Pisot family self-affine tilings, discrete spectrum, and the Meyer property

We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $\phi$ for the tiling. Assuming that $\phi$ is diagonalizable over $\C$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $\phi$ is a"Pisot family". Moreover, this is equivalent to the Meyer property of the associated discrete set of"control points"for the tiling.


Introduction
Given a self-affine tiling T of R d , we consider the tiling space, or "hull" X T , defined as the orbit closure of T in the "local" topology (please see the next section for precise definitions and statements). The translation action by R d is uniquely ergodic, so we get a measurepreserving tiling dynamical system (X T , R d , µ). We are interested in its spectral properties, specifically, in the discrete component of the spectrum which may be defined as the closed linear span of the eigenfunctions in L 2 (X T , µ). In particular, we would like to know when the tiling system is weakly mixing, which means absence of non-trivial eigenfunctions.
Our results give a complete answer to these questions in terms of the expansion matrix φ of the tiling, under the assumption that it is diagonalizable over C and its eigenvalues are algebraic conjugates of the same multiplicity. Let Λ = {λ 1 , . . . , λ d } = Spec(φ) be the set of (real and complex) eigenvalues of φ. It is known [19,26] that all λ i are algebraic integers.

Definitions and statement of results
We briefly review the basic definitions of tilings and substitution tilings (see [27,34] for more details). We begin with a set of types (or colors) {1, . . . , κ}, which we fix once and for all. A tile in R d is defined as a pair T = (A, i) where A = supp(T ) (the support of T ) is a compact set in R d which is the closure of its interior, and i = l(T ) ∈ {1, . . . , κ} is the type of T . We let g + T = (g + A, i) for g ∈ R d . We say that a set P of tiles is a patch if the number of tiles in P is finite and the tiles of P have mutually disjoint interiors. A tiling of R d is a set T of tiles such that R d = ∪{supp(T ) : T ∈ T } and distinct tiles have disjoint interiors. Given a tiling T , finite sets of tiles of T are called T -patches. For A ⊂ R d , let We always assume that any two T -tiles with the same color are translationally equivalent.
(Hence there are finitely many T -tiles up to translation.) We say that a tiling T has finite local complexity (FLC) if for each radius R > 0 there are only finitely many translational classes of patches whose support lies in some ball of radius

R.
A tiling T is said to be repetitive if translations of any given patch occur uniformly dense in R d ; more precisely, for any T -patch P , there exists R > 0 such that every ball of radius R contains a translated copy of P .
Given a tiling T , we define the tiling space as the orbit closure of T under the translation action: X T = {−g + T : g ∈ R d }, in the well-known "local topology": for a small ǫ > 0 two tilings S 1 , S 2 are ǫ-close if S 1 and S 2 agree on the ball of radius ǫ −1 around the origin, after a translation of size less than ǫ. It is known that X T is compact whenever T has FLC. Thus we get a topological dynamical system (X T , R d ) where R d acts by translations. This system is minimal (i.e. every orbit is dense) whenever T is repetitive. Let µ be an invariant Borel probability measure for the action; then we get a measure-preserving system (X T , R d , µ).
Such a measure always exists; under the natural assumption of uniform patch frequencies, it is unique, see [25]. Tiling dynamical system have been investigated in a large number of papers; we do not provide an exhaustive bibliography, but mention a few: [32,7,15,16].
They have also been studied as translation surfaces or R d -solenoids [6,12]. Definition 2.1. A vector α = (α 1 , . . . , α d ) ∈ R d is said to be an eigenvalue for the R daction if there exists an eigenfunction f ∈ L 2 (X T , µ), that is, f ≡ 0 and for all g ∈ R d and µ-almost all S ∈ X T , (2.1) f (S − g) = e 2πi g,α f (S).
Here ·, · denotes the standard scalar product in R d .
Note that this "eigenvalue" is actually a vector. In physics it might be called a "wave vector." We can also speak about eigenvalues for the topological dynamical system (X T , R d ); then the eigenfunction should be in C(X T ) and the equation (2.1) should hold everywhere.
