TILINGS ASSOCIATED WITH NON-PISOT MATRICES

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1 TILINGS ASSOCIATED WITH NON-PISOT MATRICES MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Abstract. Suppose A Gl d (Z) has a 2-dmensonal expandng subspace E u, satsfes a regularty condton, called good star, and has A 0, where A s an orented compound of A. A morphsm θ of the free group on {1, 2,..., d} s called a non-abelanzaton of A f t has structure matrx A. We show that there s a tlng substtuton Θ whose boundary substtuton θ = (Θ) s a non abelanzaton of A. Such a tlng substtuton Θ leads to a self-affne tlng of E u R 2 wth A u := A Eu GL 2 (R) as ts expanson. In the last secton we fnd condtons on A so that A has no negatve entres. 1. Introducton A tlng substtuton Θ s a mappng from tles n R 2 to fnte tlng patches that has enough regularty to be appled to tlng patches. Ths permts the teraton of Θ, startng wth a sngle tle, or a small tlng patch, to obtan larger tlng patches, and ultmately tlngs of the plane. A tlng substtuton Θ nduces a mappng θ = Θ on tle boundares. Assumng there are fntely many polygonal prototles, havng d dfferent boundary segments, ths boundary map θ can be vewed as a 1-dmensonal substtuton on B := {1, 2,..., d}. The structure matrx of θ s the matrx A whose, th entry counts the number of tmes the edge occurs n the substtuton θ(). Frequently, ths matrx A wll be hyperbolc, and have a 2-dmensonal expandng subspace, correspondng to how the tlng substtuton expands R 2. The process of tertng a (suffcently nce) tlng substtuton Θ yelds an uncountable collecton X Θ of tlngs of R 2. Ths set X Θ can be regarded as a compact metrc space (see e.g., [16]), on whch Θ acts as a homeomorphsm. Typcally Θ acts on a tlng y X Θ as follows. Frst y s replaced by A u y, for A u a lnear expanson of the plane. Then the new tlng Θ(y) s obtaned by applyng a local mappng (see [16]) to A u y, whch essentally tles certan regons of A u y accordng to ther shapes. The matrx A u s the restrcton of the structure matrx A of θ to ts expandng subspace. In the best crcumstances the acton of Θ on X Θ s semconugate to the acton of A on T d as a toral automorphsm. In fact, the tlngs y X θ are closely related to Markov parttons for A (see [11], [9]). In ths paper we seek to reverse the process descrbed above. Startng wth a hyperbolc matrx A GL d (Z) havng a 2-dmensonal expandng subspace, we want (1) to fnd a 1-dmensonal substtuton θ, or more generally a free group endomorphsm θ, that has A as ts structure matrx, and (2) fnd a tlng substtuton Θ wth Θ = θ. As t turns out, not every A s works, at least not for the method we descrbe here. However, we fnd suffcent condtons: namely, that a certan orented compound matrx, whch we denote by A, satsfes A 0. In the last part of the paper we gve an easy condton on A to check for ths. 1

2 2 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Once we have a sutable A we need to carefully choose one of the many endomorphsms θ wth structure matrx A. We descrbe a condton called properly ordered that guarentees θ s the boundary of a tlng substtuton. The dea of startng wth a matrx, makng a correspondng substtuton, and usng t geometrcally frst appears n G. Rauzy [15], where the Rauzy fractal s ntroduced (a nce update on ths theory appears n [3], hghlghtng ts connectons to number theory, dynamcs and fractal geometry). The dea of usng A and a 1-dmensonal substtuton to construct a tlng substtuton appears many works by P. Arnoux, S. Ito and ther co-workers (see e.g., [11], [4], [6], and [7]). An approach related to the one descrbed here appears n [9]. However, all of the works mentoned above assume some form of the Psot condton, a codmenson-1 condton, whch when expressed n the context of ths dscusson would say A has a 1-dmensonal contractng 1 subspace. Recall that a real algebrac nteger λ > 1 s a Psot number f all ts algebrac conugates satsfy λ < 1. The role of the Psot condton n most prevous approaches to smlar problems cannot be overstated. In ths paper we also drop the common assumpton that A 0. A program smalar to ours, (n partcular, wthout the Psot assumpton) appears n the dssertaton [13] of R. Kenyon, but wth few detals. For other approaches the non-psot case see [12], [9] and [8]. The thrd author wshes to thank Tsuda College, Japan, where he carred out much of hs work on ths paper as a frequent vstor. 2. Lnear theory 2.1. The non-psot property. Let A Gl d (Z) have λ 1 λ 2 > 1 > λ 3 λ d > 0,.e., A s nonsngular and hyperbolc and has a 2-dmensonal expandng subspace, denoted E u. Let E s be the contractng subspace, whch has dmenson d 2, and assume d 2 1. We say A satsfes a Psot condton f d = 1. A matrx A satsfyng (2.1) s called non- Psot of order n f n = d 3 > 1. Although we do not need to assume the non-psot condton, we don not need to exclude t ether. In partcular, our examples typcally have d = 4, so that n = d 2 = The good star property. Let v 1,v 2 be an ordered bass for E u (for example, we could take v 1 and v 2 to be egenvectors). We call a matrx P R 2 d a proecton to E u f (2.1) Pv 1 = e 1 and Pv 2 = e 2. It wll often be convenent to dentfy E u wth ts P-mage, whch we thnk of as the plane R 2. If P s another proecton, correspondng to a dfferent bass for E u, then (2.2) P = MP for M Gl 2 (R). Let us wrte P = (p 1,p 2,...,p d ) n terms of ts columns. Then Pe = p. In other words, the columns p are proectons of e to R 2. When we draw these vectors n R 2, we refer to the pcture as a star of vectors (see Fgure 1). 1 In most cases (e.g, [4],[9]) the Psot condton expressed n terms of what would be A 1 n our treatment, and would say that the expandng subspace s one dmensonal.

3 TILINGS 3 Defnton 2.1. We say a matrx A that satsfes (2.1) has the good star property f (2.3) p = ωp for some real ω 0 mples =. In other words, the vectors p are parwse non-parallel. By (2.2), the good star property depends on A, but not on the choce of P. Defnton 2.2. Let A satsfy (2.1) and (2.3), and let P be a proecton to E u. Defne A u R 2 2 be the matrx conugate to A E u va P. That s, (2.4) A u := PA(v 1,v 2 ). It s easy to see that A u has egenvalues λ 1, λ The compound. Let B C p q (the set of p q complex matrces), wth p, q n. The n th compound of B s the matrx C n (B) C (p n) ( p n) whose entres are the n n mnors of B (see [2]). In ths paper we wll always assume n = 2. We ndex C 2 (B) by pars, k l, where < and k < l, and we arrange such pars n lexcographc order. Theorem 2.3 (Bnet-Cauchy, see [2]). If B = B 1 B 2 then C n (B) = C n (B 1 )C n (B 2 ). If B s non-sngular then C n (B 1 ) = C n (B) 1. Let P R 2 d be the proecton for a matrx A satsfyng (2.1) and (2.3). The compound C 2 (P) R 1 (d 2) has entres (2.5) p := det(p,p ) = sn( p p ) 0, where the last nequalty follows p p ( π, 0) (0, π) by (2.3). For p 0, let sgn(p) = p/ p. For a vector a C n, let dag(a) denote the n n matrx wth the entres of a along the dagonal, and zeros everywhere else. Defne (2.6) S(A) = dag ( (sgn(p 1 2 ), sgn(p 1 3 ),...,sgn(p (d 1) d ) ) { 1, 0, 1} (d 2) ( d 2). The Bnet-Cauchy Theorem and (2.2) mply that S(A) s well defned up to a change of sgn. Defne (2.7) A = S(A)C 2 (A)S(A). Ths s the matrx wth entres a,k l sgn(p )sgn(p k l ), where a,k l are the entres of C 2 (A). We call A the orented compound of A. From now on, n addton to (2.1) and (2.3), we wll assume: (2.8) A 0,.e., A has no negatve entres. It s easy to see that ths does not depend on the choce of the bass for E u.

