Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ2

Discrete & Computational Geometry, Feb 2011

Betseygail Rand

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Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ2

Discrete Comput Geom Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in R2 Betseygail Rand 0 0 B. Rand ( ) 1000 W Court St, Seguin, TX 78155 , USA Given a tiling T , one may form a related tiling, called the derived Voronoi tiling of T , based on a patch of tiles in T . Similarly, for a tiling space X, one can identify a patch which appears regularly in all tilings in X, and form a derived Voronoi space of tilings, based on that patch. Each tiling or tiling space is defined with an associated group of rigid motions, G. In the case where G is a group of translations, a series of results have been proved demonstrating the equivalence of a tiling to its derived Voronoi tilings. Here we generalize these results: first, to groups which include both translations and rotations, and second, to tiling spaces as well. We say that two tilings, or two tiling spaces, are mutually locally derivable, or MLD, if they are related by local operations. A tiling or tiling space with a high degree of hierarchical structure is called a self-similar tiling or self-similar tiling space, respectively. A tiling or tiling space is pseudo-self-similar if it has a weaker degree of hierarchical structure. Finally, given a tiling, we are interested in the set of all derived Voronoi tilings, and whether it contains two elements related by a similar expansion. If so, the set contains a scaled pair (and the set of derived Voronoi spaces contains a scaled pair of spaces, as well). In this paper, we show: 1. A tiling and its derived Voronoi tiling are MLD, as are tiling space and its derived Voronoi space. 2. If T or X is self-similar, then its derived Voronoi tiling or derived Voronoi space has a scaled pair. 3. If a derived Voronoi tiling, or derived Voronoi tiling space, contains a scaled pair, then the original tiling or tiling space is pseudo-self-similar. 4. If a tiling or tiling space is pseudo-self-similar, then it is MLD to a self-similar tiling or tiling space. - 1 Introduction In 2000 and 2001, Natalie Priebe and Boris Solomyak published a series of results on self-similar and pseudo-self-similar tilings [ 2, 3 ]. Basically, a tiling T has an associated family of tilings, called the derived Voronoi family, F (T ). If this family is highly structured, we say it is σ -finite. Priebe [2] proved that a self-similar tiling has a σ finite family F (T ), and that any tiling with a σ -finite derived Voronoi family must be pseudo-self-similar. Priebe–Solomyak [ 3 ] showed that given any pseudo-self-similar tiling of R2, one can construct a self-similar tiling which is locally equivalent to the original. In these results, tiles are identified if they occur as translates of one another. This precludes the possibility of tilings like the pinwheel tiling, where a given tile appears in infinitely many orientations. At the time of their work, recognizability of aperiodic self-similar tilings (a necessary condition for their results) had only been established for tilings whose tiles are identified under translations. Since then, recognizability has been generalized by Holton et al. [ 1 ], in R2, for tilings with rotations. We seek to generalize the results of Priebe and Solomyak to include tilings with rotations. (Furthermore, [ 1 ] showed that self-similar tilings are not necessarily recognizable in dimension 3 and higher, and so results here are limited to R2.) Second, Priebe and Solomyak’s results involve tilings; we look to also make statements about tiling spaces. (In particular, a self-similar tiling is a fixed point of a substitution map applied to a substitution tiling space. Although we do not make use of the dynamics of a substitution tiling space here, we phrase results in terms of spaces so that they might be useful in that context.) Complications arise as we consider tiling spaces with rotations. It turns out that σ -finiteness is no longer an appropriate notion of structure for a derived Voronoi family. We develop a weaker condition, that the derived Voronoi family contains a scaled pair of tilings to replace being σ -finite. In this paper, we show that a tiling T , or space X, is locally related to all of its derived Voronoi tilings. Second, if X is a self-similar tiling space, then its derived Voronoi family has a scaled pair of orbits. If a given space X has a derived Voronoi family containing a scaled pair of orbits, then X is a pseudo-self-similar space. Finally, given a pseudo-self-similar space, we can construct a self-similar tiling space which is locally equivalent to the original. (All tilings are tilings of R2 with rotations.) In Sect. 2, we give basic definitions and conditions on tilings, and construct the derived Voronoi family of a tiling and tiling space. In Sect. 3, we state our main results. In Sect. 4, we prove that a tiling space is locally equivalent to its derived Voronoi family. In Sect. 5, we prove that a self-similar space has a scaled pair of orbits. In Sect. 6, we show that if a tiling space has a derived Voronoi family with a scaled pair of orbits, then the tiling space is a pseudo-self-similar space. Finally, in Sect. 7 we show that a pseudo-self-similar space is locally equivalent to a self-similar space. 2 Preliminaries 2.1 Tilings and Tiling Spaces Definition 1 A prototile τ , is a pair (A, l) where A is a compact set in R2 which is the closure of its interior, and l is its label. We say that the support of τ is A, supp(τ ) = A. Let Gˆ be the subgroup of the Euclidean group which is generated by rotations and translations (but not reflections.) Let G be any closed subgroup of Gˆ . G acts naturally on prototiles. Definition 2 Given a set of prototiles {τi }, a tile t is a pair {g · A, l}, where (A, l) ∈ {τi } and g ∈ G. Supp(t ) = g · A, and the label of t is l. A tiling T is a covering of R2 by tiles which overlap only on their boundaries. Definition 3 Given a tiling T , a T -patch P is a finite subset of tiles of T . In particular, for a subset A ⊂ R2 and a tiling T , we define the outer patch of A as [A]T = {t ∈ T : Supp(t ) ∩ A = ∅}. We will most often use the outer patch of a point, [x]T , where x ∈ R2, and the outer patch of an open ball of radius r , centered at a point x ∈ R2, i.e. [Br (x)]T . In Sect. 7, we will use [NR(A)]T := [∪x∈ABR(x)]T to refer to the outer patch of a neighborhood of radius R around a subset A ⊆ R2. Given a patch P , let the radius of the patch be the largest r such that Br (y) ⊂ P for some y ∈ P . Let dm be the minimum length of any edge of a tile in T . Let dM := {sup |x1 − x2| : x1, x2 ∈ ∂ti , where ti is any tile in T .} Let dθ be the minimum angle of any two incoming edges to any vertex. Two patches P1 and P2 are equivalent if there is a g ∈ G such that P1 = g · P2. We next define a metric on tilings. Two tilings are close if they match on a large ball around the origin, up to a small action g ∈ G. To be precise, define first the function ρ˜: ρ˜(T1, T2) = inf : there exists P ⊂ T1 and Q ⊂ T2 where B1/ 0 ⊂ (Supp(P ) ∩ Supp(Q)) and P = g · Q for some g ∈ G with g < where g < if, when written uniquely as its translation and rotational components such that g · P = |θ |P + |d|, both |θ | < and |d| < . We then define the distance, ρ, between two tilings: ρ(T1, T2) = min(1, ρ˜(T1, T2)). Definition 4 Given a tiling T and associated group G, a tiling space X is the orbit closure of T under G (i.e., the closure under the topology defined by ρ, of the orbit of T under G). At times, we will use the notation T to emphasize that the tiling space has been formed as the orbit closure of a particular tiling, T . 2.1.1 Useful Conditions on Tiling Spaces We restrict our attention to tiling spaces which meet the following conditions: 1. Nonperiodicity: For any T ∈ X, and any infinite order g ∈ G, we have g · T = T . 2. Almost Periodicity: X is almost periodic if, for any patch P in a tiling T ∈ X, there is a radius R > 0 such that for any y ∈ R2 and any T ∈ X, there is a g ∈ G such that g · P ⊂ [BR(y)]T . Definition 5 Given a tiling space X, the almost periodicity radius of a patch P is the least R such that for any y ∈ R2 and any T ∈ X, there is a g ∈ G such that g · P ⊂ [BR(y)]T . In other words, although our tilings never repeat exactly, every patch occurs with statistic regularity. Within any ball of radius R, we are guaranteed to find a copy of P . Notice that if a tiling space is almost periodic, then every tiling contains every patch that appears in any tiling. This implies that a tiling space is the orbit closure of every tiling it contains. 