Next we define substitution tilings. Let φ be an expanding linear map in R d , which means that all its eigenvalues are greater than one in modulus. The following definition is essentially due to Thurston [40].
we will call them prototiles. Denote by A + the set of patches made of tiles each of which is a translate of one of T i 's. We say that ω : A → A + is a tile-substitution (or simply Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the D ij to be empty. The substitution (2.2) is extended to all translates of prototiles by ω(x+T j ) = φx+ω(T j ), and to patches and tilings by ω(P ) = ∪{ω(T ) : T ∈ P }. The substitution ω can be iterated, producing larger and larger patches ω k (T j ). To the substitution ω we associate its κ × κ substitution matrix with the entries ♯(D ij ). The substitution ω is called primitive if the substitution matrix is primitive. We say that T is a fixed point of a substitution if An important question, first raised by Thurston [40], is to characterize which expanding linear maps may occur as expansion maps for self-affine (self-similar) tilings. It is pointed out in [40] that in one dimension, λ is an expansion factor if and only if θ = |λ| is a Perron number, that is, an algebraic integer greater than one whose Galois conjugates are all strictly less than θ in modulus (necessity follows from the Perron-Frobenius theorem and sufficiency follows from a result of Lind [28]). In two dimensions, Thurston [40] proved that if λ is a complex expansion factor of a self-similar tiling, then λ is a complex Perron number, that is, an algebraic integer whose Galois conjugates, other than λ, are all less than |λ| in modulus.
The following theorem was stated in [19], but complete proof was not available until recently.
Theorem 2.5. [19,20] Let φ be a diagonalizable (over C) expansion map on R d , and let T be a self-affine tiling of R d with expansion φ. Then (i) every eigenvalue of φ is an algebraic integer; (ii) if λ is an eigenvalue of φ of multiplicity k and γ is an algebraic conjugate of λ, then either |γ| < |λ|, or γ is also an eigenvalue of φ of multiplicity greater or equal to k. Remark 2.6. 1. Note that if |γ| = |λ| in part (ii) of the theorem, then the multiplicities of γ and λ are the same.
2. It is conjectured that the condition on φ in the theorem is also sufficient. There are partial results in this direction [18]; see [20] for a discussion.
For a self-affine tiling T , the corresponding tiling dynamical system (X T , R d ) is uniquely ergodic, see [27,34]. Denote by µ the unique invariant probability measure. There is a rich structure associated with self-affine tiling dynamical systems. As a side remark, we mention that the substitution map ω extends to an endomorphism of the tiling space, which is hyperbolic in a certain sense, see [3]. The partition of the tiling space according to the type of the tile containing the origin provides a Markov partition for ω. The situation is especially nice when T is non-periodic, which is equivalent to ω being invertible [36]. In order to state our results we need the following.
In this paper we assume that: • all the eigenvalues of φ are algebraic conjugates with the same multiplicity.
Let Spec(φ) be the set of all eigenvalues of φ (the spectrum of φ). By assumption, there exists a monic irreducible polynomial p(t) ∈ Z[t] (the minimal polynomial) such that p(λ) = 0 for all λ ∈ Spec(φ).
Theorem 2.8. Let T be a self-affine tiling of R d with a diagonalizable expansion map φ. Suppose that all the eigenvalues of φ are algebraic conjugates with the same multiplicity.
Then the following are equivalent: (i) The set of eigenvalues of (X T , R d , µ) is relatively dense in R d .
(iii) The system (X T , R d , µ) is not weakly mixing (i.e., it has eigenvalues other than 0). Remark 2.9. 1. In part (i) we could equally well talk about the topological dynamical system (X T , R d ) since every eigenfunction may be chosen to be continuous [37].
2. The necessity of the Pisot family condition for self-affine tiling systems that are not weakly mixing was proved by Robinson [34] in a more general case; it is a consequence of [35]. (ii) The assumption of equal multiplicity cannot be dropped from Theorem 2.8. Indeed, consider the tiling T which is a "direct product" of T 1 defined in (i) and a self-similar tiling T 2 of R with expansion λ 1 . Such a tiling T 2 exists by [28] (see [38] for more details) since λ 1 is a Perron number. Direct product substitution tilings have been studied by S. Mozes [31] and N. P. Frank [10]. It is easy to see that the set of eigenvalues for the dynamical system (X T , R 3 ) is obtained as a direct sum of those which correspond to the systems (X T 1 , R 2 ) and (X T 2 , R). By [35], the system (X T 2 , R) is weakly mixing, because λ 1 is not a Pisot number. Thus, the tiling T has expansion map φ = Diag[λ 1 , λ 2 , λ 1 ] for which Spec(φ) is a Pisot family, but the associated dynamical system does not have a relatively dense set of eigenvalues.