4 4 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR Example: The Ammann matrx. Consder the matrx (2.9) A = , whch s related to the tlng sometmes called the Ammann-Beenker tlng (see [17], [12]). Note that A s symmetrc and has characterstc polynomal p(x) = (x 2 + 2x 1) 2. The egenvalues are λ 1 = λ 2 = 1 2 and λ 3 = λ 4 = 2 1, so A satsfes (2.1). Let Q = ( ) q 1,q 2,q 3,q 4 = , where the columns q are egenvectors for λ. Defne the proecton P to be the the frst two rows of Q 1 : (2.10) P = 2 4 ( ) = (p 1,p 2,p 3,p 4 ), 2 1 The columns of P are plottedn Fgure 1. Clearly A satsfes the good star property (2.3). p 3 p 2 p 4 Fgure 1. The good star of vectors correspondng to the Ammann matrx A n (2.9). We have A u = λ 1 Id = (1 + 2)Id, whch we nterpret as a composton of an expanson of R 2 by 1 + 2, and a rotaton by π. Applyng (2.6) we fnd (2.11) S(A) = dag(1, 1, 1, 1, 1, 1), so that by (2.7) (2.12) A = Thus A satsfes (2.8). p 1

5 TILINGS 5 3. Proto-obects and obects In the next few sectons we are gong to consder geometrc obects g n R 2, ncludng curves, closed curves, tles, tlng patches and tlngs. Each obect g wll have a partcular locaton n R 2, and we can move an obect by a translaton. In partcular, g + w denotes the translaton of g by the vector w R 2. We call two obects g 1 and g 2 translatonally equvalent f g 1 = g 2 + w for some w R 2. We denote the translatonal equvalence class of the obect g by G, and refer to G as a proto-obect. Thus, we wll refer to protocurves, prototles, etc.. It wll be convenent to represent each proto-obect G by an arbtrary fxed choce of a geometrc obect from ts translatonal equvalence class. We thnk of ths obect as beng located at the orgn n R 2. Let G denote the set of all proto-obects G. Each obect g s then a translaton g = G + w := (w, G) of the correspondng proto-obect G G by a unque vector w R Words and substtutons. Let B = {1, 2,..., d}. Let B denote the free semgroup of nonempty fnte words n B. Let B ± = { ±1 : B}, and let F B denote the free group on B, whch we thnk of as the set of reduced words n (B ± ), together wth the empty word ɛ. A substtuton s a mappng θ : B B. Snce B s a bass both for both B and F B, a substtuton unquely defnes both a semgroup and a group endomorphsm. More genrally, any mappng θ : B F B defnes an endomorphsm of F B. We wll always assume an endomorphsm s non-erasng, whch means θ() ɛ. We thnk of a non-erasng endomorphsm a sort of generalzed substtuton, but refran from usng ths language to avod confuson wth ts other uses. The abelanzaton homomorphsm f : F B Z d s defned on B by f() = e, and extended to F B. A endomorphsm of Z d s gven by a matrx M Z d d. For any endomorphsm θ of F B, the abelanzaton f nduces an (abelan group) endomorphsm of Z d, denoted by L θ, satsfyng L θ f = f θ. In partcular, (3.1) L θ = (f(θ(1)), f(θ(2)),..., f(θ(d))). Ths matrx s also called the structure matrx of θ, or more formally the abelanzaton of θ. We note that f each θ() s an effcent word (see Defnton 5.6 below), then entry l, of L θ equals the sgned number tmes that appears n θ() (n partcular, ths always holds f θ s a substtuton). For a gven matrx A, we call any endomorphsm θ that satsfes L θ = A a non-abelanzaton of A Curves. A curve s contnuous, pecwse C 1 mappng w : [a, b] R 2. Two curves are consdered the same f they dffer by an orentaton preservng pcewse C 1 change of parameterzaton. The reverse of a curve w, denoted w, s gven by w(t) = w( t+a+b). For x 0,x 1 R 2 the lnear curve from x 0 to x 1 s defned to be w(t) = (1 t)x 0 + tx 1, for t [0, 1]. Smlarly, for a sequence of ponts x 0,x 2,...,x l we defne the pecwse lnear curve (or broken lne ) connectng the ponts by w : [0, l] R 2 where w(t) = (1 (t ))x + (t )x +1 for t [, + 1]. We wll usually restrct our attenton to the followng case. Let P = (p 1,p 2,...,p d ) be a good star of vectors. For each k = 1,...,l, suppose there exsts k {1,..., d} and

6 6 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. a nonzero a k R so that (3.2) x k x k 1 = a k p k. In effect, we consder pecwse lnear curves whose segments are parallel to the vectors p 1,...,p d. Now suppose W (B ± ) and x R 2. In partcular, W = a 1 1 a a l l, where k B and a k Z, a k 0. Defne a fnte sequence of ponts x 0,x 1,...,x l nductvely, by puttng x 0 = x, and applyng (3.2) for k = 1,...,l. We defne the curve w := (x, W) to be the pecwse lnear curve connectng the ponts x 0,x 1,...,x l. In ths way a word s a proto-curve. A curve w : [a, b] R 2 s closed f w(a) = w(b) (we sometmes thnk of t as mappng w : S 1 R 2 ). A smple closed curve s a closed curve that s nectve on S 1. We say W (B ± ) s a cyclc f the correspondng curve w = (x 0, W) s closed. Ths s equvalent to x l = x 0. Snce (3.3) x l x 0 = f(w), the curve w s closed, or equvalently the word W s a cyclc, f and only f f(w) = 0. A cyclc word s a proto-closed curve. We defne the commutator of two words W 1, W 2 by (3.4) W = [W 1, W 2 ] := W 1 W 2 W 1 1 W 1 2. Note that f([w 1, W 2 ]) = f(w 1 W 2 W 1 1 W 1 2 ) = 0, so that a commutator s always cyclc A topologcal proposton. Unfortunately, t s not easy to tell f a cycle W corresponds to a smple close curve. However, n ths secton we obtan a partal result n the case W = [W 1, W 2 ]. The Jordan curve theorem (see e.g., [1]) says that f w : S 1 R 2 s a smple closed curve then R 2 \w(s 1 ) conssts of two open sets: one bounded, called the nsde of w, and one unbounded, called the outsde. More generally, f w s pecwse lnear closed curve that s not necessarly smple, then R 2 \w(s 1 ) conssts of a fnte collecton of open sets: the outsde, whch s unbounded, and and a fne number of bounded open sets, called the components of the nsde. A chan s a fnte sum of the form (3.5) w = n 1 w 1 + n 2 w n k w k, where each w k s a dfferent (not necessarly closed) curve n the plane, and n k Z (see [1]). There s a farly obvous equvalence relaton on chans, whch n addton to orentaton preservng reparameterzaton, allows the combnaton of curves by followng a sucesson of them. Lne ntegrals of real or complx functons f n the plane are defned over chans. It can be shown that two chans are equvalent f and only f they yeld the same lne ntegral for every functon f, (see [1]). Equvalent chans are consdered to be equal. A chan s called a cycle f (up to equvalence) each w s a closed curve. We assume n addton that each w s pecewse lnear satsfyng (3.2). A cycle s a generalzaton of a closed curve, and lke n that case, R 2 \ k =1 w k(s 1 ) conssts of a fnte collecton of open sets: an unbounded outsde, and a fnte number of bounded nsde components. The