3. Finite Local Complexity, or FLC: A tiling space X has FLC if for any R > 0, there is a finite list of patches P1, . . . PN such that for any y ∈ R2, T ∈ X, [BR(y)]T = g · Pi for some i ∈ [1, . . . , N ] and g ∈ G. Lemma 6 If a tiling space has FLC, then it is compact. Proof Let {Ti } be a sequence of tilings in a space X. By FLC, there are finitely many possibilities for {[Br (0)]T }T ∈X, up to the action of G. Therefore, there is a subsequence {Tij } such that [Bij−1 (0)]Tij−1 = gij · [Bij (0)]Tij , and gij < for some > 0. Since the group G is compact on regions of bounded translations, we can take a new subsequence of the first subsequence, which converges in G as well. Therefore limk→∞ gijk [Bijk (0)]T is a tiling in X. 2.1.2 Comparing Two Tilings In the case of sequences, there is the concept of a sliding block code, enabling the conversion of one sequence into another by local operations. For two tilings, being locally derivable is the analogue of the existence of sliding block codes. Definition 7 Given two tilings, T1 and T2, we say that T2 is locally derivable from T1 if there exists an R > 0 such that if [BR(x)]T1 = g · [BR(y)]T1 with x = g · y, for g ∈ G, then [x]T2 = g · [y]T2 . If both T1 and T2 are locally derivable from each other, we say they are mutually locally derivable, or MLD. If T1 is locally derivable from T2, then the radius of derivability of T1 onto T2 is the least R for which the definition holds. Lemma 8 Being MLD is an equivalence relation. Proof First, a tiling is trivially MLD with itself. Second, if T1 and T2 are MLD tilings, then they are each locally derivable from the other, and so the property is symmetric. Finally, if T2 is locally derivable from T1 with radius R1, and T3 is locally derivable from T2 with radius R2, then T3 is locally derivable from T1 with radius R1 + R2. Lemma 9 If T1 and T2 are MLD tilings, and T1 is nonperiodic, almost periodic, and has finite local complexity, then T2 has these properties as well. Proof Suppose that T2 is periodic, and let g ∈ G be an infinite order element such that g · T2 = T2. Let T1 be locally derivable from T2 with radius R. For all x ∈ R2, g · [BR(x)]T2 = [BR(g · x)]T2 , and so g · [x]T1 = [g · x]T1 . Therefore T1 is periodic, which is a contradiction. Let T2 be locally derivable from T1 with radius R . Let [Bd (x)]T2 be a patch in T2. Since T1 is almost periodic, patches equivalent to [B(d+R )(x)]T1 appear regularly throughout T . If [B(d+R )]T1 (y) = g · [B(d+R )(x)]T1 for some element g ∈ G, then [Bd (y)]T2 = g · [Bd (x)]T2 . Thus [Bd (x)]T2 occurs throughout T2 with the same regularity as [B(d+R )(x)]T1 occurs in T , and so T2 is almost periodic. Finally, again let T1 be locally derivable from T2 with radius R, and T2 be locally derivable from T1 with radius R . Suppose T2 lacks finite local complexity: there is a d > 0 such that {[Bd (xi )]T2 } is an infinite sequence of patches, no two of which are equivalent. Since T1 has finite local complexity, {[Bd+R (xi )]T1 } has a finite list of representatives, up to action by G. Therefore there are patches [Bd (xi )]T2 and [Bd (xj )]T2 which are not related by an action by G even though [Bd+R (xi )]T1 = g · [Bd+R (xj )]T1 , which is a contradiction. Definition 10 (MLD Tiling Space) Given two tiling spaces X and Y with the same associated group G, we say Y is locally derivable from X if there exists a map f : X → Y which commutes with the action of the group, and a radius r such that for any T1, T2 ∈ X, [Br (0)]T1 = [Br (0)]T2 implies [0]f (T1) = [0]f (T2). Two tiling spaces are MLD if each is locally derivable from the other. The following theorem will allow us to readily extend many results about tilings to results about the spaces containing them, throughout this paper. Theorem 11 Let T be the space formed by taking the orbit closure of T under a group G, and S be the space formed by taking the orbit closure of S, under the same group G. If T and S are MLD, then T and S are MLD. Proof Let T and S be MLD with radius R and define f (T ) = S. We extend f : T → S as follows: if T1 ∈ T , then T1 = limi→∞ gi · T for some sequence {gi } ∈ G. Definef (T1) := limi→∞ gi · S. Note that this limit exists, and is independent of the sequence {gi }: for any > 0, eventually ρ(gi · T , gj · T ) < 1+R , which implies that ρ(gi · S, gj · S) < . By construction, f commutes with the action of G. Suppose T1 = limi→∞ gi · T and T2 = limj→∞ hj · T are tilings in T , and suppose [BR(0)]T1 = [BR(0)]T2 . Then ∀ > 0 there exists some N such that n > N implies that [BR(0)]gn·T = kn · [BR(0)]hn·T , where kn < . Therefore [BR(gn · 0)]T = gnknhn−1 · [BR(hn · 0)]T , which implies [gn · 0]S = gnknh−1 · [hn · 0 S n ] , and so [0]gn·S = kn · [0]hn·S for all n > N . We conclude that 0 f (T1) = [0]f (T2) as n → ∞. [ ] Definition 12 Let Ψ : A → R2 be a function for some A ⊂ R2. The map Ψ is locally derivable from a tiling T if there is some radius r such that whenever [Br (x)]T = g · [Br (y)]T with x = g · y, Ψ (x) = g · Ψ (y). 2.1.3 Self-similarity and Pseudo-self-similarity of Tilings and Tiling Spaces We next describe tilings with a high degree of hierarchical structure. Definition 13 A tiling T is self-affine if there exists a linear expansion φ : R2 → R2 such that 1. φ (t ), the image of the support of any tile t ∈ T , is composed of a union of tiles in T . 2. If, for two tiles in T , t1 = g · t2, then φ (t1) = g · φ (t2), where g = φgφ−1, and φ (ti ) represents the union of tiles in T which compose the image of ti under φ. We note that if we label the image φ (ti ) by the label of ti , then φ (T ) is also a tiling, composed of supertiles. In particular, we will be concerned with a subset of self-affine tilings: Definition 14 A self-affine tiling T is self-similar if all eigenvalues of the expansion map φ have equal modulus. In this case, we can define the expansion factor λ = √| det φ|. Definition 15 A self-similar tiling, T , with expansion map φ, is recognizable if there is an R > 0 such that [BR(x)]T = g · [BR(y)]T implies [x]φ(T ) = g · [y]φ(T ). The radius R is called the recognizability radius. Theorem 16 [ 1 ] In dimensions ≤ 2, self-similar tilings are recognizable. Next, we define tilings which are almost self-affine and self-similar: Definition 17 A tiling T is pseudo-self-affine if there exists a linear expansion φ such that φ (T ) is MLD with T . If φ is a similarity, then T is pseudo-self-similar. 2.2 Construction of Derived Voronoi Tilings First, let “cell” and “tile” be synonymous words, where “cell” is reserved for the tiles of a derived Voronoi tiling. This will simplify discussions involving both a tiling and a derived Voronoi tiling. Given a tiling and a patch in that tiling, the derived Voronoi tiling is a new tiling formed which “sees” the frequency of the patch throughout the tiling. Formally, we first fix a tiling T (with associated group G), a point s ∈ R2, and a radius r > 0. Let P refer to the patch [Br (s)]T . The associated derived Voronoi tiling will be denoted V(T ,s,r). We locate each copy of the patch P throughout T , by defining the locator set L(T ,s,r): L(T ,s,r) = q ∈ R2 such that [Br (q)]T = g · [Br (s)]T and q = g · s, where g ∈ G . Note that if P has m-fold rotational symmetry, then either the point s will be the locus of symmetry, or there will be m locator dots, all indicating the presence of the same patch. Also note that since T is almost periodic, points in L(T ,s,r) occur regularly. The almost periodicity radius, R(T ,s,r), of P , guarantees that every disc of radius R(T ,s,r) will contain an element of L(T ,s,r). The derived Voronoi tiling is formed as the dual of the locator set. Recall that a prototile t is an ordered pair (A, l), where A is a compact set, and l is the associated label. We need to describe the construction of both the compact sets and labels of the prototiles of a derived Voronoi tiling. Let p ∈ L(T ,s,r). Define Ap, the compact set containing p, by: Ap = x ∈ R2 such that d(x, p) ≤ d(x, p ) for all p ∈ L(T ,s,r) . This set is compact, and the set of cells will cover R2 and overlap only on their boundaries. Each cell is a convex polytope. Observe that the edges of the cell are determined by the locations of all neighboring locator dots. If p, q ∈ L(T ,s,r) belong to adjacent cells, then d(p, q) ≤ 2(R(T ,s,r) + dM ). (Recall