i}})$ to the ith standard basis vector in $C^{| V_{k}^{+} |}$ for all $i = 1, \dots, | V_{k}^{+} |$ ⁠, where $M_{0}^{σ^{k} (x)} = M_{ℓ_{1}} \oplus \dots \oplus M_{ℓ_{| V_{k}^{+} |}}$ ⁠. As such, we can think of each $V_{Γ}^{k} (f, \bar{e})$ as an element of ${Tr}^{*} (M_{0}^{σ^{k} (x)})$ ⁠, and we can compare $V_{Γ}^{k + 1} (f, \bar{e})$ with $σ_{*} V_{Γ}^{k} (f, \bar{e})$ ⁠. The ith component of the difference is

|{(V_{Γ}^{k + 1} (f, \bar{e}) - σ_{*} V_{Γ}^{k} (f, \bar{e}))}_{i}| = |\int_{P_{v_{i}}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s - \sum_{e^{'} \in r^{- 1} (v_{i})} \int_{P_{s (e^{'})}} f ◦ φ_{s} (T_{e_{s (e^{'})}^{+}}) d s| .

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Let $ε \in (0, λ_{1}^{+})$ ⁠. Since each patch $P_{v}$ for $v \in V_{k + 1}^{+}$ is the union of patches given by level-k supertiles, for any edge $e \in E_{k + 1}^{+}$ ⁠, the transverse distance between the plaques $P_{r (e)}^{'}$ and $P_{s (e)}^{'}$ is

d (P_{r (e)}^{'}, P_{s (e)}^{'}) \leq C_{ε} e^{- (λ_{1}^{+} - ε) k},

(23)

where the constant C_ɛ is independent of e and only depends on the family $F, μ$ and ɛ. For $v_{i} \in V_{k}^{+}$ and e ∈ r⁻¹(v_i), let

P_{v_{i}, e} ≔ f_{{\bar{e}}_{v} |_{k}}^{- 1} ◦ f_{e} (⋃_{\begin{array}{c} e^{'} \in E_{0, k - 1}^{+} : \\ r (e^{'}) = s (e) \end{array}} f_{e^{'}} (t_{s (e^{'})})) .

As such, there are the decompositions of each $P_{v_{i}}$ as patches tiled by level-k − 1 supertiles,

P_{v_{i}} = ⋃_{e \in r^{- 1} (v_{i})} P_{v_{i}, e} and P_{v_{i}}^{'} = ⋃_{e \in r^{- 1} (v_{i})} P_{v_{i}, e}^{'},

(24)

so it follows that

\begin{aligned} \int_{P_{v_{i}}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s - \sum_{e^{'} \in r^{- 1} (v_{i})} \int_{P_{s (e^{'})}} f ◦ φ_{s} (T_{e_{s (e^{'})}^{+}}) d s \\ = \sum_{e \in r^{- 1} (v_{i})} \int_{P_{v_{i}, e}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s - \int_{P_{s (e)}} f ◦ φ_{s} (T_{e_{s (e)}^{+}}) d s . \end{aligned}

(25)

Since both of the terms

\int_{P_{v_{i}, e}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s and \int_{P_{s (e)}} f ◦ φ_{s} (T_{e_{s (e)}^{+}}) d s

are integrating f along pieces of leaves, which correspond to the patches given by level-(k − 1) supertiles, and the distance between these pieces is at most $C_{ε} e^{- (λ_{1}^{+} - ε) k}$ ⁠, we can use the Lipschitz property to bound

|\int_{P_{v_{i}, e}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s - \int_{P_{s (e)}} f ◦ φ_{s} (T_{e_{s (e)}^{+}}) d s| \leq L_{f} C_{ε} e^{- (λ_{1}^{+} - ε) k} \int_{P_{v_{i}, e}} |f ◦ φ_{s} (T_{e_{v_{i}}^{+}})| d s

(26)

for any e ∈ r⁻¹(v_i). Returning to (22) and using (23)–(26),

\begin{aligned} |{(V_{Γ}^{k + 1} (f, \bar{e}) - σ_{*} V_{Γ}^{k} (f, \bar{e}))}_{i}| \leq \sum_{e \in r^{- 1} (v_{i})} |\int_{P_{v_{i}, e}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}}) d s - \int_{P_{s (e)}} f ◦ φ_{s} (T_{e_{s (e)}^{+}}) d s| \\ \leq \sum_{e \in r^{- 1} (v_{i})} L_{f} C_{ε} e^{- (λ_{1}^{+} - ε) k} |\int_{P_{v_{i}, e}} f ◦ φ_{s} (T_{e_{v_{i}}^{+}})| \leq \sum_{e \in r^{- 1} (v_{i})} L_{f} C_{ε} e^{- (λ_{1}^{+} - ε) k} V o l (P_{s (e)}) ‖ f ‖_{\infty} \\ \leq \sum_{e \in r^{- 1} (v_{i})} C_{f, ε} e^{- (λ_{1}^{+} - ε) k} e^{(λ_{1}^{+} + ε) k} ‖ f ‖_{\infty} \leq C_{f, ε, F} e^{2 ε k}, \end{aligned}

(27)

where we have used that $V o l (P_{s (e)}) \leq e^{(λ_{1}^{+} + ε) k}$ ⁠, which follows from the fact that $V o l (P_{s (e)})$ is roughly the number of tiles in the level-k supertile $P_{s (e)}$ ⁠, which is exactly τ_s(e) (i_k, …, i₁(Id)), and this is bounded by the largest growth rate of the trace cocycle.

By the estimate above, we have that for any ɛ > 0,

‖V_{Γ}^{k + 1} (f, \bar{e}) - σ_{*} V_{Γ}^{k} (f, \bar{e})‖ \leq C_{ε}^{*} e^{2 ε k}

for all k > 0, so we can now invoke Bufetov’s approximation Lemma (Bufetov, 2014, Lemma 2.8), which says that given a sequence of matrices {Θ_k} defined by a cocycle and sequence of vectors {V_k} such that ‖V_k+1 − Θ_kV_k‖ ≤ C_ϵe^ϵk, then there exists a vector v_* on the first vector space whose orbit shadows the vectors V_k at an exponential scale: $‖ Θ_{k} \dots Θ_{1} v_{*} - V_{k + 1} ‖ \leq C_{ϵ}^{'} e^{ϵ k}$ ⁠.

Applied to our situation, by (26) and Bufetov’s approximation Lemma, there exists a $a_{f, Γ} \in E_{x}^{+} \subset {Tr}^{*} (B_{x}^{+}) \subset {Tr}^{*} (M_{0}^{x})$ with the property that

‖ j_{σ^{k + 1} (x)}^{+} (σ_{*}^{(k)} a_{f, Γ}) - V_{Γ}^{k + 1} (f) ‖ \leq C_{ε}^{'} e^{2 ε k}

(28)

for all k > 0. Thus, we get a map

i_{Γ}^{+} : L (Ω_{x}) \to {Tr}^{*} (B_{x}^{+})

with $i_{Γ}^{+} (f) = a_{f, Γ}$ as defined above for any $f \in L (Ω_{x})$ ⁠. By composition with the map $j_{x}^{+}$ in (18), we get a map $j_{x}^{+} ◦ i_{x}^{+} : L (Ω_{x}) \to L F (B_{x}^{+})$ ⁠.

A. Proof of the upper bound (2)

For a tiling $T$ of $R^{d}$ of finite local complexity and a good Lipschitz domain B with nonempty interior, we denote by T · B the set (TId)B and by $O_{T}^{-} (B)$ all the tiles of $T$ ⁠, which are completely contained in B.

Given x ∈ Σ_N, denote by $θ_{{(n)}_{x}} = θ_{x_{n}} θ_{x_{n - 1}} \dots θ_{x_{1}}$ the product of the contracting constants from the substitution maps. In other words, $θ_{x_{i}}$ is the contraction constant of the substitution map $F_{x_{i}}$ ⁠. The following was proved by Schmieding and Treviño (2021) (Lemma 8).

Lemma 2.

