Topological properties of a class of cubic Rauzy fractals
TTOPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZYFRACTALS
BENOˆIT LORIDANT
Abstract.
We consider the substitution σ a,b defined by σ a,b : 1 (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) a (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) b (cid:55)→ a ≥ b ≥
1. The shift dynamical system induced by σ a,b is measure theoreticallyisomorphic to an exchange of three domains on a compact tile T a,b with fractalboundary.We prove that T a,b is homeomorphic to the closed disk iff 2 b − a ≤
3. Thissolves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we constructa H¨older continuous parametrization C a,b : S → ∂ T a,b of the boundary of T a,b . Asa by-product, this parametrization gives rise to an increasing sequence of polygonalapproximations of ∂ T a,b , whose vertices lye on ∂ T a,b and have algebraic pre-imagesin the parametrization. Introduction
In 1982, G. Rauzy studied the dynamical system generated by the substitution σ (1) =12 , σ (2) = 13 , σ (3) = 1 and proved that it is measure theoretically conjugate to a domainexchange on a compact subset T of the complex plane [35]. Moreover, it has pure discretespectrum and it is isomorphic to translation on the two dimensional torus. T has a self-similar structure and induces both a periodic and an aperiodic tiling of the plane. Theresults of Rauzy were generalized. A Rauzy fractal
T ⊂ R d − can be attached to eachirreducible unimodular Pisot substitution σ on d letters. The shift dynamical systemgenerated by σ is measure theoretically isomorphic to a domain exchange on d subtilesof T , provided that σ satisfies the combinatorial strong coincidence condition [6, 16].If σ satisfies the super coincidence condition , the shift dynamical system has even purediscrete spectrum and is measure theoretically isomorphic to a translation on the ( d − T induces a periodic tiling and thesubtiles T ( i ) for i ∈ { , . . . , d } an aperiodic self-replicating tiling of R d − [26]. In fact,the outstanding Pisot conjecture states that the dynamical system generated by everyirreducible unimodular Pisot substitution has pure discrete spectrum.There is a vast literature on Rauzy fractals, as they appear naturally in many do-mains. In β -numeration ([43]), finiteness properties of digit representations are relatedto the fact that 0 is an inner point of the Rauzy fractal, and the intersection of theRauzy fractal with lines allows to characterize the rationals numbers with purely peri-odic expansion [4]. In Diophantine approximation, best simultaneous approximations areobtained by computing the size of the largest ball inside the Rauzy fractal [23]. Rauzy Date : October 22, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Substitutions, Rauzy fractals, Tilings, Automata, Homeomorphy to a disk.This research was supported by the project P22-855 of the Austrian Science Fund (FWF) and by theproject FAN-I1136 of the FWF and the ANR (Agence Nationale de la Recherche). a r X i v : . [ m a t h . D S ] N ov BENOˆIT LORIDANT fractals also play an important rˆole in the construction of Markov partitions for toralautomorphisms. It is known that every hyperbolic automorphism of the d -dimensionaltorus admits a Markov partition [39, 12]. For d = 2, the partition is made of rectan-gles [1]. However, for d >
2, the partition can not have a smooth boundary [13]. Markovpartitions for hyperbolic toral automorphisms were explicitly constructed in [34, 33, 25]using cylinders whose bases are the original subtiles of the Rauzy fractals. Whenever theRauzy fractal is homeomorphic to the closed disk, the situation remains close to the case d = 2, as the Markov partition consists in topological 3-dimensional balls.In their monograph [38], Siegel and Thuswaldner give algorithms to check topologicalproperties such as tiling property, connectedness or homeomorphy to the closed disk forany given Pisot unimodular substitution. These criteria use graphs and rely on the self-similar structure of the Rauzy fractals. However, it is usually more difficult to describethe topological properties for whole families of Rauzy fractals.In this paper, we consider the Rauzy fractals T a,b associated with the substitutions σ a,b : 1 (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) a times (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) b times (cid:55)→ { , , } , where a ≥ b ≥
1. For every such parameters a, b , σ a,b isan irreducible primitive unimodular Pisot substitution. Moreover, it satisfies the supercoincidence condition [7, 41]. Therefore, T a,b induces a periodic tiling and its subtiles T a,b ( i ) ( i = 1 , ,