that the definition of the almost periodicity radius is based on a patch of tiles: for any y ∈ R2, there is a g ∈ G such that g · P ⊂ [BR(T,s,r) (y)]. Therefore, a point on the boundary of two Voronoi cells could be within [BR(T,s,r) (p)] and [BR(T,s,r) (q)] but within neither BR(T,s,r) (p) nor BR(T,s,r) (y). Therefore d(p, q) ≤ 2(R(T ,s,r) + dM ).) Therefore the shape of the Voronoi cell containing the locator dot p is determined by [B4(R(T,s,r)+dM )(p)]T . By finite local complexity of T , there are finitely many shapes of cells in V(T ,s,r). Next we label the cells. One of the key features of a derived Voronoi tiling is that the labels of the prototiles carry significant information about the behavior of the tiling around that prototile. Because of this, the labeling procedure is somewhat involved. Labeling the cells of a derived Voronoi tiling Let T be a tiling, and let d be a translation in G with |d | > 0. Since the action is strictly translation, and has no rotation, we will temporarily denote the operation by +. [B(r−|d|)(x)]T + d = [B(r−|d|)(x + d )]T . Definition 18 Let [Br (x)]T be a patch of a tiling T . The translational symmetry of [Br (x)]T is the smallest absolute value |d |, where d is a translation such that Lemma 19 For a tiling T and real number m, there is an upper bound on the radius r of patches [Br (x)]T with translational symmetry of size |d | ≤ m. Proof Suppose not. Then there is a sequence of patches [Bri (xi )]T with translational symmetry of size ≤ m, such that ri → ∞. Since T is compact, we can find a convergent subsequence [Brij (xij )]T , and define T = limj →∞[Brij (xij )]T . Then T is periodic and in the orbit closure of T , which contradicts the requirement that our tiling spaces be aperiodic. Fix T , a point s, a radius r , and let R(T ,s,r) be the almost periodicity radius of [Br (s)]T . Let M(T ,s,r) be the least upper bound on the radius of any patch in T with translational symmetry of size ≤ 16(R(T ,s,r) + dM ). We will use R˜(T ,s,r) := M(T ,s,r) + 5(R(T ,s,r) + dM ) as the labeling radius for V(T ,s,r). By finite local complexity, there are a finite number of patches {H1, . . . , Hn}, where each Hi = [BR˜(T ,s,r) (pi )]T , such that for all x ∈ R2, [BR˜(T ,s,r) (x)]T = g · Hi for some g ∈ G, 1 ≤ i ≤ n. We will use these patches {Hi } to label our cells. Let a cell containing a locator point p ∈ L(T ,s,r) have label Hi , where Hi = g · [BR˜(T ,s,r) (p)]T for some g ∈ G. V(T ,s,r) is now a covering of R2 by tiles. (We point out that in Priebe [ 2 ], derived Voronoi cells are labeled based on a much smaller radius, defined to be twice the almost periodicity radius of the patch. We are not claiming that our labeling radius is optimal, but it is sufficient to address the complications arising from including rotations. Note that relabeling the cells of a derived Voronoi tilings with a larger labeling radius creates a second tiling which is locally derivable from the first.) In Sect. 4, we will show that V(T ,s,r) and T are MLD. By Lemma 9, it will follow that V(T ,s,r) is nonperiodic, almost periodic, and has finite local complexity. 2.3 Derived Voronoi Families In [ 2 ], Priebe defined the derived Voronoi family of a tiling to be the set {V(T ,0,r)}r>0, the set of all derived Voronoi tilings formed by taking patches centered at the origin. We will also use this set, but we need to distinguish it from larger sets of derived Voronoi tilings. We define the derived Voronoi family of a tiling to be F (T ) = V(T ,s,r) : s ∈ R2, r > 0 . Fig. 1 Two discs whose outer patches are equivalent We define the derived Voronoi family of a tiling space, X, as F (X) = V(T ,s,r) : s ∈ R2, r > 0, T ∈ X . Definition 20 Two tilings, T and S, are a scaled pair if there exists a linear expansion φ such that for every tile ti ∈ T , φ (ti ) is a tile in S, and φ induces a bijection between the labels of tiles in T and the labels of tiles in S. We denote this T = φS. We do not require that φ be a similarity. Next, we introduce a hierarchical structure to derived Voronoi families. In [ 2 ], the hierarchical structure is that {V(T ,0,r)}r>0 be σ -finite. We restate the definition here: Definition 21 A set of tilings, F , is σ -finite if there exists a finite subset of tilings {S1, . . . , Sn}, and a linear expansion φ, such that for every T ∈ F , T = φk (Si ), for some k and i. In [ 2 ], it was shown that for self-similar tilings T , the associated set {V(T ,0,r)}r>0 is σ -finite. However, this property is no longer a reasonable measure of hierarchy when we consider F (X), of a tiling space X that includes rotations. Derived Voronoi tilings built under very similar specifications may look wildly different. We explain this point fully: Consider two patches in the same tiling, centered apart, with the same radius such that [Br (s)]T = [Br (s + )]T . This should be readily available; see Fig. 1 for illustration. The locator sets L(T ,s,r) and L(T ,s+ ,r) will both locate the same patches throughout R2. Elements of these locator sets occur in pairs, one from each set, spaced -apart. (See Fig. 2.) If G contains an infinite rotation group, then copies of this patch will be identified in infinitely many different orientations. Therefore, the positions of the -pairs of locator dots will be rotated according to the orientation of the patch containing them. But this will vastly affect the edges of cells in V(T ,s,r) and V(T ,s+ ,r). (See Fig. 3.) Because of this, the hierarchical structure that a family be σ -finite is no longer reasonable. Instead we will use the condition that a derived Voronoi family contains a scaled pair of tilings. Fig. 2 Elements of L(T ,s,r) and L(T ,s+ ,r) occur in pairs There are three key findings in [ 2 ] and [ 3 ] which we seek to generalize. The following were all proved for tilings whose associated group G is a translation group. In [ 2 ], Priebe shows that a self-similar tiling has a σ -finite derived Voronoi family of tilings. Second, any tiling with a σ -finite derived Voronoi family must be pseudo-self-similar. In [ 3 ], Priebe and Solomyak showed that every pseudo-self-similar tiling is MLD to a self-similar tiling. In this paper, we will prove the following, for tilings and tiling spaces whose associated group G may include both translations and rotations. Theorem 22 For any patch [Br (s)]T , V(T ,s,r) and T are MLD. Theorem 23 Let T be a self-similar tiling, with associated group G. Then the set {V(T ,0,r)} is σ -finite. Theorem 24 Let T be a self-similar tiling, and X the space formed by taking the orbit closure of T under the associated group G. Then both F (T ) and F (X) contain a scaled pair of tilings. Theorem 25 Let F (T ) have a scaled pair of tilings. Then T is a pseudo-self-affine tiling. If φ, the expansion map for the scaled pair, is a similarity, then T is a pseudoself-similar tiling. Likewise, let F (X) have a scaled pair. Then X is a pseudo-selfaffine tiling space, and if φ is a similarity, then X is pseudo-self-similar. Theorem 26 If X is a pseudo-self-similar tiling space, then it is MLD to a selfsimilar tiling space, X . 4 A Tiling is MLD to Any of Its Derived Voronoi Tilings We now prove the close relationship of a tiling to any of its derived Voronoi tilings. Theorem 22 Let P = [Br (s)]T , and V(T ,s,r) = VP . Then VP and T are MLD. Proof of Theorem 22 Throughout the proof, let R(T ,s,r), R˜(T ,s,r), and L(T ,s,r) be abbreviated by RP , R˜P , and LP , respectively, with the understanding that P = [Br (s)]T is fixed. Recall that RP is the almost periodicity radius of P , R˜P is the labeling radius of VP , and LP is the set of locator dots of P in VP . 