For a good Lipschitz domain B with nonempty interior, tiling

T_{\bar{e}} \in Ω_{x}

and T > 0, there exists an integer n = n(T, B) and a decomposition

O_{T_{\bar{e}}}^{-} (T \cdot B) = ⋃_{i = 0}^{n} ⋃_{j = 1}^{M (i)} ⋃_{k = 1}^{κ_{j}^{(i)}} t_{j, k}^{(i)},

(29)

where

t_{j, k}^{(i)}

is a level-i supertile of type j with

$κ_{j}^{(n)} \neq 0$ for some j and $V o l (T \cdot B) \leq K_{1} θ_{{(n)}_{x}}^{- d}$ ,
$\sum_{j = 1}^{M (i)} κ_{j}^{(i)} \leq K_{2} V o l (\partial (T \cdot B)) θ_{{(i)}_{x}}^{d - 1}$ for i = 0, …, n − 1, and
$R_{1} θ_{{(n)}_{x}}^{- 1} < T < R_{1}^{- 1} R_{2} θ_{{(n)}_{x}}^{- 1}$ and n(R₂T, B) > n(T, B)

for some K₁, K₂, R₁, R₂ > 0.

Let B be the Lipschitz domain and T > 1. For $T_{\bar{e}} \in Ω_{x}$ ⁠, consider a level-i super tile $t_{j, k}^{(i)}$ of type j given by the decomposition given in Lemma 2 and $f \in L (Ω_{x})$ ⁠. For any ɛ > 0 and spanning system Γ, as in (27), one has that

|\int_{t_{j, k}^{(i)}} f ◦ φ_{s} (T_{\bar{e}}) d s - \int_{P_{v_{j}}} f ◦ φ_{s} (T_{{\bar{e}}_{v_{j}}}) d s| \leq C_{ε} L_{f} e^{2 ε i}

(30)

with $v_{j} \in V_{i}^{+}$ ⁠. Combining this with (28), we have that

|\int_{t_{j, k}^{(i)}} f ◦ φ_{s} (T) d s - j_{σ^{i} (x)}^{+} {(σ_{*}^{(i)} i_{Γ}^{+} (f))}_{j}| \leq C_{ε}^{''} e^{2 ε i},

(31)

where $C_{ε}^{''}$ only depends on ɛ and $F$ ⁠.

For any Oseledets regular x and a generating trace $τ_{ℓ}^{x, k} \in Tr (M_{0}^{σ^{k} (x)})$ ⁠, there is a decomposition

τ_{ℓ}^{x, k} = \sum_{m^{\pm} = 1}^{d_{μ}^{\pm}} b_{m^{\pm}, ℓ, k} τ_{m, k}^{\pm} + b_{\infty, ℓ, k} τ_{\infty, k},

where $τ_{m, k}^{\pm} \in T_{m}^{\pm} (σ^{k} (x))$ and τ_∞,k ∈ T^∞(σ^k(x)) are unit vectors. Note that in such decomposition, there is a $N$ such that $| b_{m^{\pm}, ℓ, k} | \leq N$ for all indices. This follows from the fact that $τ_{m, k}^{\pm}$ are unit vectors, $τ_{ℓ}^{x, k}$ are generating traces (i.e., unit vectors), and we are dealing with finite dimensional vector spaces. Since $i_{Γ}^{+} (f) \in E_{x}^{+}$ ⁠, using (29) and (20), it follows that

\begin{aligned} \int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s = \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} \sum_{k = 1}^{κ_{j}^{(i)}} \int_{t_{j, k}^{(i)}} f ◦ φ_{s} (T_{\bar{e}}) d s \\ = \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} κ_{j}^{(i)} (j_{σ^{i} (x)}^{+} {(σ_{*}^{(i)} i_{Γ}^{+} (f))}_{j} + O (e^{2 ε i})) \\ = \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} κ_{j}^{(i)} (σ_{(i)}^{*} τ_{j}^{x, i} (j_{x}^{+} (i_{Γ}^{+} (f))) + O (e^{2 ε i})) \\ = \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} κ_{j}^{(i)} (\sum_{m = 1}^{d_{μ}^{+}} b_{m, ℓ, i} σ_{(i)}^{*} τ_{m, i}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) + O (e^{2 ε i})) \\ = \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} κ_{j}^{(i)} (\sum_{m = 1}^{d_{μ}^{+}} b_{m, ℓ, i} ‖σ_{*}^{(i)} |_{V_{m}^{+} (x)}‖ τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) + O (e^{2 ε i})) . \end{aligned}

(32)

For any ɛ > 0, the bounds in Lemma 2 give

\begin{aligned} |\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s| \leq \sum_{i = 0}^{n (T)} \sum_{j = 1}^{M (i)} κ_{j}^{(i)} (C_{1} \sum_{m = 1}^{d_{μ}^{+}} τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) e^{(λ_{m}^{+} + ε) i}) \\ \leq C_{2} \sum_{i = 0}^{n (T)} V o l (\partial (T \cdot B)) θ_{{(x)}_{i}}^{d - 1} \sum_{m = 1}^{d_{μ}^{+}} τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) e^{(λ_{m}^{+} + ε) i} \\ \leq C_{3} \sum_{i = 0}^{n (T)} {(θ_{x_{i + 1}}, \dots, θ_{x_{n}})}^{1 - d} \sum_{m = 1}^{d_{μ}^{+}} τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) e^{(λ_{m}^{+} + ε) i}, \end{aligned}

(33)

where the fact that $V o l (\partial (T \cdot B)) {(θ_{e_{1}} \dots θ_{e_{i}})}^{d - 1} \leq C_{3} {(θ_{x_{i + 1}} \dots θ_{x_{n}})}^{1 - d}$ was used. This last estimate is a straightforward consequence of the estimates in Lemma 2 and the fact that $V o l (\partial (T \cdot B)) \sim V o l {(T \cdot B)}^{\frac{d - 1}{d}}$ for Lipschitz domains B and large T. If $τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) = 0$ for all m = 1, …, r − 1 but $τ_{r, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) \neq 0$ for some $r \leq d_{μ}^{+}$ ⁠, then

|\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s| \leq C_{4} \sum_{i = 0}^{n (T)} {(θ_{x_{i + 1}} \dots θ_{x_{n}})}^{1 - d} τ_{r, 0}^{+} ([a_{f}]) e^{(λ_{r}^{+} + ε) i} .

(34)

Now, for any ɛ > 0, we have that

{(θ_{x_{i + 1}} \dots θ_{x_{n}})}^{1 - d} \leq C_{ε}^{'} e^{(\frac{λ_{1}^{+}}{d} + ε) (n - i) (d - 1)}

for some C_ɛ′ > 0. Indeed, for an Oseledets-typical $x \in X_{F}$ ⁠, the leading exponent $λ_{1}^{+}$ gives the exponential rate of increase in the number of paths starting from $V_{0}^{+}$ of length k > 0 in $B_{x}^{+}$ ⁠. Since the paths of length k are in bijection with tiles in k-approximants, the number of paths of length k also gives estimates on the volumes of patches for level-k supertiles. Thus, $λ_{1}^{+}$ gives the exponential rate of increase in the volume of supertiles. Hence,

\lim_{n \to \infty} \frac{\log θ_{x_{1}} \dots θ_{x_{n}}}{n} = - \frac{λ_{1}^{+}}{d} .

(35)

Therefore, we can continue with (33),

\begin{aligned} |\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s| \leq C_{5} \sum_{i = 0}^{n (T)} e^{(\frac{λ_{1}^{+}}{d} + ε) (n - i) (d - 1)} e^{(λ_{r}^{+} + ε) i} = C_{5} \sum_{i = 0}^{n (T)} e^{(\frac{λ_{1}^{+}}{d} + ε) (i - n) (1 - d)} e^{(λ_{r}^{+} + ε) i} \\ = C_{5} \sum_{i = 0}^{n (T)} \exp [(λ_{r}^{+} - λ_{1}^{+} \frac{d - 1}{d} + ε (2 - d)) i + (λ_{1}^{+} \frac{d - 1}{d} + ε (d - 1)) n] \\ = C_{5} \exp [(λ_{1}^{+} \frac{d - 1}{d} + ε (d - 1)) n] \sum_{i = 0}^{n (T)} \exp [(λ_{r}^{+} - λ_{1}^{+} \frac{d - 1}{d} + ε (2 - d)) i] \\ = C_{6} \exp [(λ_{1}^{+} \frac{d - 1}{d} + ε (d - 1)) n] |1 - \exp [(λ_{r}^{+} - λ_{1}^{+} \frac{d - 1}{d} + ε (2 - d)) n + 1]| \\ \leq C_{7} \exp [\max \{λ_{1}^{+} \frac{d - 1}{d} + ε (d - 1), λ_{r}^{+} + ε\} n] . \end{aligned}

(36)

Defining

{\bar{λ}}_{r, ε}^{+} ≔ \max \{λ_{1}^{+} \frac{d - 1}{d} + ε (d - 1), λ_{r}^{+} + ε\}

and using (iii) from Lemma 2, we have that

\frac{\log |\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s|}{\log T} \leq \frac{\log (C_{7}) + {\bar{λ}}_{r, ε}^{+} n (T)}{\log (R_{1}^{- 1}) + \log θ_{{(x)}_{n}}^{- 1}} .