Next we state our result on Meyer sets. Recall that a Delone set is a relatively dense and uniformly discrete subset of R d .
There is a standard way to choose distinguished points in the tiles of a self-affine tiling so that they form a φ-invariant Delone set. They are called control points. Then define the control point for a tile T ∈ T by The control points have the following properties: , for any tiles T, T ′ of the same type; Control points are also fixed for tiles of any tiling S ∈ X T : they have the same relative position as in T -tiles. Note that the choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map.
where C i is the set of control points of tiles of type i. Equivalently, Ξ is the set of translation vectors between two T -tiles of the same type.

Preliminaries
Recall that φ is assumed to be diagonalizable over C. For a complex eigenvalue λ of φ, the can assume, by appropriate choice of basis, that φ is in the real canonical form of the linear map, see [14,Th. 6.4.2]. This means that φ is block-diagonal, with the diagonal entries equal to λ corresponding to real eigenvalues, and diagonal 2 × 2 blocks of the form a j −b j b j a j corresponding to complex eigenvalues a j + ib j .
Let m := s + 2t; this is the size of the matrix ψ. For each 1 ≤ j ≤ J, let Further, for each H j we have the direct sum decomposition Let P j be the canonical projection of R d onto H j such that We define α j ∈ H j such that for each 1 ≤ n ≤ d, The next theorem is a key result of the paper; it is the manifestation of rigidity alluded in the Introduction. Then there exists an isomorphism ρ : R d → R d such that where α j , 1 ≤ j ≤ J, are given as above.
The reason we call this "rigidity" is by analogy with [17, Th. 9] (see the discussion at the beginning of the proof in [17]).
We give a proof of Theorem 3.1 in Section 5 below and make use of it in proving the main theorem in Section 4. Note that the choice of α j is rather arbitrary; it is "hidden" in the linear isomorphism ρ. Now we continue with the preliminaries; we need to handle the real and complex eigenvalues a little bit differently. Consider the linear injective map F : R m → R s ⊕ C 2t given by In other words, identifying H j with R m , we apply the transformation S from (3.1) in every subspace E jk , k = s + 1, . . . , s + t. In view of (3.1), we have is a diagonal matrix.
The following lemma is well-known and easy to prove using the Vandermonde matrix.
Proof. We have  Proof. Identifying H j with R m , we have φ j = φ| H j ≈ ψ and use the isomorphism F defined above. In view of (3.4), all the components of z j = F(α j ) are non-zero, so the claim follows from Corollary 3.3.
For x, y ∈ R m we use the standard scalar product x, y = m k=1 x k y k , and for z, u ∈ R s ⊕ C 2t the scalar product is given by Observe that Recall also that for any m × m matrix A,

Proof of the main theorem (proof of Theorem 2.8)
Here we deduce Theorem 2.8 from Theorem 3.1. Recall that a set of algebraic integers Θ = {θ 1 , · · · , θ r } is a Pisot family if for any 1 ≤ j ≤ r, every Galois conjugate γ of θ j with |γ| ≥ 1 is contained in Θ. We denote by dist(x, Z) the distance from a real number x to the nearest integer.
Proposition 4.1. Let T be a self-affine tiling of R d with a diagonalizable expansion map φ. Suppose that all the eigenvalues of φ are algebraic conjugates with the same multiplicity.
If Spec(φ) is a Pisot family, then the set of eigenvalues of (X T , R d , µ) is relatively dense.