3) an aperiodic self-replicating tiling of the plane.We will show that T a,b is homeomorphic to the closed disk if and only if 2 b − a ≤ T a,b . A standard method forthe boundary parametrization of self-affine tiles was proposed by Shigeki Akiyama andthe author in [5]. We will be able to extend this construction for the boundary of oursubstitution tiles, as it mainly relies on the graph-directed self-similar structure of theboundary. A by-product of the parametrization is a sequence of boundary approximationswhose way of generation is analogous to Dekking’s recurrent set method [17, 18].We mention existing results. In the case b = 1, the tiles T a, were shown to be disk-like and the Hausdorff dimension of their boundary was computed by Messaoudi [29, 30]via a boundary parametrization, but the technique used to parametrize would not gen-eralize to the non disk-like tiles. In [24], Ito and Kimura produced the boundary of T , by Dekking’s fractal generating method, making use of higher dimensional geomet-ric realizations of the Tribonacci substitution. This also allowed the computation ofthe Hausdorff dimension of the boundary. They could generalize their method in [36].In [44], Thuswaldner computed the so-called contact graph, related to the aperiodictilings induced by T a,b , for the whole class of substitutions σ a,b and deduced the Haus-dorff dimension of the boundary of T a,b . This graph will be of great importance in ourparametrization procedure. In [27], the non-disk-likeness for the parameters satisfying2 b − a > T a,b , for all parameters a ≥ b ≥ b − a >
3, the number of states in this graph, which is also thenumber of neighbors of T a,b in the periodic tiling, is strictly larger than 8. However, in aperiodic tiling induced by a topological disk, the tiles have either 6 or 8 neighbors [21].Therefore, T a,b is not homeomorphic to a disk. We will recover this result by anothermethod based only on the contact graphs, showing that the parametrization is not injec-tive for these parameters. The proof of the counterpart is more intricate, as it consists OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 3 in showing the injectivity of the parametrization for 2 b − a ≤
3: this requires ratherinvolved computations on B¨uchi automata.The paper is organized as follows. In Section 2, we recall basic facts concerning ourclass of substitutions and formulate our main results. In Section 3, we introduce twographs that are essential in our work: the boundary graph G ,a,b , that describes thewhole language of the boundary of T a,b , and a subgraph G ,a,b ⊂ G ,a,b , whose languageis large enough to cover the boundary. In Section 4, we use the graph G ,a,b to constructthe boundary parametrization, proving Theorem 2.2. Section 5 is devoted to the proof ofTheorem 2.1. If 2 b − a ≤
3, then G ,a,b = G ,a,b and we can show that the parametrizationis injective. Therefore, ∂ T a,b is a simple closed curve and T a,b is disk-like. Otherwise,the complement of G ,a,b in G ,a,b is nonempty and we can find a redundant point in theparametrization. Finally, in Section 6, we add some comments and questions for furtherwork. Acknowledgements.
The author is grateful to Shigeki Akiyama and Shunji It¯o formentioning the conjecture and for the motivating discussions on this subject.2.
Main results
We wish to study the topological properties of a class tiles arising from a family ofsubstitutions.2.1.
Substitutions σ a,b . Let A := { , , } be the alphabet . We denote by A ∗ the set offinite words over A , including the empty word ε . For a ≥ b ≥
1, we call σ = σ a,b : A ∗ →A ∗ the mapping(2.1) σ : 1 (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) a times (cid:55)→ . . . (cid:124) (cid:123)(cid:122) (cid:125) b times (cid:55)→ , extended to A ∗ by concatenation.For a word w ∈ A ∗ , we write | w | its length and | w | a the number of occurrences of aletter a in w . We define the abelianization mapping l : w ∈ A ∗ (cid:55)→ ( | w | a ) a ∈A ∈ N The incidence matrix M of the substitution σ is the 3 × l ( σ ( w )) = Ml ( w )for all w ∈ A ∗ . Thus we have M = a b
11 0 00 1 0 . M is a primitive matrix, i.e. , M k has only strictly positive entries for some power k ∈ N (here, k = 3). We denote by β the corresponding dominant Perron-Frobeniuseigenvalue, satisfying β = aβ + bβ + 1. The substitution σ has the following properties.It is • primitive : the incidence matrix M is a primitive matrix; • unimodular : β is an algebraic unit; • irreducible : the algebraic degree of β is exactly |A| = 3; BENOˆIT LORIDANT !, , . . . , · · · ! " a − !, , . . . , · · · ! " b − ! · · · ! " a · · · ! " b Figure 1.