(37)

Recall that by (35), we have that

\lim_{n \to \infty} \frac{\log θ_{{(x)}_{n}}^{- 1}}{n} = \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \log θ_{x_{i}}^{- 1} = \frac{λ_{1}^{+}}{d},

so it follows from (37) that

\begin{aligned} \underset{T \to \infty}{lim sup} \frac{\log |\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s|}{\log T} & \leq \underset{n \to \infty}{lim sup} \frac{\log (C_{7}) + {\bar{λ}}_{r, ε}^{+} n (T)}{\log (R_{1}^{- 1}) + \log θ_{{(x)}_{n}}^{- 1}} \\ = \frac{{\bar{λ}}_{r, ε}^{+}}{λ_{1}^{+}} d . \end{aligned}

(38)

Now, since

\begin{aligned} |\int_{T \cdot B} f ◦ φ_{s} (T_{\bar{e}}) d s| & \leq |\int_{O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s| + |\int_{T \cdot B - O_{T_{\bar{e}}}^{-} (T \cdot B)} f ◦ φ_{s} (T_{\bar{e}}) d s| \\ \leq C_{8} e^{{\bar{λ}}_{r, ε}^{+} n (T)} + O (\partial (T \cdot B)) = C_{8} e^{{\bar{λ}}_{r, ε}^{+} n (T)} + O (T^{d - 1}), \end{aligned}

(39)

this completes the proof of the bound (2).

B. Special averaging sets and proof of (1)

Let $x \in X_{F}$ be a Poincaré-recurrent, Oseledets-regular point, and $T_{\bar{e}} \in Ω_{x}$ for some $\bar{e} = ({\bar{e}}^{-}, {\bar{e}}^{+}) = (\dots, e_{- 2}, e_{- 1}, e_{1}, e_{2}, \dots) \in {\overset{̊}{X}}_{B_{x}}$ ⁠. For any ɛ > 0, there exists a T_ɛ > 0 such that

$T^{- 1} \cdot O_{T_{\bar{e}}}^{-} (T \cdot B)$ is ɛ-close to B in the Hausdorff metric,
$O_{T_{\bar{e}}}^{-}$ contains a ball of radius twice the minimal radius so that every ball of such radius contains a copy of every prototile in its interior

for all T > T_ɛ. Pick some T_* > T_ɛ, and define $B_{ε} = T_{*}^{- 1} \cdot O_{T_{\bar{e}}}^{-} (T_{*} \cdot B) and P_{ε} (\bar{e}) = T_{*} B_{ε} = O_{T_{\bar{e}}}^{-} (T_{*} \cdot B)$ ⁠, which is a patch for all tilings in Ω_x. The set B_ɛ is at most ɛ close to B in the Hausdorff metric.

Let k_i → ∞ denote the recurrence times, $σ^{k_{i}} (x) \to x$ ⁠, and suppose that $σ^{k_{i}} (\bar{e})$ converges to ${\bar{e}}^{*} = (\dots, e_{- 2}^{*}, e_{- 1}^{*}, e_{1}^{*}, e_{2}^{*}, \dots) \in {\overset{̊}{X}}_{B_{x}}$ along these times. Let k_x be the smallest integer so that for all $v \in V_{0}^{+}$ and $w \in V_{k_{x}}^{+}$ ⁠, there is a path $p \in E_{0, k_{x}}^{+}$ with s(p) = v and r(p) = w. It follows that there is a k_ɛ′ ≥ k_x and finite set of paths $E_{B_{ε}}^{'} \subset E_{0, k_{ε}^{'}}^{+}$ such that for all $p \in E_{B_{ε}}^{'}$ ⁠, one has that $r (p) = r (\bar{e} |_{k_{ε}^{'}})$ and such that the patch $P_{ε} (\bar{e})$ decomposes as

P_{ε} (\bar{e}) = φ_{τ_{\bar{e}}} (⋃_{{\bar{e}}^{'} \in E_{B_{ε}}^{'}} f_{\bar{e} |_{k_{ε}^{'}}}^{- 1} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})})) \subset φ_{τ_{\bar{e}}} (P_{k_{ε}^{'}} ({\bar{e}}^{+})),

(40)

where $τ_{\bar{e}} \in R^{d}$ is completely determined by the negative part of $\bar{e}$ ⁠. By the choice of T_*, the patch $P_{ε} (\bar{e})$ is decomposed as the union of tiles

P_{ε} (\bar{e}) = ⋃_{ℓ = 1}^{M} ⋃_{j = 1}^{κ (ℓ)} t_{ℓ, j},

(41)

where t_ℓ,j is a translate of the prototili t_ℓ. Note that the number of tiles in the decomposition (41) is $| E_{B_{ε}}^{'} |$ from (40).

By minimality, there is a smallest k_ɛ > k_ɛ′ such that there is a path $p^{'} \in E_{k_{ε}^{'}, k_{ε}}$ with $s (p^{'}) = r (\bar{e} |_{k_{ε}^{'}})$ and $r (p^{'}) = r ({\bar{e}}^{*} |_{k_{ε}})$ ⁠. This gives a finite set of paths $E_{B_{ε}} \subset E_{0, k_{ε}}^{+}$ obtained by concatenating p′ to every path $e^{'} \in E_{ε}^{'}$ ⁠. As such, the patch decomposes as

P_{ε} (\bar{e}) = φ_{τ_{\bar{e}}} (⋃_{{\bar{e}}^{'} \in E_{B_{ε}}} f_{\bar{e} |_{k_{ε}^{'}} p^{'}}^{- 1} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})})) .

Considering the patch

P_{ε}^{*} (\bar{e}) = ⋃_{{\bar{e}}^{'} \in E_{B_{ε}}} f_{{\bar{e}}^{*} |_{k_{ε}}}^{- 1} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})}) \subset P_{k_{ε}} ({\bar{e}}^{* +}),

by Lemma 1, there is a $τ_{\bar{e}, {\bar{e}}^{*}} \in R^{d}$ such that $P_{ε}^{*} (\bar{e}) = P_{ε} (\bar{e}) + τ_{\bar{e}, {\bar{e}}^{*}}$ ⁠.

Let me take the time here to describe what is about to be done. So far, we have constructed a set B_ɛ that is ɛ-close to B, but it is of a special type: when dilated by T_*, it becomes a patch that has been denoted by $P_{ε} (\bar{e})$ ⁠. Now, since x is Poincaré recurrent, there is a sequence of times k_i → ∞ such that all the tilings in $Ω_{σ^{k_{i}} (x)}$ admit $P_{ε} (\bar{e})$ as a patch since $σ^{k_{i}} (x) \to x$ ⁠. Recall that by Proposition 4 patches in $Ω_{σ^{k_{i}} (x)}$ correspond to “superpatches” in Ω_x, that is, patches in Ω_x made up of level-k_i supertiles. Hence, we want to dilate $P_{ε} (\bar{e})$ along a sequence of times T_i so that up to a small translation, it becomes a patch made up of only level-k_i supertiles, unlike general dilations of sets that, as Lemma 2 shows, involve supertiles of all levels. We do all this because the integrals along this sequence of superpatches can be controlled very well.