Proof. Recall that Ξ = {x ∈ R d : ∃ T ∈ T , T + x ∈ T } is the set of "return vectors" for the tiling T , and let K = {x ∈ R d : T − x = T } be the set of translational periods. Clearly, Let α j ∈ H j be the vectors from (3.4). Consider them as vectors in R m , and let F be the linear map R m → R s ⊕ C 2t given by (3.5). Recall that φ j = φ| H j has s real and 2t complex eigenvalues, and m = s + 2t. Define β j ∈ H j ≈ R m so that More explicitly, Note that β j ∈ H j are well-defined, and F(β j ) have all non-zero coordinates in H j . Thus, We will show that all elements of the set (ρ T ) −1 (φ T ) K B are eigenvalues for the tiling dynamical system, for K sufficiently large.
By the definition of β j , in view of (3.8) and (3.6), for any n ∈ Z ≥0 and 0 ≤ l < m, Here D is the diagonal matrix from (3.7). Since Spec(φ) is a Pisot family, it follows that dist( φ n α j , (φ T ) l β j , Z) → 0, as n → ∞. (This is a standard argument: the sum of (n+l)-th powers of all zeros of a polynomial in Z[x] is an integer, hence the distance from the sum in (4.2) to Z is bounded by the sum of the moduli of (n + l)-th powers of their remaining conjugates, which are all less than one in modulus. Thus, this distance tends to zero exponentially fast.) Observe also that φ n α u , β j = 0 if u = j, hence lim n→∞ e 2πi φ n y,(φ T ) l β j = 1 for all y ∈ Z[φ]α 1 + · · · + Z[φ]α J . Therefore, by Theorem 3.1, using that Ξ ⊂ C − C, we obtain Furthermore, by [37,Cor. 4.4], the convergence is uniform in x ∈ Ξ, that is, Recall that K ⊂ Ξ, and K is a discrete subgroup in R d . So for every x ∈ K, It follows that there exists K l ∈ Z + such that for any n ≥ K l , for all x ∈ K, However, unless φ n x, (ρ T ) −1 (φ T ) l β j ∈ Z for all x ∈ K, (4.4) does not hold. Thus e 2πi φ n x,(ρ T ) −1 (φ T ) l β j = e 2πi x,(ρ T ) −1 (φ T ) n+l β j = 1 for all x ∈ K and all n ≥ K l .
Let K = max{K l : 0 ≤ l < m}. Then So from (4.3) and (4.5) it follows that (ρ T ) −1 (φ T ) K+l β j is an eigenvalue of (X T , R d , µ) for l = 0, . . . , m−1. We have shown that all vectors of the set (ρ T ) −1 (φ T ) K B, where B = j B j and B j are given by (4.1), are eigenvalues of (X T , R d , µ). We know φ T is invertible (it is expanding), ρ is a linear isomorphism, and B is a basis of R d , hence we obtain a basis of R d consisting of eigenvalues. Integer linear combinations of eigenvalues are eigenvalues as well, so the set of eigenvalues of (X T , R d , µ) is relatively dense in R d .
The next lemma is essentially due to Robinson [34] in a more general case; we provide a proof for completeness.

Lemma 4.2.
If γ is a non-zero eigenvalue of (X T , R d , µ), then Spec(φ) is a Pisot family.
Proof. Let x ∈ Ξ. By Theorem 3.1 we have x = ρ( J j=1 p j (φ)α j ) for some polynomials p j ∈ Z[x]. Let (ρ T γ) j = p j (ρ T γ). We again use the linear injective map F : H j ≈ R m → R s ⊕ C 2t defined by (3.5) and obtain, using (3.8) and (3.6), , and c k are some complex numbers. By the assumption that γ is an eigenvalue and [35,Th. 4 Since Ξ is relatively dense in R d and γ = 0, we can easily make sure that x, γ = 0, and hence not all coefficients c k in (4.6) are equal to zero. Then we can apply a theorem of   (ii) the set of eigenvalues of (X T , R d , µ) is relatively dense; (iii) (X T , R d , µ) is not weakly mixing; (ii) ⇔ (iv) by [26,Th. 4.14].
Theorem 2.8 is contained in Theorem 4.3, so it is proved as well.

Structure of the control point set (proof of Theorem 3.1)
Now we make an isomorphic transformation τ of the tiling T into another tiling whose control point set contains α 1 , . . . , α J such that τ commutes with φ. This gives the structure of the control point set of T that we use in proving the main theorem in Section 4.