7 TILINGS 7 unon of the components s called the nsde of the curve or the cycle. The set k =1w k (S 1 ) s called the trace. Gven a cycle w, and x R 2 not n the trace, the wndng number s defned (3.6) n w (x) = 1 2π w dz z x, Here we thnk of x and z as a complex numbers. The ntegral s carred out separately over each closed curve n (3.5), and the results are added. The basc property of the wndng number functon n w s that t s s nteger valued and constant on each component of the nsde of w, ond zero outsde (see [1]). It can be shown that two cycles are equvalent f and only f they assgn the same wndng numbers to all non-trace ponts (ths follows from the generalzed Cauchy ntegral formula, [1]). The change of varables formula for ntegrals shows that n w+y (x + y) = n w (x), and n w (x) = n w (x). It s easy to see that n w1 +w 2 (x) = n w1 (x) + n w2 (x) provded x s not n the trace of w 1 or w 2. Let w be a smple closed curve. Then n w (x) = ±1 for x nsde w (see [1]). If n w (x) = 1 for x nsde w, we say w s postve (.e., t s postvely orented). More generally, we say a (not necessarly smple) closed curve w, or even a cycle w, s postve f n w (x) 0 for all x R 2 not n the trace. We call a closed curve or cycle w postve sem-smple f n w (x) {0, 1} for all x R 2 not n the trace. Proposton 3.1. Let W = [W 1, W 2 ] be the commutator of W 1, W 2 (B ± ) and defne the closed curve w = (0, W). Assume w s postve and (3.7) det ( [Pf(W 1 ), Pf(W 2 )] ) > 0. Then w s postve sem-smple. Proof. For smplcty we dentfy W = w. Let v 1 = Pf(W 1 ) and v 2 = Pf(W 2 ), whch are lnearly ndependent vectors by (3.7). Let R be the pecwse lnear curve n R 2 connectng the ponts 0,v 1,v 1 +v 2,v 2,0, and let R 1, R 2, R 3, and R 4 denote the four lnear segments that make up R. It follows from (3.7) that R s a postve smple closed curve, and thus n R (x) = 1 for x nsde R and n R (x) = 0 outsde. We also dvde W nto four segments: W R Fgure 2. The curve W = [W 1, W 2 ] and ts lnearzaton R. W 1, W 2, W 1 1, and W 1 2, correspondng to the four factors of W wth the same names. Note that for each, the curves W and R start end end at the same place.

8 8 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Startng at 0, follow W 1. For a whle t may follow R 1, but assumng, W 1 R 1, the two curves eventually part. Contnue followng W 1 untl ts frst return to R 1. Call ths pont z 1. Then follow R 1 or R 1 back to 0. Call the resultng closed curve Z 1. Let Z 1 = Z 1 + v 2 be the reflecton of Z 1 across R. Next, startng at z 1, repeat the prevous constructon. Follow W 1 untl t leaves R 1 and then returns for the frst tme at z 2. Then follow R 1 or R 1 back to z 1 Call the resultng closed curve Z 2, and defne ts reflecton Z 2 = Z 2 + v 2. Contnung n ths fashon we get a sequence Z 1, Z 2,..., Z l1 of closed curves. We stop when z l1 = v 1. We also obtan ther reflectons Z 1, Z 2,...,Z l 1 Repeat constructon wth W 2 and R 2 to obtan smple closed curves Z l1 +1,..., Z l2, and ther reflectons Z l 1 +1,...,Z l 2, where Z k = Z k v 1 for k = l 1 + 1,...,l 2. Let Λ = {nv 1 + mv 2 : n, m Z} be the lattce n R 2 generared by v 1 and v 2. Consder the tesselaton R of R 2 by the parallelograms Λ + R. We can assume wthout loss of generalty that each closed curve Z les nsde one of the parallelograms n R. If not, we can subdvde Z nto an sum of closed curves (.e., a cycle) that follow Z and the lnes n R. Smultaneously, we do the same thng to each Z n reverse. Now for convenence, we renumber all the reflectons as Z l 1, Z l 1 1,...,Z 1, Z l 2 +l 1, Z l 2 +l 1 1,...Z l 1 +1 Z l1 +l 2 +1, Z l1 +l 2 +2,...,Z 2(l1 +l 2 ). Thus the closed curves Z 1,..., Z 2(l1 +l 2 ) come n pars of reflectons (wth opposte orentatons), and each les n ust one parallelogram of R. Defne the chan (3.8) Z := 2(l 1 +l 2 ) Because of the way the curves Z were constructed, W = R + Z. It follows that =1 Z. (3.9) n W (x) = n R (x) + n Z (x) for (non-trace) x R 2. By (3.8) (3.10) n Z (x) = 2(l 1 +l 2 ) =1 n Z (x) for (non-trace) x R 2. Let x 0 R 2 be such that no x Λ + x 0 s n the trace of W, Z, R or R. Let I x0 = { : x Λ + x 0 wth n Z (x) 0}.

9 TILINGS 9 For each I x0 there s a unque x Λ + x 0 so that n Z (x ) 0. Ths s because each Z les n a sngle parallelogram form R. Usng (3.10) we have (3.11) x Λ+x 0 n Z (x) = = x Λ+x 0 2(l 1 +l 2 ) =1 = 2(l 1 +l 2 ) =1 I x0 n Z (x ). n Z (x) x Λ+x 0 n Z (x) For each I x0 there exsts so that Z s the reflecton of Z. Snce Z = Z + (x x ), t follows that n Z (x ) = n Z (x ). Ths mples the last sum n (3.11) s zero, so that (3.12) n Z (x) = 0. x Λ+x 0 Suppose that n W (x 0 ) > 1, and consder two cases: () x 0 s nsder and () x 0 s not nsde R. In case (), (3.9) and the fact that R s a postve smple closed curve mples n Z (x 0 ) 1. It also follows that n W (x) = n Z (x) for x Λ +x 0, x x 0. By (3.12) there exsts such an x so that n W (x) = n Z (x) 1 < 0. Ths contradcts the fact that W s postve. In case (), n Z (x 0 ) = n W (x 0 ) 2. It follows that ether: (a) there exsts x Λ + x 0, x x 0, so that n Z (x) 2, or (b) there exst two dstnct x 1,x 2 Λ+x 0, not equal to x 0, so that n Z (x 1 ) 1 and n Z (x 2 ) 1. In case (a), (3.9) mples n W (x) < 1 snce n R (x) 1. In case (b) we have n R (x ) = 1 for = 1 or = 2 only f x s nsde R. Ths can happen for at most one of the two, snce they dffer by Λ. Otherwse n R (x ) = 0. Then (3.9) mples that n W (x ) < 0 for at least one of the x. In both cases we agan contradct the fact that W s postve Tles, tlngs and tlng patches. Let B 2 denote the set of all pars for, B. Defne (3.13) sgn( ) := sgn(p ) = sgn(sn( p p )). By (2.5), sgn( ) { 1, 1} f and sgn( ) = 0. For <, (3.13) shows that the entres of the matrx S(A) n (2.6) are sgn( ). In ths casewe wrte = ( ), and more generally, for a, b { 1, 1} we smplfy a b = ab( ). We call a postve prototle f sgn( ) = 1, and a negatve prototle f sgn( ) = 1. In the latter case, = ( ) s postve. The set of all postve prototles s denoted by B 2 +. Prototles are nether postve or negatve, and are called trval. We smplfy a b =. Thus B 2 denotes the set of all prototles. Defnton 3.2. Let W F B be a postve sem-smple word (.e., any (x, W) s a postve sem-smple curve). For x R 2, defne the tle t = (x, W) to be the unon of the the nsde of the curve (x, W) wth ts trace. We wrte (x, W) = (x, W) for the boundary of the tle.