For all i large enough, the set $E_{k_{i}, k_{i} + k_{ε}}^{+}$ is a copy of $E_{0, k_{ε}}^{+}$ ⁠, and as such, it contains a copy $E_{B_{ε}}^{i}$ of $E_{B_{ε}}$ ⁠. In other words, since $E_{k}^{+}$ is determined by $F_{x_{k}}$ and $x_{k_{i} + j} = x_{j}$ for all large i and 0 < j ≤ k_ɛ (by Poincaré recurrence), we can make the identification $E_{k_{i} + j}^{+} = E_{j}^{+}$ for all large i. Moreover, since k_ɛ ≥ k_x, for i large enough, there is a path from v to $r ({\bar{e}}_{k_{i} + k_{ε}})$ for all $v \in V_{k_{i}}^{+}$ ⁠. Define the patches

P_{ε}^{i} (\bar{e}) ≔ ⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}}^{i} \\ {\bar{e}}^{'} \in E_{0, k_{i}} \\ s (e_{v}) = r ({\bar{e}}^{'}) \end{array}} f_{\bar{e} |_{k_{i} + k_{ε}}}^{- 1} ◦ f_{{\bar{e}}^{'} e_{v}} (t_{s ({\bar{e}}^{'})}),

(42)

and note that

\begin{aligned} P_{ε}^{i} (\bar{e}) & = ⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}}^{i} \\ {\bar{e}}^{'} \in E_{0, k_{i}} \\ s (e_{v}) = r ({\bar{e}}^{'}) \end{array}} f_{\bar{e} |_{k_{i} + k_{ε}}}^{- 1} ◦ f_{{\bar{e}}^{'} e_{v}} (t_{s ({\bar{e}}^{'})}) = ⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}}^{i} \\ {\bar{e}}^{'} \in E_{0, k_{i}} \\ s (e_{v}) = r ({\bar{e}}^{'}) \end{array}} f_{\bar{e} |_{k_{i} + k_{ε}}}^{- 1} ◦ f_{e_{v}} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})}) \\ = ⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}}^{i} \\ {\bar{e}}^{'} \in E_{0, k_{i}} \\ s (e_{v}) = r ({\bar{e}}^{'}) \end{array}} f_{\bar{e} |_{k_{i}}}^{- 1} ◦ f_{(e_{k_{i} + 1}, \dots, e_{k_{i} + k_{ε}})}^{- 1} ◦ f_{e_{v}} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})}) \\ = f_{\bar{e} |_{k_{i}}}^{- 1} (⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}}^{i} \\ {\bar{e}}^{'} \in E_{0, k_{i}} \\ s (e_{v}) = r ({\bar{e}}^{'}) \end{array}} f_{{\bar{e}}^{*} |_{k_{ε}}}^{- 1} ◦ f_{e_{v}} ◦ f_{{\bar{e}}^{'}} (t_{s ({\bar{e}}^{'})})) = f_{\bar{e} |_{k_{i}}}^{- 1} (⋃_{\begin{array}{c} e_{v} \in E_{B_{ε}} \end{array}} f_{{\bar{e}}^{*} |_{k_{ε}}}^{- 1} ◦ f_{e_{v}} (t_{s (e_{v})})) \\ = f_{\bar{e} |_{k_{i}}}^{- 1} (P_{ε}^{*} (\bar{e})) = f_{\bar{e} |_{k_{i}}}^{- 1} (P_{ε} (\bar{e}) + τ_{\bar{e}, {\bar{e}}^{*}}) = f_{\bar{e} |_{k_{i}}}^{- 1} (T_{*} \cdot B_{ε} + τ_{\bar{e}, {\bar{e}}^{*}}) . \end{aligned}

(43)

Thus, setting $t_{ℓ, j}^{(i)} ≔ f_{\bar{e} |_{k_{i}}}^{- 1} (t_{ℓ, j} + τ_{\bar{e}, {\bar{e}}^{*}})$ ⁠, by (41), it follows that

P_{ε}^{i} (\bar{e}) = ⋃_{ℓ = 1}^{M} ⋃_{j = 1}^{κ (ℓ)} t_{ℓ, j}^{(i)},

which expresses $P_{ε}^{i} (\bar{e})$ as the union of level-k_i supertiles of $T_{\bar{e}}$ ⁠.

Lemma 3.

There is a compact set $K \subset R^{d}$ such that for all i large enough, there exists a T_i > 0 and $τ_{i} \in K$ such that $T_{i} \cdot (P_{ε} (\bar{e}) + τ_{i}) = P_{x}^{i} (\bar{e})$ .

Proof.

By (), we have that

P_{ε}^{i} (\bar{e}) = f_{\bar{e} |_{k_{i}}}^{- 1} (T_{*} \cdot B_{ε} + τ_{\bar{e}, {\bar{e}}^{*}}) = θ_{{(k_{i})}_{x}}^{- 1} T_{*} \cdot B_{ε} + τ_{\bar{e}, {\bar{e}}^{*}}^{i}

for some

τ_{\bar{e}, {\bar{e}}^{*}}^{i} \in R^{d}

⁠. Defining

T_{i} ≔ θ_{{(k_{i})}_{x}}^{- 1} T_{*}

⁠, we have that

P_{ε}^{i} (\bar{e}) = T_{i} \cdot B_{ε} + τ_{\bar{e}, {\bar{e}}^{*}}^{i}

⁠. By our assumption of recurrence, we have that there exists a R_ɛ such that for all i large enough, the patch

P_{ε} (\bar{e})

is found in any ball of radius R_ɛ around any point in

R^{d}

for any

T \in Ω_{σ^{k_{i}} (x)}

for all i large enough. The scaling T_i relates the scales of Ω_x and those of

Ω_{σ^{k_{i}} (x)}

⁠. In other words, level-k_i supertiles in Ω_x correspond to tiles in

Ω_{σ^{k_{i}} (x)}

⁠, and the difference in scales is precisely T_i. By this relationship of scale and repetitivity, any ball of radius T_iR_ɛ in

T_{σ^{k_{i}} (\bar{e})}

contains a copy of the patch

P_{ε}^{i} (\bar{e})

⁠. Hence, without loss of generality, we can assume that

τ_{\bar{e}, {\bar{e}}^{*}}^{i} \in B_{T_{i} R_{ε}} (0)

⁠. Letting

τ_{i} = T_{i}^{- 1} τ_{\bar{e}, {\bar{e}}^{*}}^{i}

⁠, we get that

T_{i} \cdot (P_{ε} (\bar{e}) + τ_{i}) = P_{x}^{i} (\bar{e})

⁠.

□

1. Implicit upper bound

Let $f \in L (Ω_{x})$ ⁠. By (43), for all i large enough, there is the decomposition

\int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s = \sum_{ℓ = 1}^{M} \sum_{j = 1}^{κ (ℓ)} \int_{t_{ℓ, j}^{(i)}} f ◦ φ_{s} (T_{\bar{e}}) d s,

which, after choosing ɛ′ > 0 and using (30) and (31), becomes, as in (32),

\begin{aligned} \int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s & = \sum_{ℓ = 1}^{M} κ (ℓ) {(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ} + O (e^{2 ε^{'} k_{i}}) \\ = \sum_{ℓ = 1}^{M} κ (ℓ) σ_{(k_{i})}^{*} τ_{j}^{x, k_{i}} (j_{x}^{+} (i_{Γ}^{+} (f))) + O (e^{2 ε^{'} k_{i}}) \\ = \sum_{ℓ = 1}^{M} κ (ℓ) \sum_{m = 1}^{d_{μ}^{+}} b_{m, ℓ, k_{i}} σ_{(k_{i})}^{*} τ_{m, k_{i}}^{+} (j_{x}^{+} i_{Γ}^{+} (f)) + O (e^{2 ε^{'} k_{i}}) \\ = \sum_{ℓ = 1}^{M} κ (ℓ) \sum_{m = 1}^{d_{μ}^{+}} b_{m, ℓ, k_{i}} ‖σ_{*}^{(k_{i})} |_{V_{m}^{+} (x)}‖ τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) + O (e^{2 ε^{'} k_{i}}) . \end{aligned}

(44)

Hence, if $τ_{m, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) = 0$ for all m < r but $τ_{r, 0}^{+} (j_{x}^{+} (i_{Γ}^{+} (f))) \neq 0$ ⁠,

\int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s = \sum_{ℓ = 1}^{M} κ (ℓ) \sum_{m = r}^{d_{μ}^{+}} b_{m, ℓ, k_{i}} ‖σ_{*}^{(k_{i})} |_{V_{m}^{+} (x)}‖ τ_{m, 0}^{+} ([a_{f}]) + O (e^{2 ε^{'} k_{i}})

(45)

for all i, from which it follows that

|\int_{T_{i} (B_{ε} + τ_{i})} f ◦ φ_{s} (T_{\bar{e}}) d s| \leq C e^{(λ_{r}^{+} + ε^{'}) k_{i}}

for all i. Since T_i is proportional to $θ_{{(k_{i})}_{x}}^{- 1}$ ⁠, we can estimate as in (34)–(38) to obtain

\underset{i \to \infty}{lim sup} \frac{\log |\int_{T_{i} (B_{ε} + τ_{i})} f ◦ φ_{s} (T_{\bar{e}}) d s|}{\log T_{i}} \leq \frac{λ_{r}^{+} + ε^{'}}{λ_{1}^{+}} d .