Proof. We first notice that τ is an isomorphism of R d , since Y and W are bases of R d . In order to show that φτ (x) = τ φ(x), x ∈ R d , it is enough to check this on the basis Y . For the vectors φ k y j , 0 ≤ k < m − 1, this holds by definition, so we only need to consider φ m−1 y j .
Applying the isomorphism τ commuting with φ, we can reduce our problem to the case when the control point set of the tiling contains α 1 , . . . , α J . Thus, in the rest of this section (except the last paragraph which proves Theorem 3.1), we assume that C contains α 1 , . . . , α J .
The following two propositions were obtained in [20] in a special case. They are needed to get the structure of control point set which we use in Section 4. In the appendix, we provide the proof, which is similar to that in [20], for completeness.
In the next two propositions we do not assume that all the eigenvalues of φ are conjugates and have the same multiplicity. Let G λ be the real φ-invariant subspace of R d corresponding to an eigenvalue λ ∈ Spec(φ).
Proposition 5.2. Let C be a set of control points for a self-affine tiling T of R d with an expansion map φ : R d → R d which is diagonalizable over C. Let C ∞ = ∞ k=0 φ −k C and let D be a finitely generated Q[φ]-module containing C ∞ . Let H be a vector space over R and A : H → H be an expanding linear map, diagonalizable over C. Let g : D → H be such that g(y 1 ) − g(y 2 ) = g(y 1 − y 2 ) for any y 1 , y 2 ∈ D.
Then the following hold: (i) The map f is uniformly continuous on C ∞ , and hence extends by continuity to a map f : (ii) For any λ ∈ Spec(φ) such that |λ| = γ, and any a ∈ R d , f | a+G λ is affine linear.
Let P λ be the canonical projection of R d to G λ commuting with φ, which exists by the diagonalizability assumption on φ. Denote by G ⊥ λ = (I − P λ )R d the complementary φinvariant subspace. We consider the set (I − P λ )Ξ, that is, the projection of Ξ to G ⊥ λ (recall that Ξ is the set of translation vectors between two T -tiles of the same type). In some directions this projection may look like a lattice, i.e. be discrete. We consider the directions in which this set is not discrete, and denote the span of these directions by G ′ . We will prove that f is affine linear on all G ′ slices. More precisely, for any ǫ > 0, define φΞ ⊂ Ξ and φP λ = P λ φ.
Proof. This is proved in [20] (although not stated there explicitly). Indeed, in the last part of [20], labeled Conclusion of the proof of Theorem 3.1, it is proved that the subspace G (denoted E there) contains, for each conjugate of λ greater or equal than λ in modulus, an eigenspace of dimension at least dim(G λ ). Note that in [20] the setting is more general, of an arbitrary diagonalizable over C matrix φ. In our case all eigenvalues are conjugates of the same multiplicity, and λ is the smallest in modulus, hence G contains the entire R d .
Since T has FLC, the Z-module generated by C, denoted by C Z , is finitely generated.  Proof. This is clearly a ring, so we just need to show that p(φ j ) has an inverse for p ∈ Q[x], if it is a non-zero matrix. We need to use that all the eigenvalues of φ j are conjugates so they have the same irreducible polynomial p(x). If q(x) ∈ Z[x] is monic, such that p(x) does not divide q(x), then we can find monic polynomials Observe that D j is a vector space over the field Q[φ j ], so we can write where a j1 = 1, a jt ∈ R with 1 ≤ t ≤ r j , and {a j1 , . . . , a jr j } is linearly independent over Q.
Using the same arguments as in [37, Lemma 5.3] (which followed [40]), we obtain the next lemma.
Lemma 5.6. For any ξ, ξ ′ ∈ C, Now we use Prop. 5.2, Prop. 5.3 and Lemma 5.4 to prove Theorem 3.1, and assume that all the assumptions of the latter hold. In addition, suppose that the set of control points contains α 1 , . . . , α J . Fix 1 ≤ j ≤ J. We consider the maps g = σ j : D → H j and f = σ ′ j : C ∞ → H j , and let A = φ j = φ| H j . Note that (5.4) holds with γ equal to the smallest absolute value of eigenvalues of φ j (or φ) and the norm defined as in (3.2). Thus, all the hypotheses of Prop. 5.2, Prop. 5.3 and Lemma 5.4 are satisfied, and we obtain that for each 1 ≤ j ≤ J, the (extended) map σ ′ j is linear on R d and commutes with φ.