10 10 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Fgure 3. The postve prototles for the matrx A n (2.9): 1 2, 1 3,1 4, 3 2, 4 2, 4 3. Now let B 2. To defne the geometrc realzaton of the tles correspondng to, we frst defne the boundary of by We defne the tle ( ) = [, ] = 1 1. t = (x, ) := (x, [, ]). We defne the boundary (t) = (x, ) := (x, 1 1 ), whch s s a pecewse lnear smple closed curve (.e., a parallelogram). The four orented segments of ths curve are called the edges of the tle, and are denoted E ( (x, ) ). Two tles t 1, t 2 are sad to be adacent f E(t 1 ) E(t 2 ),.e., the two tles share an oppostely orented edge. Now consder a fnte sum 2 (3.14) y = n 1 t 1 + n 2 t n l t l of dfferent tles t k = (x k, k k ). Assume that f any two tles n (3.14) ntersect at more than a vertex, they are adacent. Defne (y) = n k (t k ), whch s a cycle. If the cycle (y) s postve sem-smple, then we say y s a (postve) tlng patch. Note that ths mples that n 1 = n 2 = = n k = 1. Tlng patches, modulo translaton are, called tlng protopatches, and are denoted by (B 2 + ) Postve tlng substtutons. Our goal n ths secton s to gve a defnton of a postve tlng substtuton analogous to the defnton θ : B B of an ordnary substtuton. Bascally t wll be a mappng Θ : B+ 2 (B2 + ). But, whereas n the case of a substtuton, essentally any mappng works, we have two addtonal requrements for a tlng substtuton: (a) Θ needs to be able to apply to tles (.e., translatons of prototles), and (b) Θ needs to apply consstently to adacent tles n a tlng patch, and stl output a tlng patch. We wll acheve these two goals usng the next defnton. Defnton 3.3. A tlng substtuton s a mappng Θ : B 2 + (B2 + ) such that there s a morphsm θ : B F B wth the property that (3.15) (Θ( )) = θ([, ]) for each B 2 +. We abbreavate (3.15) θ = (Θ). In effect, ths says that Θ( ) should be a tlng of the nsde of the curve θ([, ]). Snce Θ( ) s a postve tlng patch, t follows that ts boundary θ([, ]) s a postve sem-smple curve. More enerally, we defne Θ(x, ) := (A u x, Θ( )). 2 Ths s n the same sprt as a cycle as a sum of closed curves.

11 TILINGS 11 Lemma 3.4. Let Θ be a postve tlng substtuton. Suppose y = t 1 + t t l (B 2 +),.e., y s a postve tlng patch. Defne Θ(y) = Θ(t 1 ) + Θ(t 2 ) + + Θ(t l ). Then Θ(y) (B 2 + ). Corollary 3.5. The terates Θ n ( ) for B 2 +, and Θn (y), for y (B 2 + ), are well defned for all n 0. Proof of Lemma 3.4. For defne A u ( ) to be the tle whose boundary s the curve connectng the ponts 0,p,p + p,p,0. For t k = (x k, k k ) n y, let A u t k = (A u x k, A u ( k k )), and let A u y = A u t A u t l. Then A u y s a tlng patch. Smlarly, defne θ(t k ) = θ(x k, k k ) = (A u x k, [θ( k ), θ( k )]), usng Defnton 3.2, and put θ(y) = θ(t 1 )+...θ(t l ). Each of these tles has four edges, whch along the boundary are labeled θ( k ), θ( k ), θ( 1 k ) and θ( 1 k ). We have (3.16) (θ(t k )) = (A u x k, [θ( k ), θ( k )]) = (A u x k, θ([ k, k ])) = (Θ(x k, k k )). Snce Θ s a postve tlng substtuton, ts boundary s a postve sem-smple closed curve. Thus (3.16) shows (θ(t k )) s a postve sem-smple closed curve, and θ(t k ) s a tle. If t h and t k are adacent n y then A u t h and A u t k are adacent n A u y. Moreover, both endponts of each edge of A u t k are endponts of the edges of θ(t k ). Thus θ(t h ) and θ(t k ) are adacent n θ(y) whenever t h and t k are adacent n y. It follows that θ(y) s a postve tlng patch by the tles θ(t k ). Now each term Θ(t k ) n Θ(y) s a tlng patch wth boundary (Θ(x k, k k )), whch by (3.16) s (θ(t k )). Snce they only overlap on ther edges (wth opposte orentatons), ther sum s postve sem-smple closed curve, and t follows that Θ(y) s a postve tlng patch. 4. de Brun dagrams 4.1. The defnton of a de Brun dagram. Startng wth a cyclc word W (B ± ), we wll construct an obect Y, called a de Brun dagram, whch depends (n a nonunque way) on W. We wll thnk of such a dagram as a combnatoral representaton of a tlng patch wth W as ts boundary. However, we wll need to drop the restrcton that a patch be a tlng by postve tles, and allow negatve and trval tles as well. Suppose W = a 1 1 a an n. Let c : [0, 1] R 2 be a postve smple closed curve. Startng at c(0) we follow c and attach n arrows normal to c, labeled 1, 2,..., n. We make the kth arrow pont n f a k = 1 and pont out f a k = 1 (see Fgure 4(a)). Ths s called the frame of the (stll to be defned) de Brun dagram Y. We denote the frame by (Y ). Up to a dffeomorphsm of R 2, (Y ) s unquely defned by W. Now we descrbe how to get from the frame (Y ) to the dagram Y. Snce W s cyclc, f(w) = 0, and thus for each B there are the same number of labeled n- and out-arrows. We choose some arbtrary matchng of these and connect each matched par by a non-self ntersectng curve through the nsde of c, called a pseudo-lne. Assume that

12 12 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR (a) 1 2 (b) 1 2 (c) Fgure 4. (a) The frame for W = [W 1, W 2 ] = where c s a square. Two correspondng de Brun dagrams: (b) the product W 1 W 2, and (c) the only other possblty n ths case. at most two pseudo-lnes cross at any pont, and any two pseudo-lnes cross transversally. The resultng pcture s a de Brun dagram Y (see Fgure 4). Each pseudo-lne n Y s labeled by some B, and s orented by ts arrows. Two dagrams that dffer by an orentaton preservng dffeomorphsm of R 2 are consdered the same. The frame (Y ) of a de Brun dagram Y determnes the cyclc word W up to a cyclc permutaton. We wrte (Y ) = W. The vertces of Y are defned to be the crossngs of ts pseudo-lnes. The pseudo-lnes dvde the nsde of c nto fntely many faces F, the edges E of whch are ether pseudolne segments or segments of c. A face wth no c segment edges s called an nternal face. A vertex that s an ntersecton of two pseudo-lnes s called an nternal vertex. Gven a de Brun dagram Y, we can draw a neghborhood of any nternal vertex as shown n Fgure 5, usng a (possbly orentaton reversng) change of coordnates. In γ(f 4 ) γ(f 3 ) F 4 F 1 F 3 F 2 p p γ(f 1 ) γ(f 2 ) (a) (b) Fgure 5. (a) A vertex v of type n canoncal coordnates, showng the four faces F 1, F 2, F 3, F 4. (b) The γ mage of F 1, F 2, F 3, F 4 n the case s postve. In partcular, ths llustrates the case = 1 3 n the Ammann matrx example. partcular, we have a crossng of a straght pseudo-lne segment labeled, pontng up, and a straght pseudo-lne segment labeled, pontng left. We say ths vertex s of type.