1. VP is locally derivable from T . [x]VP = g · [y]VP . Suppose [B(R˜P +RP )(x)]T = g · [B(R˜P +RP )(y)]T and x = g · y, for some g ∈ G, and x, y points in R2. First, assume that there is a unique nearest locator dot to each of x and y. Let px and py be the nearest locator dots, respectively. Then [BR˜P (px )]T = g · [BR˜P (py )]T . Since R˜P > 2RP + 2dM , we know that these patches contain all neighboring locator dots to px and py , and hence [px ]VP and [py ]VP have the same shape. Since the discs around px and py match out to the labeling radius R˜P , the cells [px ]VP and [py ]VP have the same label. Hence [x]VP = g · [y]VP . If x and y are each equidistant to n locator dots, let {pxi } be the set of locator dots nearest to x, and {pyi } the set nearest to y, ordered so that pxi = g · pyi . Then for each i, [BR˜P (pxi )]T = g · [BR˜P (qxi )]T , and so [pxi ]VP = g · [pyi ]VP . Therefore 2. T is locally derivable from VP . We want to show that T is locally derivable from VP with local derivability radius R˜P + 2RP . In other words, that [B(R˜P +2RP )(x)]VP = g · [B(R˜P +2RP )(y)]VP and x = g · y imply that [x]T = g · [y]T . First, assume x and y are interior to the cells which contain them. Let [x]VP and [y]VP each have label Hi and contain locator dots px and py , respectively. The label Hi represents a fixed patch [BR˜P (pi )]T , for some pi ∈ LP , and a cell with locator dot pj has label Hi if [BR˜P (pj )]T is equivalent to [BR˜P (pi )]T . Therefore, [BR˜P (px )]T and [BR˜P (py )]T are both equivalent to [BR˜P (pi )]T , and so to each other. In other words, we know that [BR˜P (px )]T = h · [BR˜P (py )]T , for h ∈ G, and we must show that h = g. If the labeling radius were not chosen with sufficient care, it might happen that h = g. This could happen if a patch of tiles in T gave rise to a patch of cells in VP , and the patch in VP had rotational symmetry that the patch of tiles in T lacked. For example, in Fig. 4, two orientations of the same set of locator dots give rise to a derived Voronoi cell with rotational symmetry. If x = g · y and [BR˜P (px )]T = h · [BR˜P (py )]T , but g = h, then [x]T and [y]T will generally be different tiles. Let Di be the set of edges which form the dual to LP |Hi , after deleting any infinite edges. We say that an element hx ∈ G aligns Hi = [BR˜P (pi )]T with [x]VP if: 1. hx · pi ∈ [x]VP , and 2. the set of edges of hx · Di is a subset of the set of edges of [B(R˜P +2RP )(x)]VP . Let {hx } be the set of elements of G which align Hi with [x]VP , and {hy } be the set of elements in G which align Hi with [y]VP . Suppose |{hx }| = |{hy }| = 1. Then hy hx−1 : [BR˜P (px )]VP → [BR˜P (py )]VP uniquely. Therefore hy hx−1 = g, and so [BR˜P (px )]T = g · [BR˜P (py )]T . Therefore [x]T = g · [y]T . Suppose |{hx }| = |{hy }| = n > 1. That is, [x]VP and [y]VP have rotational symmetry of order n. Let [x ]VP share an edge with [x]VP and have label Hj . Then [y ]VP shares an edge with [y]VP and also has label Hj , where y = g · x (since [B(R˜P +2RP )(x)]VP = g · [B(R˜P +2RP )(y)]VP ). Let {hx } ∈ G and {hy } ∈ G be the set of rigid motions that align Hj with [x ]VP and [y ]VP , respectively. If |{hx }| = |{hy }| = 1, then hy hx−1 : [BR˜P (px )]VP → [BR˜P (py )]VP uniquely. Therefore hy hx−1 = g, and so [BR˜P (px )]T =g ·[BR˜P (py )]T . Since x ∈ [BR˜P (px )]T and y ∈ [BR˜P (py )]T , we can conclude that [x]T = g · [y]T . Lemma 27 It is not possible that both |{hx }| = |{hy }| > 1 and |{hx }| = |{hy }| > 1. Assuming the proof of Lemma 27, Theorem 22 follows immediately for any points x and y interior to cells in VP . If x and y lie on the boundary of more than one cell, then Lemma 27 implies that one of the cells lacks rotational symmetry. The argument above can be applied using the label of that cell. During the proof of Lemma 27, we say that hx · Hi and hx · Hj are consistent if for every w ∈ (hx · Hi ) ∩ (hx · Hj ), hx [hx−1(w)]T = hx [hx−1(w)]T . In other words, all edges, vertices and labels of hx · Hi match edges, vertices, and labels of hx · Hj , on their intersection. Proof of Lemma 27 A region with two loci of rotational symmetry has a translational symmetry. Suppose T is a tiling with two points of rotational symmetry of the entire plane: rotating T by θ1 around p1 maps T to itself, and rotating T by θ2 around p2 maps T to itself. Then applying the composition θ2−1θ1−1θ2θ1 to the tiling T results in a translation of length ≤ 4|p1 − p2|. In other words, θ2−1θ1−1θ2θ1(T ) = T + d , where |d| ≤ 4|p1 − p2|. (For this reason, a tiling cannot have two points of rotational symmetry of the whole plane, because the tiling would then be periodic.) Suppose |{hx }| = |{hy }| > 1 and |{hx }| = |{hy }| > 1. Let hx1 , hx2 ∈ {hx } and ahnx1d,hhxx22 ·∈H{jh,xh}x.2T·hHeni athnedrehxa1re· Hfoju,racnodnshixs2te·nHtpiaainrsd: hhxx21 ·· HHji .a(nTdhehrxe1 a·rHe jf,ouhrx1m·oHrei pairs near y ∈ R2 with which we are not concerned.) First, hx1 · Hj is consistent with both hx1 · Hi and hx2 · Hi . Therefore (hx1 · Hj )|hx1 ·Hi has rotational symmetry under hx1 hx−21. Let hx1 hx−21 be a rotation of angle θx around qx . Similarly, hx1 · Hi is consistent with both hx1 · Hj and hx2 · Hj . Therefore (hx1 · Hi )|hx1 ·Hj also has rotational symmetry under hx1 hx−1. Let hx h−1 2 1 x2 be a rotation of angle θx around qx . Let F = (hx1 · Hi ) ∩ (hx1 · Hj ) ∩ (hx2 · Hi ) ∩ (hx2 · Hj ). Since the cells Hi and Hj share an edge in VP , we know that |px − px | < 2RP + 2dM . Therefore F has radius > R˜P − 4(RP + dM ). Since hx1 hx−21 : [x]VP → [x]VP , the locus of rotation qx is within RP + dM of the locator point px . Likewise, [x ]VP is fixed under hx h−1, and so qx is within RP + dM 1 x2 of px . Therefore the two loci of rotational symmetry of F are at most 4(RP + dM ) apart, and so F has translational symmetry of size ≤ 16(RP + dM ). We have, then, that F has radius > R˜P − 4(RP + dM ) in T and translational symmetry of size ≤ 16(RP + dM ). Recall that MP is the least upper bound on the radius of any patch in T with translational symmetry of size ≤ 16(RP + dM ). Therefore R˜P − 4(RP + dM ) < MP . However, by definition R˜P = MP + 5(RP + dM ), which is a contradiction. 5 A Self-similar Tiling Space has a Scaled Pair of Orbits The next two theorems explore the hierarchical structure associated with the derived Voronoi families of a self-similar tiling or a self-similar tiling space. Recall that there are three variations of families. First, we will use the set {V(T ,0,r) : T is fixed, r > 0}. (Note that Priebe, in [ 2 ], referred to this as the derived Voronoi family of the tiling T . We are reserving that term for the next set, and will simply refer to this set as {V(T ,0,r)}.) Second, the derived Voronoi family of a tiling T is the set F (T ) := {V(T ,s,r) : s ∈ R2, r > 0}. Finally, the derived Voronoi family of a tiling space X will refer to the set F (X) := {V(T ,s,r) : T ∈ X, s ∈ R2, r > 0}. Once we have that {V(T ,0,r)} is σ -finite, it will be immediate to show that F (T ) and F (X) both have a scaled pair of tilings. We now will prove Theorem 23, which we restate here: Theorem 23 Let T be a self-similar tiling. Then the set {V(T ,0,r)} is σ -finite. We wish to emphasize that the first half of this proof—that there are finitely many locator sets up to scale—is a modified version of the proof by Priebe [ 2 ], that {V(T ,0,r)} is σ -finite when G is a translation group. We begin with a lemma, which appears first in [ 2 ] for tilings defined under a translation group. Lemma 28 Let T be a self-similar tiling with expansion map φ, and let λ be the scaling factor of φ, and let T have recognizability radius ρ. Let l be the least integer such that λl (λ − 1) > 1. Then the recognizability radius of the image of T under the map φk is bounded above by λk+l ρ. Proof of Lemma 28 Because T is recognizable, [Bρ (x)]T = g · [Bρ (y)]T with x = g · y implies [x]φ(T ) = g · [y]φ(T ). Applying φk−1 to the tiling gives us that for all x, y ∈ R2, [Bλk−1ρ (x)]φk−1T = g · [Bλk−1ρ (y)]φk−1T with x = g · y implies that [x]φkT = g · [y]φkT . In other words, a radius of λk−1ρ around x is sufficient in φk−1T to identify the kth order tile containing x. Likewise, a radius of λk−2ρ around x is sufficient in φk−2T to identify the (k − 1)st order tile containing x. Therefore, a radius of λk−1ρ + λk−2ρ is sufficient in φk−2T to identify the kth order tile containing x. Continuing to break each tile into smaller pieces, we get that a radius of jk=−01λj ρ around x in T is sufficient to determine which kth order supertile contains x. Finally, note that jk=−01λj = λλk−−11 < λλ−k1 = (λλ−k1λ)lλl . Since l was chosen such that the denominator is greater than 1, we get k−1λj ρ < λk+l ρ. j=0 0 T be the elementary patch containing the oriProof of Theorem 23 Let E = [ ] gin, and L(T ,0,0) be denoted LE . Consider the outer patch of radius λl+1ρ around each locator dot q ∈ LE . By finite local complexity, there is a maximal list {[Bλl+1ρ (q1)]T , . . . , [Bλl+1ρ (qn)]T }, such that [Bλl+1ρ (qi )]T = g · [Bλl+1ρ (qj )]T for any g ∈ G, i = j . Let the patch [Bλl+1ρ (qi )]T be denoted Fi , and let the locator set L(T ,qi ,λl+1ρ) be denoted LFi . Subsets of {LF1 , . . . , LFn } will form the locator sets for the finite set of tilings that yield all of {V(T ,0,r)}, under expansion by powers of φ. Let r > 0, and let k be the integer such that λk+l ρ < r ≤ λk+l+1ρ. (If r < λl+1, let k = 0.) Since r is larger than the recognizability