Now we do not assume that the control point set of T contains α 1 , . . . , α J in order to prove Theorem 3.1. Instead, we apply the above propositions and lemmas to τ (C).
Proof of Theorem 3.1. By Lemma 5.7, for each ξ ∈ C, Since C is finitely generated, we multiply (5.7) by a common denominator b ∈ Z + to get where ρ is an isomorphism of R d which commutes with φ.

Appendix
We give the proofs of Prop. 5.2 and Prop. 5.3 after a sequence of auxiliary lemmas. The arguments are similar to those in [20], but we present them in a more general form for our purposes.
Denote by B R (a) the open ball of radius R centered at a and let B R := B R (0). We will also write B R (a) for the closure of B R (a). Let r = r(T ) > 0 be such that for every a ∈ R d the ball B r (a) is covered by a tile containing a (which need not be unique) and its neighbors.
Let λ max be the largest eigenvalue of φ.
Lemma 6.1. The function f is uniformly continuous on C ∞ .
Proof. This is very similar to [20,Lem. 3.4]. It is enough to show that for α = log γ log |λmax| and some L > 0 (that is, f is Hölder continuous on C ∞ ). Let ξ 1 , ξ 2 ∈ C ∞ satisfy ||ξ 1 −ξ 2 || = δ ≤ r. Then there exist y 1 , y 2 ∈ C such that φ −s y 1 = ξ 1 and φ −s y 2 = ξ 2 for some s ∈ Z ≥0 . We choose the smallest l ∈ Z ≥0 such that which is equivalent to φ s−l B δ (φ −s y 1 ) ⊂ B r (φ −l y 1 ). Since δ ≤ r, we have l ≤ s and hence l is the smallest integer satisfying |λ max | s−l δ ≤ r. Thus, Observe that y 2 ∈ φ s B δ (φ −s y 1 ) ⊂ φ l B r (φ −l y 1 ), hence φ −l y 1 and φ −l y 2 are in the same or in the neighboring tiles of T by the choice of r. It is shown in the course of the proof of [20,Lem. 3.4] that we can write y 1 − y 2 = l h=1 φ h w h , where w h ∈ W for some finite set W ⊂ φ −1 Ξ which depends only on the tiling T (a similar statement, but without precise value of l is proved in [26,Lemma 4.5]). So for some L ′ > 0 independent of l. Notice that γ l−s = (|λ max | l−s ) α , where α = log γ log |λmax| . Thus and (6.1) is proved.
Since C ∞ is dense in R d , we can extend f to a map f : R d → H by continuity, and moreover, Proof. It is enough to show that (6.3) holds for a dense subset of supp(T ), namely, C ∞ ∩ supp(T ). Suppose that ξ = φ −k c(S), where S ∈ ω k (T ). Note that Recall that A is diagonalizable over C. For θ ∈ Spec(A) let p θ : H → H be the canonical projection onto the real A-invariant subspace for A corresponding to θ, so that we have Suppose that λ ∈ Spec(φ) satisfies |λ| = γ. Lemma 6.3. For θ ∈ Spec(A) and a ∈ R d , Moreover, the Lipschitz constant is uniform in a ∈ R d (equal to C from (5.3)).
Remark 6.4. First note that |λ| = γ ≤ min{|θ| : θ ∈ Spec(A)} by (5.4). The last lemma implies that for any ξ ∈ R d and w ∈ G λ , the vector f (ξ + w) − f (ξ) is in the subspace generated by eigenspaces of A corresponding to eigenvalues θ for which |θ| = |λ|. We make use of this observation to show (6.10) in Lemma 6.6 below. From Lemma 6.3 and (6.5), we get the following corollary. Corollary 6.5. f | a+G λ is Lipschitz for any a ∈ R d .
We now prove furthermore that f is affine linear on G λ slices of R d . Lemma 6.6. f | a+G λ is affine linear for any a ∈ R d .
Proof. This is analogous to [20,Lem. 3.7], but in some places the presentation is sketchy, so we provide complete details for the readers' convenience.