2. Implicit lower bound

We partition the set of indices {1, …, M} into two sets I⁺, I⁰. An index ℓ is in I⁺ if

\underset{i \to \infty}{lim sup} \frac{|{(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ}|}{‖σ_{*}^{k_{i}} π_{ℓ, x}^{+} i_{Γ}^{+} f‖} > 0

and ℓ ∈ I⁰ otherwise, where we recall that $π_{ℓ, x}^{+} : {Tr}^{*} (M_{0}^{x}) \to V_{ℓ}^{+} (x)$ is the corresponding projection to the ℓth positive Oseledets subspace. The set I⁺ is not empty because (1) by assumption, $τ_{ℓ}^{+} ([a_{f}]) = 0$ for all ℓ < r and $τ_{r}^{+} ([a_{f}]) \neq 0$ ⁠, meaning that $π_{ℓ, x}^{+} i_{Γ}^{+} f = 0$ for all ℓ < r but $π_{r, x}^{+} i_{Γ}^{+} f \neq 0$ ⁠; and (2) all norms are equivalent in finite dimensional vector spaces.

Now, we recall (44) and express it with indices according to the partition I⁺, I⁰,

\int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s = \sum_{ℓ \in I^{+}} κ (ℓ) {(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ} + \sum_{ℓ \in I^{0}} κ (ℓ) {(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ} + O (e^{2 ε^{'} k_{i}}),

which, after rearranging, using the triangle inequality, and rearranging again, we get

\begin{aligned} |\int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s| & \geq |\sum_{ℓ \in I^{+}} κ (ℓ) {(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ}| \\ - |\sum_{ℓ \in I^{0}} κ (ℓ) {(j_{σ^{k_{i}} (x)}^{+} (σ_{*}^{(k_{i})} i_{Γ}^{+} (f)))}_{ℓ} + O (e^{2 ε^{'} k_{i}})| \\ \geq C^{+} ‖ σ_{*}^{k_{i}} π_{r, x}^{+} i_{Γ}^{+} f ‖ \end{aligned}

(46)

for all i and some C⁺ > 0 small enough. Recalling that T_i is proportional to $θ_{{(k_{i})}_{x}}^{- 1}$ and using (35),

\begin{aligned} \underset{i \to \infty}{lim sup} \frac{\log |\int_{P_{ε}^{i} (\bar{e})} f ◦ φ_{s} (T_{\bar{e}}) d s|}{\log T_{i}} & \geq \underset{i \to \infty}{lim sup} \frac{\log C^{+} ‖ σ_{*}^{k_{i}} π_{r, x}^{+} i_{Γ}^{+} f ‖}{\log T_{i}} \\ = \underset{i \to \infty}{lim sup} \frac{k_{i}}{\log T_{i}} \frac{\log ‖σ_{*}^{k_{i}} π_{r, x}^{+} i_{Γ}^{+} f‖}{k_{i}} = \frac{d}{λ_{1}^{+}} λ_{r}^{+} . \end{aligned}

(47)

VII. SOLENOIDS AND THE DENJOY–KOKSMA INEQUALITY

For a function $f : X \to R$ on a Cantor set X and a clopen subset C ⊂ X, define

Var (f, C) ≔ \sup_{x, y \in C} | f (x) - f (y) |,

and for a partition P = {P₁, …, P_n} of X into disjoint clopen subsets, define

Var (f, P) = \sum_{i = 1}^{n} Var (f, P_{i}) .

A Cantor set naturally carries a metric structure. In fact, Cantor sets carry ultrametric structures, and so any ball B_ϵ(x) ⊂ X is a clopen set. Let $d_{X} : X^{2} \to R$ be an ultrametric on X. For any set C ⊂ X, let diam C = sup{d_X(x, y) : x, y ∈ C}. Let $P_{ϵ}$ be the set of all partitions {P₁, …, P_k} of X by clopen sets with diam P_i ≤ ϵ for all i. Finally, let

{Var}_{ϵ} (f) = \sup_{P \in P_{ϵ}} Var (f, P) and Var (f) = \sup_{ϵ > 0} {Var}_{ϵ} (f) .

A function $f : X \to R$ on a Cantor set X has bounded variation if Var(f) < ∞. Note that if f is a locally constant function on a Cantor set, then Var (f) = 0, so it is of bounded variation.

Definition 12.

Let Ω be the tiling space of an aperiodic, repetitive tiling of finite local complexity. A continuous function $f : Ω \to R$ has bounded variation if there is a V_f < ∞ such that Var(f|_℧) ≤ V_f for all transversals ℧ ⊂ Ω, which are Cantor sets.

The set of continuous functions on Ω with bounded variation is denoted by BV(Ω). Note that if f is a transversally locally constant function, then it is in BV(Ω). Let $\bar{q} = (q_{1}, q_{2}, \dots) \in N^{N}$ ⁠, where q_k > 1 for all k. For any such $\bar{q}$ we will denote q_(n) = q₁, …, q_n.

Definition 13.

A d-dimensional solenoid is the tiling space $Ω_{\bar{q}}$ associated with a family of substitutions $F$ on a single prototile $t_{1} = {[- \frac{1}{2}, \frac{1}{2}]}^{d}$ ⁠. The Bratteli diagram $B_{\bar{q}}$ for such tiling spaces have a single vertex at every level and $| E_{k} | = q_{k}^{d}$ for all $k \in Z$ ⁠, and it is also required here that $θ_{e} = q_{k}^{- d}$ for any $e \in E_{k}$ ⁠. In this case, the family $F$ is allowed to be infinite.

Remark 5.

The definition for a solenoid above is slightly more general than the usual definition of a solenoid as an inverse limit of $T^{d}$ under maps of the form q_n · Id.

The goal of this section is to prove a type of bound known as a Denjoy–Koksma inequality (Herman, 1979, Sec. VI 3) for solenoids.

Theorem 5

(Denjoy–Koksma inequality for solenoids). Let

Ω_{\bar{q}}

be a d-dimensional solenoid. Then, for any

f \in B V (Ω_{\bar{q}})

and

p \in Ω_{\bar{q}}

,

|\int_{{[0, q_{n}]}^{d}} f ◦ φ_{s} (p) d s - q_{(n)}^{d} \int_{Ω_{\bar{q}}} f d μ| \leq Var (f)

for all n > 0.

Remark 6.

It seems reasonable to conjecture that a Denjoy–Koksma inequality holds for any tiling space Ω_x obtained from compatible and uniformly expanding substitutions with $A F (B_{x}^{+})$ is UHF [see Davidson (1996), Sec. III 5]]. It seems like for d = 1, the proof below can be combined with the usual intertwining arguments to give a proof.

Proof.

Let $X_{\bar{q}}^{+} ≔ X_{B_{\bar{q}}}^{+}$ ⁠. Since any substitution in the family $F$ forces the border, the map $Δ_{\bar{q}} : X_{\bar{q}}^{+} \to ℧_{\bar{q}}$ is a homeomorphism of Cantor sets. As such, the topology of $℧_{\bar{q}}$ is generated by the image of cylinder subsets of $X_{\bar{q}}^{+}$ under the map $Δ_{\bar{q}}$ ⁠, and the ultrametric structure of $℧_{\bar{q}}$ is inherited from that of $X_{\bar{q}}$ ⁠. As such, for every k > 0, there are $q_{(k)}^{d}$ pairwise-disjoint cylinder sets $C_{i}^{k} \subset ℧_{\bar{q}}$ ⁠, parameterized by $i \in {1, \dots, q_{(k)}}^{d}$ ⁠, one for each path $p \in E_{0, k}$ ⁠, whose union is $℧_{\bar{q}}$ ⁠. Moreover, since it is well known that $X_{B_{\bar{q}}}^{+}$ admits a unique tail-invariant Borel probability measure, by Proposition 1, we have that $d i a m C_{i}^{k} = ν (C_{i}^{k}) = q_{(k)}^{- d}$ for all i for the unique holonomy-invariant measure ν on $℧_{\bar{q}}$ ⁠.