radius of φk , we have that q ∈ L(T ,0,r) implies that [q]φkT = g · [0]φkT = g · φk[0]T for some g ∈ G. Therefore q is in φkLE . Since LE has been partitioned into {LF1 , . . . , LFn }, let {LFi1 , . . . , LFim } be the subset of LE such that L(T ,0,r) ∩ φkLFij = ∅. We have then that L(T ,0,r) ⊆ φk{LFi1 ∪ · · · ∪ LFim }. Next we show that φk{LFi1 ∪ · · · ∪ LFim } ⊆ L(T ,0,r). Let q ∈ φkLFij for some j . Since L(T ,0,r) ∩ φkLFij = ∅, let p be a point in the intersection. Thus [Br (p)]T = g · [Br (0)]T for some g, and [Bλk+l+1ρ (p)]T = g · [Bλk+l+1ρ (q)]T for some g . Since r ≤ λk+l+1ρ, [Br (q)]T = g −1g · [Br (0)]T , and so q ∈ L(T ,0,r). Therefore L(T ,0,r) = φk{LFi1 ∪ · · · ∪ LFim }. Since there are finitely many subsets of {LF1 , . . . , LFn }, there are finitely many possibilities for locator sets L(T ,0,r), up to expansion by a power of φ. Finally, we must show that this set of locator sets yields finitely many tilings, up to scale by φk , when we label the derived Voronoi cells. We claim that there is an upper bound on the number of labels of any tiling in {V(T ,0,r)}. Lemma 29 For r > 1, R˜(T ,0,r) < Cr for some constant C. Proof Recall that R˜(T ,0,r) := M(T ,0,r) + 5R(T ,0,r) + 5dM , where M(T ,0,r) is the least radius of a patch with translational symmetry ≤ 16(R(T ,0,r) + dM ). First we claim that R(T ,0,λr) < λR(T ,0,r). Note that [Bλr (0)]T ⊂ φ[Br (0)]T as subsets of R2, and so [Bλr (0)]T ⊂ [φ[Br (0)]T ]T as patches in T . Therefore the almost periodicity radius of [Bλr (0)]T is less than or equal to the almost periodicity radius of [φ[Br (0)]T ]T . While the latter does not lend itself to R(T ,0,∗) notation, any patch of radius λR(T ,0,r) is guaranteed to contain a copy of [φ[Br (0)]T ]T . We conclude that R(T ,0,λr) < λR(T ,0,r). Let k1 = max{ R(Td,0,d) : 0 < d ≤ λ}. Since R(T ,0,r) is an increasing function of r , we have R(T ,0,r) < k1r . Next we claim that M(T ,0,r) grows linearly with r . The following proposition is from [ 1 ]. A patch is called admissible if it is contained in T , and has period g ∈ G if P ∩ g · P is a patch. Proposition 30 There is a constant k2 > 0 such that if P is an admissible patch whose support contains a ball of radius r , then every non-identity period g of P satisfies gb − b > k2r for some b ∈ supp(P ). Clearly, if g is a translation and P is a patch with period g, then gb − b is constant for all b ∈ supp(P ). By Proposition 30, if P is an admissible patch whose support contains a ball of radius M(T ,0,r), then every translation of P has size > k2M(T ,0,r). Since some such patch exists with size M(T ,0,r) and translational symmetry d ≤ 16(R(T ,0,r) + dM ), we have 16(R(T ,0,r) + dM ) ≥ d > k2M(T ,0,r). Therefore M(T ,0,r) < (16/ k2)(R(T ,0,r) + dM ) < (16/ k2)(k1r + dM ). When r > 1, (16/ k2)(k1r + dM ) < (16/ k2)(k1 + dM )r . Therefore for r > 1, R˜(T ,0,r) < Cr , where C = (16/ k2)(k1 + dM ) + 5k1 + 5dM . We now prove that there is an upper bound on the number of labels of any tiling in {V(T ,0,r)}. Recall that ρ is the recognizability radius of T , and let N be the number of patches of radius ρ in T , up to action by G. Pick some derived Voronoi tiling V(T ,0,s) ∈ {V(T ,0,r)}, and let {q1, . . . , qm} ∈ L(T ,0,s) be a maximal set of locator dots such that every tile in V(T ,0,s) is represented by a unique qi . Let n be such that ρλn−1 < R˜(T ,0,s) ≤ ρλn, and consider the set of patches {[Bλ−nR˜(T,0,s) (φ−nqi )]T }. Since λ−nR˜(T ,0,s) ≤ ρ, this set contains at most N distinct elements, up to action by G. If [Bλ−nR˜(T,0,s) (φ−nqi )]T = g · [Bλ−nR˜(T,0,s) (φ−nqj )]T for some i = j , then [BR˜(T,0,s) (qi )]φnT = φngφ−n · [BR˜(T,0,s) (qj )]φnT . Therefore there are at most N distinct patches in the set {[BR˜(T,0,s) (qi )]φnT } up to action by G. It does not follow that [BR˜(T,0,s) (qi )]T = φngφ−n · [BR˜(T,0,s) (qj )]T , since there may be several patches in T whose outer patch in φnT is composed of the same set of supertiles. We claim that there is a bound on the density of locator dots, and so we can cap the number of possible labels available within [BR˜(T,0,s) (qi )]φnT . Fix qi , and let {h1, . . . , hz} ∈ G be all distinct elements such that hi · [Bs (0)]T is admissible and contained within [BR˜(T,0,s) (qi )]φnT . Claim: 256π M2/ k23 is a (crude) upper bound on z, where M = (C + Cλ2dM /ρ), and C and k2 are from above. (The following is due to James B. Shearer, personal communication, August 2010.) First, [BR˜(T,0,s) (qi )]φnT is contained within the disk BR˜(T,0,s)+λn+1dM (qi ), and R˜(T ,0,s) + λn+1dM < Cs + λ2CsdM /ρ = s(C + Cλ2dM /ρ) = sM . Next, BsM (qi ) is contained in the square of side 2sM , centered at qi . Suppose z > 256π M2/ k23. If we can show that there are two patches, hi · [Bs (0)]T and hj · [Bs (0)]T , such that hi · 0 − hj · 0 < k2s/2, with relative angle θ < k2/2, then we will have demonstrated a contradiction. (If g : hi · [Bs (0)]T → hj · [Bs (0)]T , then for all b ∈ hi · [Bs (0)]T , gb − b < k2s/2 + s(k2/2), which violates Proposition 30.) If there are more than 4π/ k2 images of {hi · [Bs (0)]T } contained within a square of side length k2s/4, then all of them will have centers which are at most k2s/2 apart, 2π and at least two of them will have relative angle < 4π/k2 = k2/2. There are at most 64M2/ k22 non-overlapping squares of side length k2s/4 contained within our square of side length 2sM , centered at qi . Therefore, we cannot have (64M2/ k2s)(4π/ k2) images of {hi · [Bs (0)]T } within our square of side length 2sM , and so we conclude that z ≤ 256π M2/ k23. In total, V(T ,0,s) must have fewer than 256π M2N / k23 labels, and so this bound does not depend on choice of s. Consider all possible subsets of {LF1 , . . . , LFn } that are actually realized by some L(T ,0,r) = φk(LFi1 ∪ · · · ∪ LFim ). For each subset, there is a finite number of ways to label the cells with fewer than 256π M2N / k23 labels. Therefore we have finitely many derived Voronoi tilings, up to expansion by φk . This concludes the proof of Theorem 23. As described in Sect. 2.3, F (T ) and F (X) are not necessarily σ -finite if the associated group contains rotations. However, since {V(T ,0,r)} ⊂ F (T ) ⊂ F (X), both F (T ) and F (X) have a scaled pair of derived Voronoi tilings. Therefore this theorem follows immediately: Theorem 24 Let T be a self-similar tiling, and X the space formed by taking the orbit closure of T under a group G. Then both F (T ) and F (X) contain a scaled pair of derived Voronoi tilings. 6 A Scaled Pair of Voronoi Tilings or Orbits Implies Pseudo-self-similarity In this section, we do not require that a tiling T be self-similar, or that a space X be a self-similar tiling space. Instead, we start with the condition that F (T ) and F (X) have a scaled pair of derived Voronoi tilings. We will show that if F (T ) and F (X) each have a scaled pair, then T and X, respectively, are pseudo-self-affine or pseudoself-similar. We restate Theorem 25 for convenience: Theorem 25 Let F (T ) have a scaled pair. Then T is a pseudo-self-affine tiling. If φ, the expansion map for the scaled pair, is a similarity, then T is a pseudo-self-similar tiling. Likewise, let F (X) have a scaled pair. Then X is a pseudo-self-affine tiling space, and if φ is a similarity, then X is pseudo-self-similar. Proof F (T ) contains tilings V(T ,s1,r1) and V(T ,s2,r2) where V(T ,s1,r1) = φ (V(T ,s2,r2)), and φ is a linear expansion. From Theorem 22, we have that T is MLD to both of these derived Voronoi tilings. Since T is MLD to V(T ,s2,r2), φ (T ) is MLD to φ (V(T ,s2,r2)). Because φ (V(T ,s2,r2)) = V(T ,s1,r1), and V(T ,s1,r1) is MLD to T , we have that φ (T ) is MLD to T . Therefore T is pseudo-self-affine, and if φ is a similarity, then T is pseudo-selfsimilar. Schematically: MLD V(T ,s1,r1) ←→ T Applying φ : φ (T ) MLD ←→ V(T ,s2,r2) ←→ φ (V(T ,s2,r2)) = V(T ,s1,r1) ←ML→D T . MLD Finally, suppose F (X) contains a scaled pair V(T1,s1,r1) = φ · V(T2,s2,r2). Then we can derive an analogous diagram for the spaces formed as the orbit closures of each space: V (T1,s1,r1) ←ML→D T 1 = X = T 2 ←→ MLD Applying φ : φ (X) V (T2,s2,r2) MLD MLD ←→ φ (V (T2,s2,r2)) = V (T1,s1,r1) ←→ X. 7 Pseudo-self-similarity Implies MLD to a Self-similar Thing Theorem 26 If T is a pseudo-self-similar tiling, then it is MLD to a self-similar tiling, T . If X is a pseudo-self-similar tiling space, then it is MLD to a self-similar tiling space. We wish to emphasize that this proof must be essentially credited to PriebeSolomyak in [ 3 ]. Here we modify the details to account for a group G involving rotations, and a tiling space X. Let ∂T refer to the boundary graph composed of edges and vertices of tiles in T , and let the degree of a vertex refer to the number of edges extending from it. Outline of Proof of Theorem 26 1. Without loss of generality, we can assume our tiling space has certain simplifying properties. 