Prefix-suffix graph: i p −→ j ∈ Γ ⇐⇒ σ ( j ) = pis . • Pisot : the Galois conjugates α , α of β satisfy | α | , | α | < Associated Rauzy fractals T a,b . We turn to the construction of the Rauzy fractalsassociated with the substitution σ .Let v β be a strictly positive left eigenvector of M for the dominant eigenvalue β and u β a strictly positive right eigenvector with coordinates in Z [ β ], satisfying (cid:104) u β , v β (cid:105) = 1.Moreover, let u α i be the eigenvectors for the Galois conjugates obtained by replacing β by α i in the coordinates of the vector u β . We obtain the decomposition R = H e ⊕ H c , where • H e is the expanding line , generated by u β , • H c is the contracting plane , generated by u α , u α (or by (cid:60) ( u α ) , (cid:61) ( u α ) when-ever α , α are complex conjugates).We denote by π : R → H c the projection onto H c along H e and by h the restriction of M on the contractive plane H c . Note that if we define the norm || x || = max {|(cid:104) x , v α (cid:105)| , |(cid:104) x , v α (cid:105)|} , then h is a contraction with || hx || ≤ max {| α | , | α |}|| x || for all x ∈ H c .Furthermore, we have(2.3) ∀ w ∈ A ∗ , h ( π ( l ( w ))) = π ( Ml ( w )) = π ( l ( σ ( w ))) . The fixed point w = w w w · · · = lim k →∞ σ k (1) ∈ A N imbeds into R as a discreteline with vertices { l ( w · · · w n ); n ∈ N } . The assumption that σ is a Pisot substitutionimplies that this broken line remains at a bounded distance of the expanding line. Pro-jecting the vertices of the discrete line on the contracting plane, we obtain the Rauzyfractal of σ (see [6]): T = T a,b = { π ◦ l ( w w . . . w n − ); n ∈ N } , ∀ i ∈ A , T ( i ) = T a,b ( i ) = { π ◦ l ( w w . . . w n − ); w n = i, n ∈ N } . For our purpose, we will need to view the Rauzy fractals as solution of a graph directediteration function system ( GIFS , see [28]). The appropriate graph is the prefix-suffixgraph , defined as in [15]: • vertices: the letters of A ; • edges: i p −→ j if and only if σ ( j ) = pis for some s ∈ A ∗ .The prefix-suffix graph Γ = Γ a,b of σ is depicted on Figure 1. OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 5
Since σ is a primitive unimodular Pisot substitution, T is the attractor of the GIFSdefined by the prefix-suffix graph (see for example [11]):(2.4) ∀ i ∈ A , T ( i ) = (cid:83) i p −→ j h T ( j ) + π l ( p ) , T = (cid:83) i =1 T ( i ) . From this GIFS structure we deduce that the Rauzy fractal and its subtiles are ageometric representation of the language of the prefix-suffix graph [16]: T = (cid:88) k ≥ h k π ( l ( p k )); i p −→ i p −→ i p −→ . . . ∈ Γ and for i ∈ A (2.5) T ( i ) = (cid:88) k ≥ h k π ( l ( p k )); i = i p −→ i p −→ i p −→ . . . ∈ Γ . There are other equivalent constructions of the Rauzy fractal. An overview of thedifferent methods can be found in [10].Fundamental topological properties of these Rauzy fractals can be found in the liter-ature.(1) T is a compact set and T = T o .(2) For i = 1 , ,
3, the subtile T ( i ) is a compact set and T ( i ) = T ( i ) o .(3) The subtiles induce an aperiodic tiling of the contracting plane. Let ( e , e , e )be the canonical basis of R . The tiling set isΓ srs := (cid:8) [ π ( x ) , i ] ∈ π ( Z ) × A ; 0 ≤ (cid:104) x , v β (cid:105) < (cid:104) e i , v β (cid:105) (cid:9) and(2.6) ∀ [ γ, i ] (cid:54) = [ γ (cid:48) , j ] ∈ Γ srs , ( T ( i ) + γ ) o ∩ ( T ( j ) + γ (cid:48) ) o = ∅ , H c = (cid:83) [ γ,i ] ∈ Γ srs T ( i ) + γ. (1) and (2) hold because σ is a primitive unimodular Pisot substitution [40]. (3) is aconsequence of the combinatorial super coincidence condition satisfied by σ . Indeed,Solomyak [41] proved in 1992 that the associated dynamical system has pure discretespectrum, and Barge and Kwapisz [7] showed in 2006 that this is equivalent to the supercoincidence condition for the substitution. By [26], the subtiles T ( i ) ( i = 1 , ,
3) inducethe aperiodic tiling of the plane (2.6). This tiling is also self-replicating (see [38, Chapter3]). Examples are depicted in Figure 2.In this paper, we will prove the following theorem.
Theorem 2.1.