For any $\bar{e} = ({\bar{e}}^{-}, {\bar{e}}^{+}) \in X_{\bar{q}}$ ⁠, the kth approximant $P_{k} ({\bar{e}}^{+})$ is a tiled cube of side length q_(k) containing the origin, and it is tiled by $q_{(k)}^{d}$ tiles isometric to [0,1]^d. For $T_{\bar{e}} = Δ_{\bar{q}} (\bar{e}) \in Ω_{\bar{q}}$ ⁠, there exists a vector $τ_{\bar{e}} \in {[- \frac{1}{2}, \frac{1}{2}]}^{d}$ such that $φ_{τ_{\bar{e}}} (T_{\bar{e}}) \in ℧_{\bar{q}}$ ⁠. By Lemma 1, there exist $q_{(k)}^{d}$ vectors $τ_{1}, \dots, τ_{q_{(k)}^{d}}$ such that $φ_{τ_{i}} (T_{\bar{e}}) \in C_{i}^{k}$ ⁠. In other words, the points ${φ_{τ_{i}} (T_{\bar{e}})}_{i} q_{(k)}^{- d}$ -equidistribute in $℧_{\bar{q}}$ ⁠.

In fact, more is true: the vectors $τ_{1}, \dots, τ_{q_{(k)}^{d}}$ can be chosen to be nice elements of $Z^{d}$ ⁠. In particular, one can choose them to be the elements of the set ${0, \dots, q_{(k)} - 1}^{d}$ ⁠. First, note that for any $s \in {0, \dots, q_{(k)} - 1}^{d}$ ⁠, $φ_{s + τ_{\bar{e}}} (T_{\bar{e}}) \in ℧_{\bar{q}}$ ⁠. This follows from the fact that there is a single prototile (a unit cube) in the tiling and its center is the puncture. Thus, since $φ_{τ_{\bar{e}}} (T_{\bar{e}}) \in ℧_{\bar{q}}$ ⁠, it follows that $φ_{s + τ_{\bar{e}}} (T_{\bar{e}}) \in ℧_{\bar{q}}$ ⁠. Moreover, for any $s \neq s^{'} \in {1, \dots, q_{(k)}}^{d}$ ⁠, it follows that $Δ_{\bar{q}}^{- 1} (φ_{s + τ_{\bar{e}}} (T_{\bar{e}})) |_{k} \neq Δ_{\bar{q}}^{- 1} (φ_{s^{'} + τ_{\bar{e}}} (T_{\bar{e}})) |_{k}$ ⁠, so they $q_{(k)}^{- d}$ -equidistribute in $℧_{\bar{q}}$ ⁠.

Now recall the proof of the classical Denjoy–Koksma inequality for the dynamics restricted to

℧_{\bar{q}}

(Herman, 1979, Theorem VI.3.1). For

\bar{e} \in ℧_{\bar{q}}

⁠,

\begin{aligned} |\sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} f ◦ φ_{i} (T_{\bar{e}}) - q_{(k)}^{d} \int_{℧_{\bar{q}}} f d ν| = |\sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} \frac{1}{ν (C_{i}^{k})} \int_{C_{i}^{k}} (f ◦ φ_{i} (T_{\bar{e}}) - f (x)) d ν (x)| \\ \leq \sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} \frac{1}{ν (C_{i}^{k})} \int_{C_{i}^{k}} |f ◦ φ_{i} (T_{\bar{e}}) - f (x)| d ν (x) \leq \sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} \sup_{y, z \in C_{i}^{k}} | f (y) - f (z) | \\ \leq \sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} Var (f, C_{i}^{k}) \leq {Var}_{q_{(k)}^{- d}} (f) . \end{aligned}

(48)

Up to this point, everything has been done with reference to the transversal at zero

℧_{\bar{q}}

⁠. It turns out that for every point

y \in {[0, 1)}^{d}

⁠, there is an associated transversal

℧_{\bar{q}}^{y} \subset Ω_{\bar{q}}

obtained by translating

℧_{\bar{q}} = ℧_{\bar{q}}^{0}

by y. By composition with this translation, the map

Δ_{\bar{q}}

is a homeomorphism between

X_{\bar{q}}^{+}

and

℧_{\bar{q}}^{y}

⁠, and so for any k > 0, there is a partition

{C_{i}^{k} (y)}_{i}

of

℧_{\bar{q}}^{y}

by

q_{(k)}^{d}

cylinder sets of measure

ν_{y} (C_{i}^{k} (y)) = q_{(k)}^{- d}

⁠, where the measure ν_y on

℧_{\bar{q}}^{y}

is the translate of the measure ν on

℧_{\bar{q}}

⁠. Thus, the same arguments leading to () hold for the transversal

℧_{\bar{q}}^{y}

⁠, and so it follows that for

T_{\bar{e}} \in ℧_{\bar{q}}^{y}

⁠,

|\sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} f ◦ φ_{i} (T_{\bar{e}}) - q_{(k)}^{d} \int_{℧_{\bar{q}}^{y}} f d ν_{y}| \leq {Var}_{q_{(k)}^{- d}} (f) .

(49)

Finally, note that if

p \in Ω_{\bar{q}}

and

y \in {[0, 1)}^{d}

such that

p \in ℧_{\bar{q}}^{y}

⁠, then

\int_{{[0, q_{k}]}^{d}} f ◦ φ_{s} (p) d s = \int_{{[0, 1]}^{d}} \sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} f ◦ φ_{i + s} (p) d s .

Putting it all together, let

p \in Ω_{\bar{q}}

and

y \in {[0, 1)}^{d}

such that

p \in ℧_{\bar{q}}^{y}

⁠. Then,

\begin{aligned} |\int_{{[0, q_{k}]}^{d}} f ◦ φ_{s} (p) d s - q_{k}^{d} \int_{Ω_{\bar{q}}} f d μ| \\ = |\int_{{[0, 1]}^{d}} \sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} f ◦ φ_{i + s} (p) d s - q_{(k)}^{d} \int_{{[0, 1]}^{d}} \int_{℧_{\bar{q}}^{φ_{s} (y)}} f d ν_{φ_{s} (y)} d s| \\ \leq \int_{{[0, 1]}^{d}} |\sum_{i \in {\{0, \dots, q_{(k)} - 1\}}^{d}} f ◦ φ_{i + s} (p) - q_{(k)}^{d} \int_{℧_{\bar{q}}^{φ_{s} (y)}} f d ν_{φ_{s} (y)}| d s \\ \leq \int_{{[0, 1]}^{d}} {Var}_{q_{(k)}^{- d}} (f) d s = {Var}_{q_{(k)}^{- d}} (f) \leq Var (f) . \end{aligned}

(50)

□

VIII. RANDOM SCHRÖDINGER OPERATORS

This section will focus on applications of the results of Sec. VI to algebras of operators coming from the tiling spaces obtained by collections of substitution rules. Although it is natural in such cases to focus on the C∗-algebras of operators obtained, here the focus is on *-algebras, which are dense in the C∗-algebras of usual interest. This is because the traces obtained are only densely defined and one loses all but one trace by going to the completion C∗-algebras. This is mentioned for the curious reader wondering how one completes the algebras constructed; it is not relevant for the work here. However, the reader can see, for example, Bellissard (1986) for how the use of operator algebras enters the study of aperiodic media from a mathematical physics point of view; see also Kellendonk and Putnam (2000) and Lenz and Stollmann (2003) for several uses of C∗-algebras in the study of tilings. The *-algebras used here will be dense subalgebras of the ones used in Lenz and Stollmann (2003).