2. If T is pseudo-self-similar with expansion φ, then there is a one-to-one, onto map Ψ : ∂T → ∂(φ−kT ), which maps edges in a piecewise linear, local manner. 3. We may construct a new map Ψn := φ−nk(φkΨ )n, which approximates edges in T with unions of edges in φ−nkT . Taking the limn→∞ Ψn := Ψ∞, we show that Ψ∞ is a continuous, injective map from ∂T to R2. 4. We construct a tiling whose edges and vertices are the image of T under Ψ∞. We call this tiling T∞ and show that T∞ is MLD to T . 5. T∞ is a self-similar tiling with expansion φk , and so T ∞ is a self-similar tiling space which is MLD to X. Step 1. Simplifying assumptions for a tiling space X Lemma 31 Let X be a pseudo-self-similar tiling space. Then X is MLD to a pseudoself-similar space X˜ , where all tiles are convex polytopes, and each vertex of X˜ has degree 3. Proof Let X have associated group G, and let T ∈ X. Then V(T ,s,r) and T are MLD, and V(T ,s,r) has the property that all cells are convex polytopes. Let V (T ,s,r) be the orbit closure of V(T ,s,r) under G. Then V (T ,s,r) is a tiling space which satisfies the first property. We will form a second tiling space, V (T ,s,r), which is MLD to V (T ,s,r), and satisfies the second property, as follows: Let {xi } be the set of vertices in V(T ,s,r) with degree > 3. To define V(T ,s,r), first let ∂V(T ,s,r) := ∂V(T ,s,r) on R2\ ∪ {[Be/3(xi )]V(T,s,r) }, where e is the minimum length of any edge in V(T ,s,r). Each cell of V(T ,s,r) whose vertices all have degree ≤ 3 has been left unchanged in V(T ,s,r), and we preserve the label of these cells, as well. In other words, away from every vertex of degree greater than 3, our new tiling agrees with the old tiling. Let {x1, . . . , xn} ⊂ {xi } be a finite set of unique representatives, such that for all {xi }, [Be/3(xi )]V(T,s,r) = g · [Be/3(xk)]V(T,s,r) , for 1 ≤ k ≤ n, and g ∈ G. We will first define V(T ,s,r) for the regions of R2 covered by {[Be/3(xk)]V(T,s,r) }, 1 ≤ k ≤ n, and then extend it by the g-action to all regions {[Be/3(xi )]V(T,s,r) }, i > n. Consider xk , 1 ≤ k ≤ n. For each edge extending from xk , introduce a vertex at a distance of e/3 from xk . Connect each of these new vertices to new vertices on either side, creating a polygon containing xk . (See Fig. 5.) Lastly, we delete xk and each edge extending from xk to the new vertices. In doing so, we have created multiple new cells, for each of which we create a new label. If [Be/3(xi )]V(T,s,r) had rotational symmetry, then the new edges of [Be/3(xi )]V(T,s,r) do as well, and we label the new cells to preserve that symmetry. Finally, we extend this procedure to all [Be/3(xi )]V(T,s,r) , xi a vertex of degree > 3. If [Be/3(xi )]V(T,s,r) = g · [Be/3(xk)]V(T,s,r) , 1 ≤ k ≤ n, define [Be/3(xi )]V(T,s,r) = g · [Be/3(xk)]V(T,s,r) . We label these cells in accordance with the label of the cell related by g, noting that if g is not unique, then the labels are the same across all possible choices of g. Note that since this procedure is reversible, V(T ,s,r) and V(T ,s,r) are MLD, and that V(T ,s,r) satisfies property 2. Finally, since T is pseudo-self-similar, T and φ (T ) are MLD. Since T is MLD with V(T ,s,r), and φT is MLD with φV(T ,s,r), we have that V(T ,s,r) and φV(T ,s,r) are MLD. Therefore V(T ,s,r) is pseudo-self-similar. Therefore V (T ,s,r) is MLD to X, and satisfies the properties of Lemma 31. Our next simplifying assumption states that, for any radius r , we can assume that all loci of rotational symmetry of size r are interior to the tiles containing them. [Br (x)]Tˆ = g · [Br (x)]Tˆ with x = g · x, then x ∈/ ∂Tˆ . Lemma 32 Let X be a pseudo-self-similar tiling space. Then for any r > 0, X is MLD to a pseudo-self-similar space Xˆ , with the following property: for all Tˆ ∈ Xˆ , if Proof We create Xˆ with a local replacement procedure much like as in the proof of Lemma 31. Let X satisfy the conditions of Lemma 31, and let T ∈ X. Let {xi } be the set of points contained in ∂T such that [Br (xi )]T = g · [Br (xi )]T with xi = g · xi , for some nontrivial g ∈ G. On R2\ ∪ [xi ]T , define ∂Tˆ := ∂T . Any tile not containing an edge or vertex which is a locus of rotational symmetry is left unchanged, and we label these tiles in Tˆ with the same label as they had in T . Let {x1, . . . xn} ⊂ {xi } be a finite set of representatives such that [Br (xi )]T = g · [Br (xj )]T with xi = g · xj , for i = j , g ∈ G. If xi is a vertex in T , then [Br (xi )]T must have rotational symmetry of angle 2π/3. We replace it with a triangular cell, as depicted in Fig. 5, and label the newly created cells so as to preserve the rotational symmetry. If xi lies on an edge in T , then [Br (xi )]T must have rotational symmetry of angle π . We introduce two new chords, related by a rotation of π around xi , across each tile of which xi is on the boundary. Now xi is a vertex of degree 4, and we have created two new vertices of degree 3 opposite xi . (If the new chords happen to land on existing vertices, offset the new chords by some angle θ , symmetrically on both sides of xi .) (See Fig. 6.) We apply the local replacement procedure depicted in Fig. 5 to the vertex xi . Note that the result is still symmetric with respect to rotation by π . We create a new label for each new cell that has been created, keeping the result symmetric with respect to rotation by π . We now define Tˆ on {[xi ]T }. For each xi , [Br (xi )]T = g · [Br (xk)]T , for some 1 ≤ k ≤ n and g ∈ G. Define [Be/3(xi )]Tˆ := g · [Be/3(xk)]Tˆ , and label accordingly. (Note that g is not unique, since these are loci of symmetry, but that the replacement procedure respects the symmetry. Also, [xi ]T and [xj ]T may not be disjoint, in which case Tˆ might lack some rotational symmetry of size r that T has.) As before, since the procedure is reversible, the tiling Tˆ is MLD to T , and so the closure of Tˆ under G is a tiling space that satisfies the lemma. Step 2. Developing Ψ : ∂T → ∂(φ−kT ) Given two points, x and y, let x−→,y be the line segment between x and y. For any k > 0, let Rk be the radius of derivability of φ−kT from T . Recall that dm is the minimum length of any edge of a tile in T , that dM := {sup |x1 − x2| : x1, x2 ∈ ∂ti , where ti is any tile in T }, and that dθ is the minimum angle of any two incoming edges to any vertex in T . Let K be a fixed integer such that λ−K < 48ddmM √2 − 2 cos dθ . Lemma 33 Given a tiling T , it is sufficient to prove Theorem 26 holds for tilings MLD to T , satisfying the conditions in Lemmas 31 and 32, with r = RK , the radius of derivability of φ−K T from T . Proposition 34 Let T be a pseudo-self-similar tiling with expansion φ, and λ = |φ|, satisfying the conditions in Lemma 33. Let > 0. For all integers k sufficiently large, there is a continuous, one-to-one map Ψ : ∂T → ∂(φ−kT ) = φ−k∂T which takes edges in ∂T onto unions of edges in φ−k∂T and which satisfies: 1. For x ∈ ∂T , |x − Ψ (x)| < . 2. If l is an edge in ImΨ , then the restriction Ψ |Ψ −1(l) is linear. Furthermore, there is a ρ ∈ (0, 1) such that if e is an edge in T , and l ∈ Ψ (e), then |Ψ −1(l)| ≤ ρ|e|. 