Consider the substitution σ a,b ( a ≥ b ≥ ) defined in (2.1) and let T a,b be its Rauzy fractal. Then T a,b is homeomorphic to a closed disk ⇐⇒ b − a ≤ . Some examples can be seen on Figure 3. The cases a = b = 1 and a ≥ b = 1 weretreated in [29, 30], where it was shown that the Rauzy fractals are quasi-circles. Also,it was proved in [27] that T a,b can not be homeomorphic to a closed disk as soon as2 b − a >
3. We will recover all these results by another method. Indeed, in order toprove Theorem 2.1, we will construct a parametrization of the boundary of T . Thisparametrization will have the following properties. BENOˆIT LORIDANT
Tribonacci substitution Substitution σ , Figure 2.
Aperiodic self-replicating tilings of the contracting plane σ , σ , σ , σ , Figure 3.
Disk-like (above) and non disk-like (below) cubic Rauzy fractals
Theorem 2.2.
Consider the substitution σ = σ a,b ( a ≥ b ≥ ) defined in (2.1) and let T be its Rauzy fractal. Let λ be the largest root of x + (1 − b ) x + ( b − a ) x − ( a + 1) x − . Then there exists a surjective H¨older continuous mapping C : [0 , → ∂ T with C (0) = C (1) and a sequence of polygonal curves (∆ n ) n ≥ such that • lim n →∞ ∆ n = ∂ T (Hausdorff metric). OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 7 • Denote by V n the set of vertices of ∆ n . Then V n ⊂ V n +1 ⊂ C ( Q ( λ ) ∩ [0 , . The H¨older exponent is s = − log | α | log | λ | , where | α | = max {| α | , | α |} .Remark . In the case α = α , the H¨older exponent is s = 1dim H ∂ T . The construction of the boundary parametrization C in Theorem 2.2 roughly readsas follows. The tile T = T a,b is the attractor of the graph directed construction (2.4).The labels of the infinite walks in the associated prefix-suffix graph Γ = Γ a,b buildup the language of the tile. The boundary ∂ T happens to be also the attractor of agraph directed construction. A finite graph G with a bigger number of states than Γdescribes the corresponding sublanguage of the language of T . This graph induces aDumont-Thomas numeration system [19], leading to the parametrization schematicallyrepresented below: C : [0 , −→ ∂ T(cid:38) (cid:37) G with C (0) = C (1). To prove Theorem 2.1, we will investigate the injectivity of C on [0 , C is injective, ∂ T is a simple closed curve and T is homeomorphic to aclosed disk by a theorem of Sch¨onflies - a strengthened form of Jordan’s curve theorem,see [45]. 3. GIFS for the boundary of T a,b In this section, we introduce two graphs that describe the boundary of the Rauzyfractals T = T a,b associated to the substitutions σ = σ a,b . First, we will focus on theboundary graph G ,a,b , that describes the whole language of the boundary of T a,b . Sec-ond, we will present a subgraph G ,a,b ⊂ G ,a,b , whose language is large enough to coverthe boundary. The latter graph will be strongly connected (see Lemma 3.11), unlike theboundary graph, and this property will allow us to perform the boundary parametriza-tion. Both graphs will be of importance to distinguish the disk-like tiles from the non-disk-like tiles. Roughly speaking, whenever the languages of these graphs are equal,the parametrization is injective and the boundary is a simple closed curve, otherwisethe parametrization fails to be injective. For our class of substitutions, slight differentversions of these graphs were computed in 2006 [44] and in 2013 [27] (see Remarks 3.4and 3.9). A crucial result will be Lemma 3.2, characterizing the boundary points of thetiles. Indeed, the “if part” will be used to prove the continuity of the parametrization C in Theorem 2.2 for all parameters a, b , the “only if part” to prove its injectivity whenever2 b − a ≤ ∂ T = (cid:91) i =1 (cid:91) [ γ,j ] ∈ Γ srs ,γ (cid:54) =0 T ( i ) ∩ ( T ( j ) + γ ) . The subtiles T ( i ) satisfy the equations (2.4). This allows to write the boundary ∂ T itselfas the attractor of a graph directed function system ( GIFS ). BENOˆIT LORIDANT
The boundary graph: the boundary language.