For a family $F$ of uniformly expanding and compatible substitutions defined on the same set of prototiles and $x \in X_{F}$ ⁠, let $B_{x}$ be the associated Bratteli diagram as constructed in Sec. III and assume $B_{x}$ is minimal. Recall that by construction, any tile t on any tiling $T \in Ω$ has a distinguished point in its interior, and they correspond to the placement of the origin inside of the prototiles {t₁, …, t_M}. These distinguished points are called punctures in Kellendonk (1995). Once the punctures have been chosen in the interior of the prototiles, there exists a ϱ > 0 such that any ball of radius less than ϱ centered at the puncture of a tile $t \in T \in Ω_{x}$ does not intersect the boundary of t, and this holds for all $x \in X_{F}$ and $T \in Ω_{x}$ ⁠. Let $Λ_{T}$ be the set of punctures of $T$ ⁠, that is, the union of all distinguished points of all tiles of $T$ ⁠, and define

G_{x} ≔ \{(p, T^{'}, q) \in R^{d} \times Ω_{x} \times R^{d} : p, q \in Λ_{T^{'}}\} .

Definition 14.

A kernel of finite range is a function $k \in C (G_{x})$ such that we have the following:

k is bounded.
k has finite range. In other words, there is a R_k > 0 such that $k (p, T, q) = 0$ whenever |p − q| > R_k.
k is $R^{d}$ -invariant: $k (p - t, φ_{t} (T), q - t) = k (p, T, q)$ for any $t \in R^{d}$ ⁠.

The set of all kernels of finite range associated with Ω_x are denoted by $K_{x}^{fi n}$ ⁠. For any $k \in K_{x}^{fi n}$ ⁠, there is a family of representations ${π_{T}}_{T \in Ω_{x}}$ in $B (ℓ^{2} (Λ_{T}))$ defined, for $k \in K_{x}^{fi n}$ ⁠, by

〈 K_{T} δ_{p}, δ_{q} 〉 = 〈 (π_{T} k) δ_{p}, δ_{q} 〉 = k (p, T, q) .

The family ${K_{T}}$ parameterized by Ω_x is bounded in the product $\prod_{T \in Ω_{x}} B (ℓ^{2} (Λ_{T}))$ ⁠. Defining a convolution product as

(a \cdot b) (p, T, q) = \sum_{x \in Λ_{T}} a (p, T, x) b (x, T, q)

and involution by $k^{*} (p, T, q) = \bar{k (q, T, p)}$ ⁠, $K_{x}^{fi n}$ has the structure of a *-algebra. It follows that the map $π : K_{x}^{fi n} \to \prod_{T} B (ℓ^{2} (Λ_{T}))$ is a faithful *-representation. The image is denoted by $A_{x}^{fi n}$ ⁠, and it is the algebra of operators of finite range. The completion of this algebra is denoted by $A_{x}$ ⁠.

Definition 15.

The set of Lipschitz kernels of finite range consists of kernels $k \in K_{x}^{fi n}$ for which there are constants R_k, L_k > 0 such that if for two $T_{1}, T_{2} \in Ω_{x}$ ⁠, one has that $B_{R_{k}} (0) \cap Λ_{T_{1}} = B_{R_{k}} (0) \cap Λ_{T_{2}}$ ⁠, then for any $p, q \in B_{R_{k}} (0) \cap Λ_{T_{1}}$ ⁠, one has that $| k (p, T_{1}, q) - k (p, T_{2}, q) | \leq L_{k} d (T_{1}, T_{2})$ ⁠.

The kernels in the above definition carry the label Lipschitz since they will be connected to Lipschitz functions on the tiling space Ω_x; see Lemma 4.

The set of Lipschitz kernels of finite range is denoted by ${L K}_{x}^{fi n} \subset K_{x}^{fi n}$ ⁠. The image of ${L K}_{x}^{fi n}$ is denoted by ${L A}_{x}^{fi n} = π {L K}_{x}^{fi n} \subset A_{x}^{fi n}$ ⁠, and it is the set of Lipschitz operators of finite range. It should be pointed out that most operators of interest in mathematical physics, such as operators of the form H = Δ + V, where V is a “localized” potential on defined on $T$ ⁠, are contained in the set ${L A}_{x}^{fi n}$ ⁠. A simple example to consider in one dimension is as follows. Let $T$ be a tiling of $R$ by N different tile types and $Λ_{T}$ be the collection of endpoints of tiles of $T$ ⁠. There is an obvious, order-preserving labeling of Λ by $Z$ ⁠. For i ∈ {1, …, N} and λ ≠ 0, consider the operator H_i,λ = Δ + λV_i, where Δ is the discrete Laplacian on $Z$ and the localized potential V_i is defined by V_i(p) = p if p ∈ Λ is a left endpoint of a tile of type i and otherwise V_i(p) = 0. Similar constructions can be made for tilings in higher dimensions by considering the graph $G_{T}$ given by the Delaunay triangulation of $Λ_{T}$ ⁠, considering the graph Laplacian $△_{G_{T}}$ on $G_{T}$ ⁠, and considering an operator of the form $H = △_{G_{T}} + V$ ⁠, where V is a localized potential depending only on the local pattern around $p \in Λ_{T}$ ⁠.

Let $u : R^{d} \to R$ be a smooth non-negative (bump) function of integral 1, compactly supported in a disk of radius less than ϱ. This defines a family of functions $w_{u, T} : {L A}_{x}^{fi n} \to C^{\infty} (R^{d})$ parameterized by Ω_x as follows: For $A = π k \in A_{x}^{fi n}$ and $A_{T} = π_{T} k \in B (ℓ^{2} (Λ_{T}))$ ⁠, let $f_{A_{T}}^{u}$ be defined by

f_{A_{T}}^{u} (t) = w_{u, T} (A) (t) = \sum_{p \in Λ_{T}} A_{T} (p, p) u (p - t) .

Lemma 4.

For any $A \in L A_{x}^{fi n}$ and $T \in Ω_{x}$ ⁠, there exists a Lipschitz function $h = h_{A}^{u} \in L (Ω_{x})$ such that $f_{A_{T}}^{u} (t) = h ◦ φ_{t} (T)$ .

Proof.

The assignment

T \mapsto f_{A_{T}}^{u} (0)

defines a function

f_{A_{\cdot}}^{u} (0) : ℧_{x} \to R

⁠. For

T, T^{'} \in ℧_{x}

⁠, one then has for any

k \in {L K}_{x}^{fi n}

⁠,

|f_{A_{T}}^{u} (0) - f_{A_{T^{'}}}^{u} (0)| = |k (0, T, 0) - k (0, T^{'}, 0)| u (0) \leq u (0) L_{k} d (T_{1}, T_{2}),

so this is a Lipschitz function on ℧_x with the Lipschitz constant L_ku(0).

The function $T \mapsto f_{A_{\cdot}}^{u} (0)$ can be extended to Ω_x by choosing a neighborhood U of ℧_x of size r_u and a product chart $ϕ_{u} : U \to B_{r_{u}} (0) \times ℧_{x}$ ⁠, and noting that the function defined by $h = ϕ_{u}^{*} \bar{u}$ ⁠, where $\bar{u} (t, T) = f_{A_{T}}^{u} (0) u (t)$ with ‖t‖ < r_u, defines a Lipschitz function on Ω_x. That this gives $f_{A_{T}}^{u} (t) = h ◦ φ_{t} (T)$ follows from the $R^{d}$ invariance of the kernel k used to define A.□

Let $M_{u} : {L A}_{x}^{fi n} \to L (Ω_{x})$ be the map given by Lemma 4, and denote the composition $ϒ_{u, Γ} ≔ j_{x}^{+} ◦ i_{Γ}^{+} ◦ M_{u} : {L A}_{x}^{fi n} \to L F (B_{x}^{+})$ ⁠. We can define functionals $τ_{i}^{'} : {L A}_{x}^{fi n} \to C$ by pullback $τ_{i}^{'} = ϒ_{u, Γ}^{*} τ_{i}^{+}$ ⁠, i.e., $τ_{i}^{'} (A) = τ_{i}^{+} (ϒ_{u, Γ} (A))$ ⁠, for $A \in {L A}_{x}^{fi n}$ ⁠, where $τ_{i}^{+} \in T_{i}^{+} (x)$ ⁠. The functionals $τ_{i}^{'}$ may or may not be traces. By Schmieding and Treviño (2018b), Proposition 1, we know some cases when they are.

Proposition 5.