3. The map Ψ is locally derivable from T with radius R˜ , for some R˜ > 0. Proof of Proposition 34 Without loss of generality, we can assume that: 1. < d4m 2. K < k 3. λ−k < 12dM √2 − 2 cos dθ . Note that K < k implies that all symmetry of size Rk has been disrupted. Also, since 0 ≤ dθ ≤ 2π/3, it follows that λ−kdM < 6 . We partition the vertices of T according to Rk + /4: let {v1, v2, . . . , vm} be a maximal list of vertices such that [BRk+ /4(vi )]T = g · [BRk+ /4(vj )]T for any i = j, g ∈ G. We will define Ψ on each B /4(vi ) ∩ ∂T , and then extend Ψ to all /4neighborhoods of vertices in T by the action of G. Let ei,1, ei,2, and ei,3 be the three incoming edges to vi in T and let the point in the intersection of ei,j and ∂B /4(vi ) be labeled vi,j . (See Fig. 7 for clarification.) For each j ∈ {1, 2, 3}, let Pi,j be any directed path from a vertex in [vi ]φ−kT to a vertex in [vi,j ]φ−kT , contained in [ei,j ]φ−kT . Let the vertices along Pi,j be labeled {wj1 , . . . , wjn }. If wj1 = wjk for k > 1, delete all segments between wj1 and wjk . Continuing along the vertices in order, delete all cycles until Pi,j has no selfintersections. Since cells are convex, ∩j {[ei,j ]φ−kT } ⊂ [vi ]φ−kT . Therefore any pair of {Pi,1, Pi,2, Pi,3} intersect only on [vi ]φ−kT , if at all. Extend or delete edges of each Pi,j until all three paths meet on a vertex of [vi ]φ−kT , without overlapping on any edge of [vi ]φ−kT . Assign Ψ (vi ) = ∩j Pi,j , and let the outer endpoint of each path be labeled vi,j . We now map each partial edge v−i−,−v→i,j linearly to the path from Ψ (vi ) to vi,j . Let kj be the number of edges contained in the path from Ψ (vi ) to vi,j . Divide v−i−,−v→i,j into kj equal segments, and map each segment linearly to the corresponding edge on the path in φ−kT . In this way Ψ : ∂T ∩ B /4(vi ) → ∂(φ−kT ) ∩ B /4(vi ). Note that |x − Ψ (x)| < for all x ∈ ∂T ∩ B /4(vi ), and that for any l ∈ Im(∂T ∩ B /4(vi )), Ψ |Ψ −1(l) is linear and |Ψ −1(l)| < /4 < dm/16, and so this map meets the first two criteria of the proposition. Extending Ψ to all vertices in ∂T Next, we extend Ψ to all ∂T ∩ B /4(v), for all vertices v ∈ T . For each vertex v in T , there is a unique j ∈ {1, 2, . . . , n} and g ∈ G such that [BRk+ /4(v)]T = g · [BRk+ /4(vj )]T and v = g · vj . Note that g is unique since T has no vertices which are loci of rotational symmetry of size Rk . We define Ψ on ∂T ∩ B /4(v) as follows: if w ∈ ∂T and |w − v| < /4, then let Ψ (w) = gΨ g−1(w). Since Rk is the radius of derivability of φ−kT from T , we have that [B /4(v)]φ−kT = g · [B /4(vj )]φ−kT . Therefore, gΨ g−1(w) ∈ ∂φ−kT . Defining Ψ on edges in T We will now partition the edges in T according to their neighborhoods of radius Rk + /2. Let {e1, . . . , em} be a maximal list of edges in T such that [NRk+ /2(ei )]T = g · [NRk+ /2(ej )]T for any i = j, g ∈ G. For a given edge ei , we have already mapped under Ψ the two segments of length /4 which are connected to each endpoint. Let v1 and v2 be the two points which are /4 away from each endpoint of ei . (In other words, Ψ (v1) and Ψ (v2) were defined in the prior section.) (See Fig. 8 for clarification.) We wish to map the line segment v−−1−,→v2 to a path from Ψ (v1) to Ψ (v2), composed of edges in [v−−1−,→v2]φ−kT . Let Pi be a path composed of edges in [ei ]φ−kT connecting Ψ (v1) and Ψ (v2). Let v1 = Ψ (v1), vn = Ψ (v2), and {v1, v2, . . . , vn} be an ordered list of all the vertices on the path Pi , directed from Ψ (v1) to Ψ (v2). As before, we begin removing cycles by checking if v1 = vj for j > 1, and if this occurs, deleting all segments between v1 and vj . We then repeat for the next available vertex until Pi is not self-intersecting. Relabel the remaining vertices {v1, . . . , vn}. For each vertex vj , choose a point wj on ei , such that 1. w1 = v1, wn = v2 2. |wi − v1| < |wi+1 − v1| 3. wj ∈ {[vj ]φ−kT ∩ ei } (which may contain disjoint segments of ei ). −−−−j−→+1. Note that |x − Ψj (x)| ≤ Let Ψj be the linear map from w−−j−,−w−j→+1 → vj , v 2λ−kdM < 3 . Concatenating the maps Ψj creates a single, piecewise linear map −−−→ Ψ : w−−1−,−w→n → v1, vn. Thus Ψ : v−−1−,→v2 → Pi satisfies the first two conditions of Proposition 34. As before, we can now extend Ψ to map all edges in T to unions of edges in φ−kT , based on [NRk+ /2(e)]T . Define Ψ (e) to be g · Ψ (ei ), where ei ∈ {e1, . . . , em} is the unique edge such that [NRk+ /2(e)]T = g · [NRk+ /2(ei )]T , and g is unique since e cannot contain any locus of rotational symmetry. Since the extension of Ψ from {e1, . . . , em} to all other edges is based on neighborhoods of radius Rk + /2, we have that Ψ (ei ) ∈ ∂φ−kT implies Ψ (e) ∈ ∂φ−kT . Fig. 9 By choice of k, edges are separated by at least 3λ−kdM , outside of B /4(v) Next we show that the images of edges under Ψ are non-intersecting. In T , suppose that two incoming edges to a vertex v are separated by an angle θ , 0 < θ < π . Let v1 and v2 be the two respective points on each edge which are /4 away from v. Then |v1 − v2| = /4√2 − 2 cos θ ≥ /4√2 − 2 cos dθ > 3λ−kdM . This ensures that edges in T which are outside of [B /4(v)]T are separated by at least 3λ−kdM (see Fig. 9) and so guarantees that the images of edges under Ψ do not intersect. Thus Ψ is injective. By construction, Ψ is continuous and linear on the preimage of all edges in ImΨ . We claim that Ψ is locally derivable from T with radius R˜ = Rk + + dM . Suppose [BRk+ +dM (p)]T = g · [BRk+ +dM (q)]T for p, q ∈ ∂T . If p and q are within /4 of a vertices vp and vq , respectively, then [BRk+3 /4(vp)]T = g · [BRk+3 /4(vq )]T , which is more than sufficient to ensure that Ψ (p) = g · Ψ (q). Otherwise, we use the fact that p and q are contained within edges ep and eq , and [NRk+ (ep)]T = g · [NRk+ (eq )]T , which ensures that Ψ (p) = g · Ψ (q). Step 3: Constructing Ψ∞, a continuous, injective map from ∂T to R2 Define Ψn := φ−nk(φkΨ )n. In other words, Ψn redraws edges in T with edges in φ−nkT . Since φ and Ψ are continuous and one-to-one, Ψn is as well. Notice that since Ψ : ∂T → ∂(φ−kT ), we have that φkΨ : ∂T → ∂T , and so Ψn : ∂T → ∂φ−nkT . With this, we define Ψ∞ : ∂T → R2 by Ψ∞(x) = limn→∞ Ψn(x). First we show that {Ψn} converges uniformly: |Ψn(x) − Ψn+1(x)| = φ−nk φkΨ n(x) − φ−(n+1)k φkΨ (n+1)(x) = φ−nk φkΨ n(x) − φ−k φkΨ (n+1)(x) = φ−nk φkΨ n(x) − Ψ φkΨ n(x) ≤ λ−nk . Therefore Ψ∞(x) is well-defined. Ψ∞ is injective We next claim that Ψ∞ is injective, for sufficiently small. Suppose x, y ∈ ∂T and Ψ∞(x) = Ψ∞(y). Since |Ψn(x) − Ψn+1(x)| ≤ λ−nk , |x − Ψ∞(x)| ≤ 1−λ−k < 2 . (The inequality that λ−k < 1/2 follows from λ−K < 48ddmM √2 − 2 cos dθ and k > K .) Therefore |x − y| < 4 . Since Ψn = φ−nk(φkΨ )n, it follows that Ψn+1 = φ−kΨnφkΨ , and so Ψ∞ = φ−kΨ∞φkΨ. (1) Therefore we have φ−kΨ∞φkΨ (x) = φ−kΨ∞φkΨ (y). Therefore, Ψ∞[φkΨ (x)] = Ψ∞[φkΨ (y)], and so resubstituting for Ψ∞ n times yields Ψ∞[(φkΨ )n(x)] = Ψ∞[(φkΨ )n(y)]. Therefore |(φkΨ )n(x) − (φkΨ )n(y)| < 4 . We can pick small enough that x and y must lie on the same or adjacent edges. Let be such that: 4 < min{|x1 − x2| : x1, x2 lie on edges e1, e2, with e1 ∩ e2 = ∅.}. This ensures that (φkΨ )n(x) and (φkΨ )n(y) lie on the same edge or adjacent edges in ∂T , since φkΨ : ∂T → ∂T . First, suppose that (φkΨ )n(x) and (φkΨ )n(y) lie on the same edge in ∂T . Since φ−k takes edges to edges, and Ψ takes edges to unions of edges, the preimage of any edge e under Ψ −1 must be contained within an edge of T . Therefore (φkΨ )n−1(x) and (φkΨ )n−1(y) lie on the same edge in ∂T . Repeating this, (φkΨ )i (x) and (φkΨ )i (y) will all lie on the same edge in ∂T for all i, and so x and y lie on the same edge in ∂T . Ψ was constructed so that if l is an edge in Im Ψ , then the restriction Ψ |Ψ −1(l) is linear. Furthermore, there is a ρ ∈ (0, 1) such that if e is an edge in T , and l ∈ Ψ (e), then |Ψ −1(l)| ≤ ρ|e|. Let eˆ = max{|e|}, e ∈ ∂T . Then |Ψ −1(l)| ≤ ρeˆ, for any edge l in Im Ψ , and so |x − y| ≤ ρeˆ. Therefore, the restriction (φkΨ )n|Ψ −n(l) is linear, and |Ψ −n(l)| ≤ ρn|e|. Therefore, |x − y| ≤ ρneˆ, and so as n → ∞, we have x = y. Suppose that (φkΨ )n(x) and (φkΨ )n(y) lie on the adjacent edges in ∂T . The preimage under Ψ of each pair of edges which share a vertex is each contained in edges which again share a vertex, and so we have that (φkΨ )n−1(x) and (φkΨ )n−1(y) lie on the adjacent edges in ∂T , and as above, eventually that x and y lie on adjacent edges in ∂T . As before, |Ψ −1(l)| ≤ ρeˆ, for any edge l in ImΨ , and so |x − y| ≤ 2ρeˆ. Therefore |x − y| ≤ 2ρneˆ, and so as n → ∞, we have x = y. Therefore Ψ∞ is injective. Let ti = (Ai , li ) be a tile in T . Since Ψ∞ is injective, by the Jordan Curve theorem, Ψ∞ maps ∂Ai to a simple curve which separates the plane into two components, of which Ψ∞(∂Ai ) is the boundary. We will now define a tiling, T , such that the ∞ closure of the support of the interior of Ψ∞(∂Ai ) is the support for a tile in T . Let ∞ this be called Ai , the compact set for a tile ti in T . ∞ We label {ti } based on a neighborhood of radius R˜ = RΨ , the radius of derivability of Ψ . Formally, let {[NR˜ (ti )]T } be a comprehensive, finite list of neighborhoods of tiles in T of radius R˜ , up to action by G, where ti has support Ai . Let li = [NR˜ (ti )]T . Given an unlabeled tile t in T∞, let t be the associated tile in T such that Ψ∞ : ∂t → ∂t . We label t with li if [NR˜ (t )]T = g · [NR˜ (ti )]T for some g ∈ G. Next we must show that the image of tiles of T under Ψ∞ overlap only on their boundaries. Let Ai be the support of a tile in T , and Ψ∞(∂Ai ) be the corresponding compact set in T∞. Since |x − Ψ∞(x)| < 4 for x ∈ ∂T , the unbounded component