13 TILINGS 13 Defnton 4.1. A vertex of type n a de Brun dagram Y s called postve f sgn( ) > 0. A de Brun dagram Y s postve f all ts vertces are postve The tlng patch correspondng to a dagram. Here we descrbe how to go from a nonsngular de Brun dagrams Y to the correspondng tlng patch y. The dea to use planar graph dualty comes from de Brun s algebrac theory of Penrose tlngs [5]. In partcular, an nternal type vertex n Y corresponds to a tle t of type. Smlarly, each face F n Y corresponds, under dualty, to a tle vertex n y. We wll start wth ths correspondence. Let F denote the set of all faces n Y. Gven a vector x 0 and some F 0 F, we defne γ : F R 2 as follows. Put γ(f 0 ) = x 0, and proceed by nducton. Suppose γ has been defned on a set F k F of k < #(F) faces. Let F F\F k be adacent to some F F k across a pseudo-lne segment labeled. Put a = 1 or a = 1 dependng on whether segment s orented to the left or to the rght n crossng from F to F. Defne γ(f) = γ(f ) + ap, and put F k+1 = F k {F }. Lemma 4.2. Suppose Y s a nonsngular de Brun dagram. Let F 1, F 2, F 3 and F 4 be faces surroundng a type vertex n Y, as shown n Fgure 5(a). Then γ(f 1 ), γ(f 2 ), γ(f 3 ) and γ(f 4 ) are the vertces of the tle (γ(f 1 ), ) (Fgure 5(b)), and the boundary of ths tle s the pecwse lnear curve correspondng to the sequence of ponts γ(f 1 ),γ(f 2 ), γ(f 3 ), γ(f 4 ), γ(f 1 ). Proof. We can assume wthout loss of generalty that γ(f 1 ) = 0. Then γ(f 2 ) = p, γ(f 4 ) = p, γ(f 3 ) = p + p (see Fgure 5(b)). Lemma 4.2 says that a postve de Brun dagram descrbes a sort of locally correct tlng patch. It gves each tle a precse locaton, and tles correspondng to adacent vertces meet across complete edges. However, there s no a pror guarantee that dfferent parts of the tlng do not overlap. The next result shows ths cannot happen f the boundary choce s correct. Proposton 4.3. Let W = [W 1, W 2 ] wth W 1, W 2 F B satsfy det ( [Pf(W 1 ), Pf(W 2 )] ) > 0. Suppose Y s a postve de Brun dagram wth (Y ) = W. Then Y corresponds to a postve tlng patch y wth (y) = (x, W) for some x R 2. Proof. By Lemma 4.2, the vertces of Y correspond to postve tles t 1, t 2,...t l. Let T = (t ). Then T s a smple closed curve, whch s postve snce Y s postve. Thus n T (x) = 1 f x s nsde T, and 0 otherwse. Also l (4.1) T = (Y ) = W, snce two adacent tles have oppostely orented boundares. Let x be nsde W. Then (4.1) mples l (4.2) n W (x) = n T (x) 0. =1 =1

14 14 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Snce det ( [Pf(W 1 ), Pf(W 2 )] ) > 0, Proposton 3.1 mples n W (x) = 1, from whch t follows that the sum n (4.2) s 1 for all non-trace x. Thus any x nsde W can be n at most one tle t, so that two tles only can ntersect along ther boundares. Ths means y = t 1 + t t l s a postve tlng patch. Conversely, f y s a postve tlng patch wth (y) = (x, W), then there exsts a de Brun dagram Y correspondng to y such that (Y ) = W. Ths fact, whch we do not use, s dscussed n [5]. In general, we thnk of a not necessarly postve de Brun dagram as a generalzed tlng protopatch by postve, negatve and trval tles. Ths s llustrated n Secton 6.1. We denote the set of all these protopatches by (B 2 ) Product dagrams and tlng substtutons. From now on we wll consder only dagrams Y where (Y ) = [V, W] for V, W F B. For the frame, we take c to be the unt square. We put V along the bottom, put W gong up the rght sde, put V 1 gong backwards along the top, and put W 1 gong down along the left sde (see Fgure 4(a)). The smplest dagram of ths type s a product dagram, denoted Y = V W. To obtan t, we start wth the frame descrbed above and connect the arrows by vertcal and horzontal lnes (see Fgure 4(b)). It follows that (4.3) (V W) = [V, W]. The followng matrx notaton for a product dagram wll be useful. Suppose V = v 1 v 1,...v l and W = w 1 w 2...w n. We wrte v 1 w n v 2 w n... v l w n V W = v 1 w 1 v 2 w 1... v l w 1 5. Cancellaton 5.1. The three moves on de Brun dagrams. Let Y be a de Brun dagram. In ths secton we wll descrbe three moves µ that can be appled to a dagram Y to obtan a new dagram Y. Snce the moves are reversble denote them Y µ Y. The frst move µ 1 s called the flp. In t, we slde a a pseudo-lne across a vertex. It k k^ k^ ^ k Fgure 6. The flp move µ 1. s called the flp move because n a (postve) tlng t mplements the Necker cube flp (see Fgure 7).

15 TILINGS 15 k Fgure 7. The Necker cube flp, mplemented by the flp move µ 1. The second move µ 2 s called cancellaton. It takes a par of opposte sgn adacent vertces, and ( ), and cancels them by uncrossng two loops (see Fgure 8). ^ K^ Fgure 8. The cancellaton move µ 2. The thrd move µ 3 s called trval tle elmnaton. It may be used to cancel a trval tle, whch occurs when two pseudo-lnes wth the same label cross (see Fgure 9). Move µ 3 s dfferent form the other two for two reasons: () pseudo-lnes are cut and reconnected n a dfferent way, and () t depends on the orentaton. ^ Fgure 9. Trval tle elmnaton move µ 3. Defnton 5.1. Two de Brun dagrams are called µ-equvalent f one can be obtaned form the other by a fnte seres of moves. Lemma 5.2. If Y µ Y the (Y ) = (Y ) Countng tles. For a de Brun dagram Y based on B = {1, 2,..., d}, let M(Y ) be the d d matrx wth, th entry m equal to the number of type vertces that occur n Y. Let e be the d d matrx wth ( )th entry equal to 1 and all other entres 0.