Let

F_{1}, \dots, F_{N}

be a collection of uniformly expanding and compatible substitution rules on a set of prototiles t₁, …, t_M and μ be a minimal, σ-invariant ergodic Borel probability measure on

X_{F}

. Then, for μ-almost every x, for a spanning system of patches Γ on Ω_x, the functional

τ_{i}^{'} = ϒ_{u, Γ}^{*} τ_{i}^{+}

is a trace if

\frac{λ_{i}^{+}}{λ_{1}^{+}} > \frac{d - 1}{d}

. Hence, ϒ_u,Γ induces a map on traces

ϒ_{u, Γ}^{*} : Tr {(B_{x}^{+})}^{+ +} \to Tr ({L A}_{x}^{fi n}),

where

Tr {(B_{x}^{+})}^{+ +}

is the subspace of

Tr (B_{x}^{+})

generated by traces

τ_{i}^{+}

⁠, which satisfy

\frac{λ_{i}^{+}}{λ_{1}^{+}} > \frac{d - 1}{d}

.

A. Proof of theorem 3

Let $d_{μ}^{+ +}$ be the dimension of the subspace $Tr {(B_{x}^{+})}^{+ +}$ ⁠. Define the $d_{μ}^{+ +}$ traces in $Tr ({L A}_{x}^{fi n})$ to be ${τ_{1}, \dots, τ_{d_{μ}^{+ +}}}$ ⁠, where $τ_{i} \in ϒ_{u, Γ}^{*} T_{i}^{+} (x)$ is any non-zero element. Now pick $A_{T} \in {L A}_{x}^{fi n}$ ⁠, a good Lipschitz domain B and T > 0. First, note that for two smooth bump functions u, u′ of compact support in a ball of radius less than ρ and integral 1, it follows that

\begin{aligned} |\int_{O_{T}^{-} (E)} M_{u} A ◦ φ_{t} (T) - M_{u^{'}} A ◦ φ_{t} (T) d t| = 0, and \\ |\int_{E} M_{u} A ◦ φ_{t} (T) - M_{u^{'}} A ◦ φ_{t} (T) d t| = O (| \partial E |) \end{aligned}

(51)

for any measurable E of finite volume. In addition, it follows that

\begin{aligned} |t r (A_{T} |_{O_{T}^{-} (E)}) - \int_{O_{T}^{-} (E)} M_{u} A ◦ φ_{t} (T) d t| & = 0, \\ |t r (A_{T} |_{E}) - \int_{E} M_{u} A ◦ φ_{t} (T) d t| & = O (| \partial E |) \end{aligned}

(52)

for any measurable set E of finite volume, where the second estimate is from Schmieding and Treviño (2018b), Eq. (22). Thus, if τ_i(A) = 0 for all i = 1, …, r for some $r < d_{μ}^{+ +}$ but τ_r(A) ≠ 0, by (36), it follows that for any ɛ > 0,

\begin{aligned} | t r (A_{T} |_{T \cdot B)}) | & = |\int_{T \cdot B} M_{u} A ◦ φ_{t} (T) d t| + O (\partial (T \cdot B)) \\ \leq \max \{C_{ε, A} τ_{r} (A) T^{d \frac{λ_{r}^{+}}{λ_{1}^{+}} + d ε}, O (T^{d - 1})\} \end{aligned}

(53)

independent of which bump function u was used by (51). Thus, by (38) and (39), if $d λ_{r}^{+} \geq (d - 1) λ_{1}^{+}$ ⁠, then

\underset{T \to \infty}{lim sup} \frac{\log | t r (A_{T} |_{T \cdot B}) |}{\log T} \leq \frac{λ_{r}^{+}}{λ_{1}^{+}} d .

For ɛ′ > 0, we choose a set B_ɛ′ as in Sec. VI B along with the sequence of times T_i → ∞ and vectors $τ_{i} \in R^{d}$ ⁠. By construction, $T_{i} \cdot (B_{ε^{'}} + τ_{i}) = O_{T}^{-} (E_{i})$ ⁠, where $E_{i} \subset R^{d}$ is some measurable subset of finite volume. Thus, the results of Sec. VI B along with (51) and (52) imply that

\underset{i \to \infty}{lim sup} \frac{\log | t r (A_{T} |_{T_{i} \cdot (B_{ε^{'}} + τ_{i})}) |}{\log T_{i}} = \frac{λ_{r}^{+}}{λ_{1}^{+}} d .

IX. VARIATIONS ON HALF HEXAGONS

Let me close by giving some experimental results. Consider the two substitution rules on the half hexagons in Fig. 1 in the Introduction and which were studied in Sec. IV. The first substitution rule depicted is the classical substitution rule in the half-hexagon with expansion constant 2. The eigenvalues of the corresponding substitution matrix are 4, 2, 1, 1, −1, −1. The second substitution rules has expansion constant 4, and the eigenvalues for the corresponding substitution matrix are $16, 7 \pm i \sqrt{3}, 2, 2, 2$ ⁠. Note that $| 7 \pm i \sqrt{3} | > 4 = \sqrt{16}$ ⁠, so the second substitution rule has “rapidly expanding” eigenvalues.

For p ∈ (0, 1), let μ_p be the Bernoulli measure on Σ₂, which gives the cylinder set μ_p(C₁) = p and μ_p(C₂) = 1 − p, where C_i = {x ∈ Σ₂ : x₁ = i}. The typical points for the measure μ_p then give tiling spaces Ω_x, which are obtained from tilings that were constructed, on average, by applications of the first substitution in Fig. 3 with probability p and the second substitution from Fig. 3 with probability 1 − p. Note that from the graphs in Fig. 3, it is easy to recover the two matrices that are used to compute the trace cocycle.

Figure 9 shows the (normalized) spectrum as a function of p. It is normalized because what is plotted are the ratios $2 λ_{i}^{+} / λ_{1}^{+}$ ⁠, which are the relevant exponents in the main results of this paper. Perhaps not surprisingly, when p > 1/2, there seems to be a pair of (normalized) Lyapunov exponents greater than 1, meaning that there are non-trivial deviations of ergodic averages for tilings in a typical tiling space Ω_x with respect to the measure μ_p. In particular, as pointed out in the first item of Remark 2, this shows the rate of convergence in the Shubin–Bellissard formula for the integrated density of states for any Lipschitz kernels of finite range.

FIG. 9.

View large Download slide

Lyapunov spectrum for the measures μ_p as a function of p.

ACKNOWLEDGMENTS

I am deeply grateful to Lorenzo Sadun who pointed out a mistake in an earlier version of this paper and to Dan Rust for helpful discussions, especially bringing Godrèche and Luck, 1989 to my attention. I am also grateful to an anonymous referee for suggestions, which made the exposition of this paper much better. This work was supported by NSF Grant No. DMS-1665100.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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2021

Author(s)

Tilings and traces

I. INTRODUCTION

A. Statement of results

II. TILINGS

A. Lipschitz domains

III. BRATTELI DIAGRAMS AND TILINGS

A. Tilings from diagrams

B. Renormalization

IV. INTERLUDE: AN EXAMPLE

V. LF ALGEBRAS AND TRACES

A. The trace cocycle

VI. ERGODIC INTEGRALS

A. Proof of the upper bound (2)

B. Special averaging sets and proof of (1)

1. Implicit upper bound

2. Implicit lower bound

VII. SOLENOIDS AND THE DENJOY–KOKSMA INEQUALITY

VIII. RANDOM SCHRÖDINGER OPERATORS

A. Proof of theorem 3

IX. VARIATIONS ON HALF HEXAGONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

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Tilings and traces

I. INTRODUCTION

A. Statement of results

II. TILINGS

A. Lipschitz domains

III. BRATTELI DIAGRAMS AND TILINGS

A. Tilings from diagrams

B. Renormalization

IV. INTERLUDE: AN EXAMPLE

V. LF ALGEBRAS AND TRACES

A. The trace cocycle

VI. ERGODIC INTEGRALS

A. Proof of the upper bound (2)

B. Special averaging sets and proof of (1)

1. Implicit upper bound

2. Implicit lower bound

VII. SOLENOIDS AND THE DENJOY–KOKSMA INEQUALITY

VIII. RANDOM SCHRÖDINGER OPERATORS

A. Proof of theorem 3

IX. VARIATIONS ON HALF HEXAGONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

Citing articles via

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