of ∂T \∂Ai must overlap with the unbounded component of R2\Ψ∞(∂Ai ). Since ∂T \∂Ai is connected, so is Ψ∞(∂T \∂Ai ). And since Ψ∞ is injective, Ψ∞(∂T \∂Ai ) must lie entirely in the unbounded component of R2\Ψ∞(∂Ai ). If Aj is the support of a different tile in T , then intAj ⊂ R2\Ai , and so int(Ψ∞(∂Aj )) is contained in the unbounded component of R2\Ψ∞(∂Ai ). Therefore, int(Ψ∞(∂Aj ))∩ int(Ψ∞(∂Ai )) = ∅. Lastly we must show that the tiles cover R2. Let z ∈ R2. Then [BdM +4 (z)]T is the union of finitely many tiles of T , and z lies in the bounded component of Ψ∞(∂[BdM +4 (z)]T ). Since ∂[BdM +4 (z)]T is a compact set, Ψ∞|∂[BdM +4 (z)]T can be extended to a homeomorphism h : [BdM +4 (z)]T → supp(int Ψ∞(∂[BdM +4 (z)]T )). Therefore z is in the image under h of some tile t ∈ [BdM +4 (z)]T , and so T∞ does in fact cover R2. We have shown that T∞ is a cover of R2 by tiles which overlap only on their boundaries, and so T∞ is a tiling. Theorem 35 T∞ and T are MLD tilings, for sufficiently small. T∞ is locally derivable from T Let ρ be the recognizability radius of T from φT . Note that Ψ is locally derivable under any radius larger than R˜ . Without loss of generality, we may assume that R˜ > λkλ(kρ−+1 ) . Lemma 36 Ψn is locally derivable from T with radius R˜ , for all n. We will proceed by induction; Ψ = Ψ1 is locally derivable with radius R˜ by construction. Assume Ψn is as well. Suppose [BR˜ (x)]T = g · [BR˜ (y)]T , with x = g · y. Since |Ψ (x) − x| < , [BR˜− (Ψ (x))]T = g · [BR˜− (g−1 · Ψ (x))]T . Applying φk , we have [Bλk(R˜− ) × (φkΨ (x))]φkT = φkgφ−k · [Bλk(R˜− )(φk(g−1 · Ψ (x)))]φkT . Since λkρ is the radius of derivability of T from φkT , [Bλk(R˜− )−λkρ (φkΨ (x))]T = φkgφ−k · [Bλk(R˜− )−λkρ (φk(g−1 · Ψ (x)))]T . Now, since R˜ is the radius of derivability of Ψ from T , and [BR˜ (x)]T = g · [BR˜ (y)]T , we have that Ψ (x) = g · Ψ (y). Therefore, [Bλk(R˜− )−λkρ (φkΨ (x))]T = φkgφ−k · [Bλk(R˜− )−λkρ (φk(Ψ (y)))]T . By the inductive hypothesis, Ψn is derivable from T with radius R˜ . Since λk(R˜ − ) − λkρ > R˜ , we have that ΨnφkΨ (x)) = φkg−1φ−k · Ψn(φk(Ψ (y))). Since Ψn+1 = φ−kΨnφkΨ , we get that φkΨn+1(x) = φkg−1Ψn+1(y), and so Ψn+1 is also locally derivable with radius R˜ . Once we have that all Ψn are LD from T with radius R˜ , it follows that limn→∞ Ψn = Ψ∞ is as well. Now we must show that T∞ is LD from T . Suppose [BR˜+dM +2 (x)]T = g · [BR˜+dM +2 (y)]T , with x = g · y. For every p ∈ ∂[x]T and q ∈ ∂[y]T with q = g · p, we have Ψ∞(p) = g · Ψ∞(q). Therefore Ψ∞(∂[x]T ) = g · Ψ∞(∂[y]T ). Since tiles in T∞ were labeled based on a neighborhood of R˜ in T , [x]T∞ and [y]T∞ have the same label. Therefore [x]T∞ = g · [y]T∞ . T is locally derivable from T∞ First, we must ensure that tiles in T∞ do not have any additional rotational symmetry that tiles in T lack. If t is a tile in T , let Ψ∞(t ) refer to the tile in T∞ whose boundary is the image of the boundary of t under Ψ∞. Suppose t = g · t but Ψ∞(t ) = Ψ∞(g · t ). Let {v1, . . . , vn} be the ordered vertices of t , and {Ψ∞(v1), . . . , Ψ∞(vn)} be the corresponding images under Ψ∞. Then {Ψ∞(v1), . . . , Ψ∞(vn)} = {Ψ∞(g · v1), . . . , Ψ∞(g · vn)}. Let vi be a vertex of t such that g · vi = vj for any j , but Ψ∞(g · vi ) = Ψ∞(vj ) for some j . Let d := |g · vi − vj |. Let dˆ be the minimum d for any vertex in any tile t which lacks symmetry that Ψ∞(t ) has. If we take < dˆ/3, then we ensure that Ψ∞(g · vi ) = Ψ∞(vj ), and so T∞ will only have rotational symmetry if T does. Suppose [B2dM + (x)]T∞ = g · [B2dM + (y)]T∞ and x = g · y. While it may be that Ψ ([x]T ) = [x]T∞ and Ψ ([y]T ) = [y]T∞ , we still have that Ψ ([x]T ) ∈ [B2dM + (x)]T∞ and Ψ ([y]T ) ∈ [B2dM + (y)]T∞ , and so Ψ ([x]T ) and Ψ ([y]T ) have matching labels in T . Therefore [NR˜ ([x]T )]T = g · [NR˜ ([y]T )]T for some g ∈ G, and so [x]T = ∞ g · [y]T . Since x = g · y, g = g only if [x]T∞ and [y]T∞ have a rotational symmetry which [x]T and [y]T lack. By choice of this cannot occur, and so g = g. Therefore T and T∞ are MLD tilings. Step 5. T∞ is a self-similar tiling; T ∞ is a self-similar tiling space which is MLD to X Theorem 37 T∞ is a self-similar tiling. We must show first that for a tile t ∈ T , φk(t ) is the union of tiles in φkT . ∞ ∞ From (1), in Step 3, Ψ∞ = φ−kΨ∞φkΨ . Therefore φk(Ψ∞(∂T )) ⊂ Ψ∞(∂T ). If we take the support A of a tile t in T , we have that ∂A is a Jordan curve, and so ∞ φk(∂A) divides R2 into two components. Since φk∂A ⊂ ∂T∞, the support A of any tile t in T∞ must lie entirely in one of the two components of R2\φk∂A. Therefore φk∂A is exactly the union of all tiles whose interiors have nontrivial intersection with the interior of φk∂A. Second, we must show that if t1 and t2 are tiles in T∞ with t1 = g · t2, then φk(t1) = g · φk(t2) as patches of tiles in φkT , where g = φkgφ−k . Suppose that t1, t2 are tiles in T∞ with the same label, and t1 = g · t2. Since the labels are based on a radius R˜ of the corresponding tile in T , we have that [NR˜ (Ψ∞−1(t1))]T = g · [NR˜ (Ψ∞−1(t2))]T . Applying φk , we have that [NλkR˜ (φk (Ψ∞−1(t1))]φk T = φk gφ−k [NλkR˜ (φk (Ψ∞−1(t2))]φk T . Since R˜ is the radius of derivability of Ψ∞ onto T , λk R˜ is a lower bound on the local derivability radius of φk Ψ∞ onto φk T . Therefore, for all points [x] ∈ φk (Ψ∞−1(t1)) and [y] ∈ φk (Ψ∞−1(t2)) with x = φk gφ−k y, [x]φk T∞ = φk · g · φ−k [y]φkT∞ . Therefore T∞ is a self-similar tiling. Theorem 26 If X is a pseudo-self-similar tiling space, then it is MLD to a self-similar tiling space, X . Proof Let T be a pseudo-self-similar tiling contained in X. Then T∞ is self-similar, and T ∞ is MLD to X. 8 Conclusion We have shown, up to MLD, a series of results detailing the relationship between self-similar tiling spaces and pseudo-self-similar spaces, and derived Voronoi spaces, in R2. First, a tiling is MLD to all of its derived Voronoi tilings, and likewise for tiling spaces. Next, given a self-similar tiling T , the associated set {V(T ,0,r)} is σ finite, which implies that a self-similar tiling must contain a scaled pair of tilings, and that a self-similar tiling space must contain a scaled pair of orbits. Having a scaled pair of tilings or orbits, in turn, implies that the underlying tiling or tiling space is pseudo-self-similar. Finally, we’ve shown that a pseudo-self-similar tiling or tiling space is MLD to a self-similar tiling or tiling space. These results couch self-similarity and pseudo-self-similarity in a context where the group structure is utilized. While many tilings whose group includes only finite order rotations group can be redefined as tilings whose group is strictly translations, much of the basic structure of the tiling is lost in this relabeling. Second, tilings whose group includes infinite order rotations cannot be dealt with at all, without incorporating rotations into the discussion. Finally, including tiling spaces in these results allows for a broader range of connections with features of tiling spaces such as their dynamics or their topology, which does not exist with individual tilings. There are large impediments to extending these results to higher dimensions. We often use the fact that self-similar tilings are recognizable, which is not necessarily true in dimension 3 and higher. Holton et al. [ 1 ] have an example of a pseudoself-similar tiling of R3 which is not recognizable. It would be interesting to explore whether there are self-similar tilings of R3 which are not recognizable, or whether there are pseudo-self-similar tilings of R3 which are not MLD to self-similar tilings. Acknowledgements I would like to acknowledge first my advisor, Lorenzo Sadun, for many hours and much guidance given in writing this paper. Also, the referee provided much indispensable help revising and improving this paper. This was made possible by funding from NSF Grant 26-1136-4712, via Karen Uhlenbeck and the geometry group at the University of Texas. 1. Holton , C. , Radin , C. , Sadun , L. : Conjugacies for tiling dynamical systems . Commun. Math. Phys. 254 , 343 - 359 ( 2005 ) 2. Priebe , N.: Towards a characterization of self-similar tilings in terms of derived Voronoi tessellations . Geom. Dedic . 79 , 239 - 265 ( 2000 ) 3. Priebe , N. , Solomyak , B. : Characterization of planar pseudo-self-similar tilings . Discrete Comput. Geom . 26 , 289 - 306 ( 2001 )


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Betseygail Rand. Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ2, Discrete & Computational Geometry, 2011, 1-28, DOI: 10.1007/s00454-011-9327-y