16 16 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Lemma 5.3. Let Y and Y be de Brun dagrams wth (Y ) = (Y ). () If Y µ 1 Y (.e., a flp) then M(Y ) = M(Y ). () If Y µ 2 Y, where µ 2 mplements a cancellaton of and, then M(Y ) = M(Y ) e e. () Y µ 3 Y, where µ 3 mplementselmnaton of a trval tle, then M(Y ) = M(Y ) e. For B 2, let = sgn( )( ). That s to say, for a nontrval tle, s ts postve verson. Now suppose Y s a de Brun dagram and defne the vector f (Y ) Z (d 2) to have entres (5.1) f (Y ) = sgn( )(m m ), < n lexcographc order. Corollary 5.4. (of Lemma 5.3) The vector f (Y ) s an nvarant of µ-equvalence. For an endomorphsm θ, defne the product tlng substtuton by Proposton 5.5. Suppose θ s an endomorphsm on B = {1,...,d}, and let and let A = L θ. Then the product tlng substtuton θ θ : B 2 + (B 2 ), defned by (5.2) (θ θ)( ) := θ() θ(), satsfes A = (f (Θ(1 2)), f (Θ(1 3)),..., f (Θ((d 1) d))), where Θ = θ θ. Proof. Frst note that by (2.7) and (3.13) that A has entres ( ) a,k l = sgn( )sgn( ) det a,k a,l, a,k a,l where a, are the entres of A. By (5.1) f (Θ(k l)) = sgn(k l) ( m m ) = sgn( )sgn(k l)(m m ), where the m are the entres of M(Θ(k l)). Snce Θ(k l) = θ(k) θ(l) and L θ = A, t follows that m = a,k a,l and m = a,k a,l. Thus ( ) a,k a m m = det,l, a,k a,l and f (Θ(k l)) = a,k l.

17 TILINGS Properly ordered words. Let (5.3) W = a 1 1 a a l l F B. We say W has the natural order f W = 1 a 1 2 a 2...d a d, where ak Z. (In effect, we have mposed an arbtrary order on B). We say W s unpotent f a = 1 for = 1,...,l, We say W s postve f a k > 0 for = 1,...,l, A postve word W has f(w) 0 (where f s the abelanzaton mappng). Defnton 5.6. We say W s effcent f whever = k, then a and a k are ether both postve or both negatve. Two effcent words W 1 and W 2 are called dsont f f(w 1 ) and f(w 2 ) have dfferent entres nonsero. We say a word W 2 s a subword of W 1 f W 2 s obtaned from W 1 by deletng some letters. We wrte W 2 W 1. Defnton 5.7. A word W F B s sad to be properly ordered f any of the followng hold: (1) W s postve, unpotent, and has the natural order, (2) W 1, where W s postve, unpotent and has the natural order, (3) W = (W 1 ) 1 W 2, where W 1 and W 2 are postve, unpotent, have the natural order, and are dsont. (4) W = W 1 W 2...W n, where each W satsfes (1), (2) or (3) above, and there s a permutaton 1, 2,..., n of 1,..., n so that (5.4) W k+1 W k, k < n. Lemma 5.8. For any a Z d, a 0, there exsts a properly ordered word W wth f(w) = a. Ths follows drectly from Defnton 5.7. We say a endomorphsm θ s properly ordered f θ() s a properly ordered word for each B. Corollary 5.9. Suppose A s a d d nteger matrx so that no row has all entres zero. Ths holds, n partcular, f A satsfes (2.1). Then there exsts a properly ordered endomorphsm θ so that L θ = A. Ths follows from Lemma 5.8. We say a non-sngular de Brun dagram Y s effcent f there are no two vertces and ( ) =. Proposton Let V, W F B be properly ordered. Then there exsts an effcent de Brun dagram Y that s µ-equvalent to V W. Proof. We frst prove a basc case n whch we assume V and W are postve, unpotent and naturally ordered. Thus V = 1 a 1 2 a 2...d a d W = 1 b 1 2 b 2... d b d where a k, b k {0, 1}.

18 18 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Let U = 1 c 1 2 c 2...d cs, where c k = mn{a k, b k }, and let l be the length of U. Thnk of V(U U) V(V W). Denote the set of l vertces along the dagonal of U U by S. All of them are trval. Let S 1 denote the set of vertces of U U above the dagonal and let S 2 denote the correspondng set of vertces below the dagonal. Let ψ : S 1 S 2 be the transpose, map, whch takes a vertex to ts flp across the dagonal. A vertex S 1 s connected to ts transpose = ψ( ) S 2 by a a sequence of two pseudo-lne segments labeled that meet at a trval vertex. The same par of vertces s also connected by a sequence of two pseudo-lne segments labeled that meet at a trval vertex (see Fgure 10(a)). (a) (b) (c) Fgure 10. (a) A vertex confguraton showng S = {, }, S 1 = { } and S 2 = { }. (b) The result of applyng move µ 3 to the trval vertces n S. (c) Here and are canceled usng µ 2. After performng move µ 3 (trval tle elmnaton) on and on, there wll be a sngle curve labeled and a sngle curve labeled connectng the two remanng vertces, and, common to the two curves (see Fgure 10(b)). If these two vertces are adacent along these curves, they can be canceled by move µ 2 (Fgure 10(c)). However, there may be other vertces between the two loops (see Fgure 11(a)). In case the two vertces must frst be brought together by a seres of µ 1 (flp) moves (see Fgure 11(b)). In general, we proceed as follows: () We remove all the trval tles n S, usng move µ 3 a total of l tmes. () We make all the trval tles adacent usng the flp move µ 1, as many tmes as necessary. () We cancel l pars of trval tles, matched by ψ. Let us call the resultng dagram Y. Step () requres some comments. (1) There s no loss of generalty n assumng that S, S 1, S 2 V(U U) V(V W). To reduce V W to Y may requre addtonal flp moves µ 3, but no addtonal moves µ 1 or µ 2 are needed. Thus we can assume Y s µ-equvalent to V W. (2) The sequence of flp moves s hghly non-unque, so the resultng dagram Y s not unque.

19 TILINGS 19 (a) (b) Fgure 11. (a) Foregn vertces nsde the loop. (b) The result of applyng eght flp moves µ 1 as well as a few cancellatons. (3) It s crucal n ths part of proof that V and W are naturally (.e., properly) ordered. In partcular, f we take the dagram V W where V = and W =, the cancellaton wll fal (see Fgure 12). (a) (b) Fgure 12. Badly ordered words lead to a dagram where cancellaton fals. Now let us consder the vertces of Y. Let V and W be the subwords of V and W that reman after removng U. Snce all the nontrval vertces that were n U U have been canceled, all the nontrval vertces n Y come from V U, U W, or V W. Snce U, V and W are all dsont, there are no pars and n these sets. It follows that Y s effcent, and the lemma s proved n ths basc case. There are three more basc cases: V 1 W, V W 1 and V 1 W 1, where V and W postve, unpotent, and naturally ordered. The proofs, n each of these cases, proceed exactly as above. Now suppose V = V1 1 V 2 and W = W1 1 W 2 are smply properly ordered as n part (3) of Defnton 5.7. That s V 1, V 2, W 1, W 2 are postve, unpotent, naturally ordered words, wth V 1 and V 2 dsont and W 1 and W 2 dsount. Then [ V 1 1 W 2 V 2 W 2 V W = V1 1 W1 1 V 2 W1 1 ].