Since f | a+G λ is Lipschitz for any a ∈ R d , it is a.e. differentiable by Rademacher's theorem, and hence f is differentiable in the direction of G λ a.e. in R d , by Fubini's theorem. Let t for u ∈ G λ and z ∈ R d .
The limit exists a.e. z ∈ R d and for all u ∈ G λ , and D(z)u is a linear transformation in u (from G λ to H). Moreover D(z) is a measurable function of z, being a limit of continuous functions. By the definition of total derivative, lim n→∞ F n (z) = 0 for a.e. z ∈ R d , where F n (z) = sup By Egorov's theorem, {F n } converges uniformly on a set of positive measure. This implies that there exists a sequence of positive integers N l ↑ ∞ such that has positive Lebesgue measure.
Our goal is proving that Ω has full Lebesgue measure. The argument is based on a kind of "ergodicity". First observe from Lemma 6.2 that Ω is "piecewise translation-invariant" in the following sense: Second, Ω is forward invariant under the expansion map φ. Indeed, let ξ ∈ Ω and u ∈ φ(B 1/N l ) ∩ G λ . Then This implies that D(φξ) exists and equals AD(ξ)φ −1 , and since φ( we also obtain that φ(Ω) ⊂ Ω.
We will need a version of the Lebesgue-Vitali density theorem where the differentiation basis is the collection of sets of the form φ −l B 1 , l ≥ 0, and their translates. It is well-known that such sets form a density basis, see [39, pp. 8-13]. Let y be a density point of Ω with respect to this density basis. Then where m denotes the Lebesgue measure. Note that By FLC and repetitivity, there exists R > 0 such that B R contains equivalence classes of all the patches [B 1 (φ l y)] T . Then for any l ∈ Z + , there exists y l ∈ B R such that [B 1 (y l )] T = [B 1 (φ l y)] T + (y l − φ l y).
By (6.9), we have m(Ω ∩ B 1 (y l )) ≥ (1 − ǫ l )m(B 1 ), hence m(Ω ∩ B 1 (y ′ )) = m(B 1 ) for any limit point y ′ of the sequence {y l }. We have shown that Ω is a set of full measure in B 1 (y ′ ). But then it is also a set of full measure in φ k B 1 (y ′ ) for k ≥ 1. By the repetivity of T , using (6.9), we obtain that Ω has full measure in R d .
Choose n l ∈ Z + so that |λ| n l > N l . Repeating the argument of (6.10) we obtain Thus f (ξ + v) = f (ξ) + D(ξ)v for any ξ ∈ ∞ l=1 φ n l Ω and v ∈ B 1 ∩ G λ . Note that ∞ l=1 φ n l Ω has full measure, hence it is dense in R d . So for any ξ ∈ R d , we can find a sequence {ξ j } ⊂ ∞ l=1 φ n l Ω such that ξ j → ξ. Since f | ξ j +G λ is Lipschitz with a uniform Lipschitz constant C, the derivatives D(ξ j ) are uniformly bounded, and we can assume that D(ξ j ) converges to some linear transformation D ξ by passing to a subsequence. Then we can let Since this holds for every point in R d , we obtain that D ξ = D ξ ′ for any ξ, ξ ′ ∈ R d with ξ − ξ ′ ∈ G λ , and f | ξ+G λ is affine linear for any ξ ∈ R d .
This concludes the proof of Proposition 5.2.
To prove the second claim, we just need to show G ′ ⊂ G ′′ since C 1 − C 1 ⊂ Ξ. There exists k ∈ Z + such that φ k Ξ ⊂ C 1 − C 1 (just choose k such that ω k (T 1 ) contains tiles of all types). Then as desired.
Proof of Proposition 5.3. This is similar to [20,Lem. 3.8], but again, there are some differences, and we provide more details here.
We consider the lattice generated by the y j 's in G ′ . It defines a grid with grid cells of diameter less than s max j ||y j || ≤ sǫ ′ . Thus there exist b j ∈ Z, 1 ≤ j ≤ s, such that Letζ := ζ 1 + s j=1 b j y j , so that ζ 1 + ζ 2 2 −ζ < sǫ ′ .
This completes the proof of Theorem 3.1.