20 20 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Note that the four entres n the matrx V W are de Brun dagrams that are dsont n the sense that they have no vertces n common. We apply the prevous argument to each of thes four entres to obtan µ-equvalent dagrams Y 1, Y 2, Y 3, Y 4 that are effcent. Then [ ] Y1 Y Y = 2. Y 3 Y 4 s effcent and µ-equvalent to V W. Fnally, suppose V = V 1 V 2...V n and W = W 1 W 2...W l are properly ordered, each satsfyng (5.4), where each V m and W k are unpotent properly ordered. Then V 1 W l... V n W l V W = V 1 W 1... V n W 1 We complete the proof by applyng the prevous argument to each entry V m W k. Consder a non erasng endomorphsm θ such that L θ = A. We say θ s effcent f the word θ() s effcent for each B. We say θ s the natural order endomorphsm f each θ() has the natural order. We say θ s a properly ordered endomorphsm f θ() s properly ordered for each B. 6. The man result Theorem 6.1. Suppose A s a d d matrx satsfyng (2.1), (2.3) and (2.8). Then there exsts a properly ordered non endomorphsm θ wth L θ = A, such that there s a postve tlng substtuton Θ wth (Θ) = θ. Proof. By Corollary 5.9, (2.1) mples there exsts a properly ordered non erasng endomorphsm θ wth L θ = A. For each B+ 2, there exsts an effcent de Brun dagram Y ( ) that s µ-equvalent to θ() θ() by Proposton By Corollary 5.4, f (Y ( )) = f (θ() θ()). By Proposton 5.5, the vector f (θ() θ()) s column of A. Thus A 0 mples f (Y ( )) 0, whch snce Y ( ) s effcent, shows that t s postve. Defne Θ( ) to be the geometrc realzaton of Y ( ). By Proposton 4.3, Θ( ) (B+) 2,.e., t s a postve tlng patch. Fnally we have (Θ( )) = ( (θ θ)( ) ), by Lemma 5.2, = ( θ() θ() ), by (5.2), = [θ(), θ()], by(4.3), = θ([, ]), endomorphsm, = θ( ( )), so (Θ) = θ. It follows that Θ s a postve tlng substtuton. We defne the structure matrx L Θ of a postve tlng substtuton θ to be the ( d 2) ( d 2) nteger matrx whose, k l entry gves the number of tmes k l occurs nsde Θ( ).

21 TILINGS 21 Corollary 6.2. The postve tlng substtuton Θ constructed above satsfes L Θ = A,.e., (L θ ) = L Θ Example. The tlng substtuton for the Ammann matrx. Let A be the Ammann matrx (2.9), whch we know from Secton 2.4 satsfes (2.1), (2.3) and (2.7). Let P be the proecton (2.10). The frst task s to defne a properly ordered endomorphsm θ. For each we want θ() = W, where W s a properly ordered word such thatf(w ) s the th column of A. Usng cases (2) and (3) of Defnton 5.7, we obtan the followng endomorphsm θ(1) = θ(2) = θ(3) = θ(4) = Next defne the product substtuton (θ θ)( ) = θ() θ() whose value on each of the sx postve prototles we represent as product de Brun dagram. In two cases, (θ θ)(1 3) and (θ θ)(4 2), the product dagrams consst of ust postve and trval tles. For example, (θ θ)(1 3) = ( ) ( ) = , n matrx form, where the entres have been smplfed to postve prototles. The resultng µ-equvalent postve de Brun dagram Y (1 4) s shown below (left), as well as ts realzaton Θ(1 4) as a tlng patch (rght). f h e b d g p2 p3 p1 p4 Y (1 3) Θ (1 3) Smlarly, for (θ θ)(4 2) we obtan:

22 22 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. p2 p3 p4 p1 Y (4 2) Now consder the case Θ (4 2) (1 2) (θ θ)(1 2) = ( ) ( ) = Notce that the upper rght 2 2 block has the form shown n Fgure 10(a), and can be cacelled as shown n Fgure 10(b) and (c) to obtan the postve dagram Y (1 2) shown below (left). The resultng geometrc realzaton Θ(1 2) s shown (rght). p2 p3 p1 p4 Y (1 2) Θ (1 2) There are two more cases that work exactly the same way: (θ θ)(3 2) and (θ θ)(4 3). These are shown below.

23 TILINGS 23 p2 p3 p4 p1 Y (3 2) Θ (3 2) p2 p3 p4 Y (4 3) Θ (4 3) p1 The most complcated case s (1 2) (θ θ)(1 4) = ( ) ( ) = Notce that the requred cancellatons are not adacent. We use the method n the proof of Proposton 5.10 as llustrated n Fgure 10. The steps are shown n detal below.

24 24 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. p2 p3 p4 p1 Product dagram (θ θ)(1 4). (θ θ) (1 4) as a sgned tlng patch. p2 p3 p1 p4 p2 p3 p1 p4

25 TILINGS 25 p2 p3 p4 p1 Y (1 4) Θ (1 4) Now we wll show how to terate Θ to obtan tlng patches and tlngs. We start wth the tlng patch y, shown below, and apply Θ to obtan a sequence of patches Θ n (y). The Fgure 13. The patches y and Θ(y). patch y has been carefully chosen so that for each B+, 2 a translaton of y appears as a subpatch of each Θ n ( ) for all n suffcently large. The fact that such a patch y exsts follows from the fact that (A ) N > 0 for all N 2. We say n such a case that Θ s a prmtve tlng substtuton (ths means that each k l appears n each Θ N ( ) for all N suffcently large). Another property of the patch y s that t appears as a subpatch of Θ 2 (y), completely surrounded by other tles, as shown n Fgure 13. It follows that Θ 2N (y) s a subpatch of Θ 2M (y) for all M > N. Thus the nfnte sum N 0 Θ2N (y) s na ncreasng sum of patches, and defnes a tlng of R 2. A swatch of ths tlng s shown n Fgure 16. It s nterestng to note that one obtans a dfferent tlng wth the sum N 0 Θ2N+1 (y) (there s a dfferent patch y around the orgn). However, snce Θ s prmtve, both of these belong to the same tlng space (see [16]) X Θ. We call ths an Ammann tlng although t s much less symmetrc than the classcal Ammann-Beenker tlng (see [17]).

26 26 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Fgure 14. The patch Θ 2 (y), showng y as a sub-patch around 0. Fgure 15. The patch Θ 3 (y). 7. The postvty of (A ) N In addton to (2.1) and (2.3), all of the tlng results n ths paper requre that A satsfy (2.8): A 0. We begn by notng that A and A N, N > 1, have the same expandng subspace E u and the same proectons P to E u. Thus A N satsfes (2.1) and (2.3) f and only f A does. In general, we are not so concerned wth the dfference between A and A N. If θ s a non-abelanzaton of A, then θ N s a non-abelanzaton of A N, and we tend to thnk of θ N and θ as the same substtuton. The man result n ths secton s that A N may satsfy (2.8) (or rather a slghtly stronger condton) even f A does not.

27 TILINGS 27 Fgure 16. A swatch of the Ammann tlng. It follows easly from the Bnet-Cauchy theorem (Theorem 2.3) that f A satsfes (2.1) then (7.1) (A ) N = (A N ) for all N 0. Instead of lookng for matrces A where A 0, t wll suffce to fnd A so that (A ) N = (A N ) 0. Unfortunately ths s not so easy. However, t turns out to be possble to fnd good condtons on A for the stronger concluson (A ) N > 0. In ths case, we say A s eventually postve. Even n the case where A 0, the stuaton that A s eventually postve s mportant because t mples a correspondng tlng substtuton Θ s a prmtve tlng substtuton. In any case, f a matrx A satsfyng (2.1) and (2.3) s such that (A ) N > 0 then (7.1) mples B = A N satsfes (2.1), (2.3) and (2.8) The eventual postvty of A. A matrx A s called eventually postve f A N > 0 for all N suffcently large. A matrc A s prmtve f A 0 and A s eventually postve. Our man result n ths secton s the followng. Theorem 7.1. Let A satsfy (2.1) and (2.3). The matrx A s eventually postve f and only f S(A) = ±S(A T ), where S(A) s the matrx defned n (2.6). Snce A and A T have the same egenvalues, A satsfes (2.1) f and only f A T does. The proof of Theorem 7.1 wll come at the end of ths secton. Corollary 7.2. If A T = A, then A s eventually postve. If A = A T, then A satsfes (2.3) f and only f A T does (they share the proecton P). As we wll show, however, there are also many non-symmetrc examples. In the 3 3 case we get the followng. Corollary 7.3. Let B Sl 3 (Z). Suppose B s eventually postve and satsfes the followng Psot condton: dm(e u (B)) = 1. Then A := B 1 satsfes (2.1), (2.3) and (2.8). Combnng ths Theorem 6.1, we obtan a new proof of the result from [9] that under the constons of Corollary 7.3 there exsts a tlng substtuton Θ wth boundary θ whose structutre matrx s A. In [9] an addtonal connectvty result a obtaned as well.

28 28 MAKI FURUKADO, SHUNJI ITO, AND E. ARTHUR ROBINSON, JR. Proof of Corollary 7.3. Assume wthout loss of generalty that B > 0. The Psot property for B mples that B has three dstnct egenvalues satsfyng ω 3 > 1 and ω 2, ω 1 < 1. It follows that the correspondng egenvalues of A, λ = ω 1, satsfy λ 1, λ 2 > 1 and λ 3 < 1, so A satsfes (2.1). Let v 1,v 2,v 3 be the correspondng egenvectors. Snce B > 0 we can assume v 3 = (1, a, b) where a, b > 0. Let w 1 = ( a, 1, 0) and w 2 = ( b, 0, 1). Snce w 1 v 3 = w 2 v 3 = 0, ( ) a 1 0 (7.2) P = (w 1,w 2 ) T = b 0 1 and C 2 (P) = (b, a, 1). It follows that S(A) = dag(1, 1, 1). Snce B T > 0, and snce A T = (B T ) 1, the same argument shows S(A T ) = S(A), and the corollary follows from Theorem Subspaces and proectons. We say v C d s real f γv R d for some nonzero γ C; otherwse we say v s complex. Let v 1,v 2 C d be lnearly ndependent over C. If both v 1 and v 2 are real then span{v 1,v 2 } R d denotes span over R. Otherwse span{v 1,v 2 } C d denotes span over C. In general, we let F = C or F = R. Gven a two dmensonal subspace E = span{v 1,v 2 } F d, we call a 2 d matrx P satsfyng (2.1) a proecton to E. As before, (2.2) holds, except now for M Gl 2 (F). A subspace E s called real f t has a real bass, or equvalently, a real proecton P. In partcular, ths means there s a nonsngular M so that MP has all real entres. Defnng R = (v 1,v 2 ), we have that (2.1) mples PR = I. We put A E = PAR, whch satsfes P E A = A E P E Jordan forms. Let V GL d (C) be such that V J = AV, where J s the upper trangular Jordan Canoncal Form of A, wth dag(j) = (λ 1, λ 2,...,λ d ) C d. The columns V = (v 1,...,v d ) are a bass of ordered generalzed egenvectors for A. If A s dagonalzable, these are actually egenvectors. Proposton 7.4. Let A satsfy (2.1) and let w 1 and w 2 be the frst two ordered generalzed egenvectors for A T (.e., correspondng to E u (A T )). Then P = (w 1,w 2 ) T s a proecton to E u (A). Proof. The unque dual bass w 1,...w d correspondng to v 1... v d s defned to satsfy v w = δ,. Thus W := (w 1,w 2,...,w d )T satsfes W = V 1. We have A T (W ) T = (W ) T J T because W A(W ) 1 = V 1 AV = J mples W A = JW. But J T s the lower-trangular Jordan form for A. There exsts a permutaton matrx U so that U T J T U = J where the frst two rows (and columns) of U are ether: (a) e 1,e 2, or (b) e 2,e 1, dependng on whether or not λ 1 and λ 2 are n a non-trval Jordan block (ths can only happen f λ 1 = λ 2 ). Let W = U T W, and express W = (w 1,...,w d ) T. Then W 1 A T W = (U T W ) 1 A T (U T W ) = J. Ths shows w 1,...,w d are the ordered generalzed egenvectors ( ) for A T. Let P = (w 1,w 2 ) T. 0 1 Then P = MP, where M = I n case (a), or M = n the case (b). 1 0

29 Corollary 7.5. The expandng subspace E u (A) s real. TILINGS 29 Proof. Ths s clear f λ 1 s real, snce ths s equvalent to λ 2 also beng real. If λ 1 s complex, ( then ) λ 2 = λ 1. Put P = (w 1,w 1 ) T. Let P = MP = (Re{w 1 }, Im{w 1 }) T 1 1 where M = 1. Then P 2 s a real proecton to E u Perron-Frobenus theory for A. Let v 1,v 2 R d. Defne v 1 v 2 R (d 2) by (7.3) v 1 v 2 = C 2 ( (v1,v 2 ) ). Lemma 7.6. If A s d d and v 1,v 2 R d then Proof. C 2 (A)(v 1 v 2 ) = (Av 1 ) (Av 2 ). C 2 (A)(v 1 v 2 ) = C 2 (A)C 2 ( (v1,v 2 ) ) = C 2 ( (Av1, Av 2 ) ) = (Av 1 ) (Av 2 ). A matrx B Z p p s called spectrally Perron (see [14]) f t has a real egenvalue ω > 0, such that for any other egenvalue ω of B, ω < ω. The egenvector u correspondng to ω s called a Perron egenvector. Theorem 7.7. (Lnd and Marcus, [14]) A matrx B s eventually postve f and only f t s spectrally Perron and the Perron egenvectors u and u for B and B T respectvely are both postve. The only f drecton follows from the the Perron-Frobenus Theorem (see [14]). The f drecton s Lnd and Marcus [14], Exercse Lemma 7.8. Let A satsfy (2.1) and let E = span{v 1,v 2 } be any real or complex 2- dmensonal A-nvarant subspace. Let P be a proecton to E, and let A E be the nduced matrx on F 2. Then u = S(A)(v 1 v 2 ) s an egenvector for A correspondng to the egenvalue ω = det(a E ). Moreover, every egenvalue of A s obtaned ths way. Proof. We have RA E = RPAR = AR. Usng ths wth (7.3) and the Bnet-Cauchy Theorem (Theorem 2.3) C 2 (A)(v 1 v 2 ) = C 2 (A)C 2 (R) = C 2 (AR) = C 2 (RA E ) = det(a E )C 2 (R) = ω (v 1 v 2 ), where ω = det(a E ). The fact that every egenvalue of C 2 (A) s obtaned ths way follows from the Jordan canoncal form for A.