Definition . The boundary graph G = G ,a,b is the largest graph satisfying the follow-ing conditions.( i ) A triple [ i, γ, j ] ∈ A × π ( Z ) × A is a vertex of G if(3.1) || γ || ≤ {|| π l ( p ) || ; p label of Γ } − max {| α | , | α |} . ( ii ) There is an edge [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ] iff i p −→ i ∈ Γ, j p (cid:48) −→ j ∈ Γ and h γ = γ + π ( l ( p (cid:48) ) − l ( p )) . ( iii ) Each vertex belongs to an infinite walk starting from a vertex [ i, γ, j ] with [ γ, j ] ∈ Γ srs and ( γ (cid:54) = 0 or i < j ).The set of vertices of G is denoted by S .An analogous definition can be found in [38, Definition 5.4]. Note that (3.1) is anupper bound for the diameter of T .For a given substitution, the computation of G is algorithmic. There are finitelymany triples satisfying (3.1). G is obtained after checking the algebraic relation of ( ii )between all pairs of triples and erasing the vertices that do not fulfill ( iii ). See also [38]. Example . G is depicted on Figure 4 for a = b = 1. See Table 1 for the verticesassociated to the letters in this graph. Here, if S = [ i, γ, j ], then S − := [ j, − γ, i ]. Thecolored states stand for triples [ i, γ, j ] with [ γ, j ] ∈ Γ srs . The labels just indicate thenumber of 1’s in p , p (0 for the prefix (cid:15) ).Boundary points are characterized as follows. Lemma 3.2.
Let ( p k ) k ≥ and ( p (cid:48) k ) k ≥ be the labels of infinite walks in the prefix-suffixgraph Γ starting from i ∈ A and j ∈ A respectively. Let γ ∈ π ( Z ) such that [ γ, j ] ∈ Γ srs and ( γ (cid:54) = 0 or i < j ) . Then (cid:88) k ≥ h k π l ( p k ) = γ + (cid:88) k ≥ h k π l ( p (cid:48) k ) =: x if and only if there is an infinite walk [ i, γ, j ] p | p (cid:48) −−−→ [ i , γ , j ] p | p (cid:48) −−−→ . . . ∈ G . In this case, x ∈ T ( i ) ∩ ( T ( j ) + γ ) .Proof. We mainly use arguments of [38, Proof of Theorem 5.6]. If the above infinite walkexists in G , then using the definition of the edges one can write for all n ≥ h n +1 γ n +1 + n (cid:88) k =0 π l ( p k ) = γ + n (cid:88) k =0 π l ( p (cid:48) k ) . As h is contracting and ( γ n ) n ≥ is a bounded sequence, letting n → ∞ gives the requiredequality.We now construct the walk by assuming the equality of the two infinite expansions.Note that γ satisfies (3.1), and by assumption there exist edges i p −→ i and j p (cid:48) −→ j inΓ. Let γ = ∞ (cid:88) k =0 π l ( p k +1 ) − ∞ (cid:88) k =0 π l ( p (cid:48) k +1 ) = h − ( γ + π l ( p (cid:48) ) − π l ( p )) . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 9 E − C − P − H − O − F − D − NBC EHM PA O FDJL K ! | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Figure 4.
Boundary graph of the Tribonacci substitution ( a = b = 1)Then again γ satisfies (3.1) and h γ = γ + π ( l ( p (cid:48) ) − l ( p )). Moreover, choosing x ∈ Z satisfying π ( x ) = γ , we can define x = M − ( x + l ( p (cid:48) ) − l ( p )) ∈ Z , that is, γ ∈ π ( Z ). Therefore, the edge [ i, γ, j ] p | p (cid:48) −−−→ [ i , γ , j ] fulfills ( ii ) of Defini-tion 3.1. The infinite sequence of edges [ i, γ, j ] p | p (cid:48) −−−→ [ i , γ , j ] p | p (cid:48) −−−→ . . . satisfying ( i )and ( ii ) of Definition 3.1 is constructed iteratively in the above way. It satisfies also ( iii ),since [ γ, j ] ∈ Γ srs and ( γ (cid:54) = 0 or i < j ). Therefore, it is an infinite walk in G . (cid:3) Lemma 3.3.
Let [ i, γ, j ] ∈ S . Then either [ γ, j ] or [ − γ, i ] belongs to Γ srs .Proof. Note that [0 , i ] ∈ Γ srs for all i ∈ A . By definition, a vertex of G belongs to aninfinite walk starting from a vertex [ i, γ, j ] with [ γ, j ] ∈ Γ srs . Thus we assume that agiven vertex [ i, γ, j ] of G satisfies [ γ, j ] ∈ Γ srs or [ − γ, i ] ∈ Γ srs , and check that as soonas there is an edge [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ] in G , then either [ γ , j ] or [ − γ , i ] belongs to Γ srs . Indeed, let x ∈ Z such that π ( x ) = γ . Then the existence of such an edge insuresthat γ = π ( x ) = π ( M − ( x + l ( p (cid:48) ) − l ( p )))for some x ∈ Z . Therefore, (cid:104) x , v β (cid:105) = (cid:104) M − ( x + l ( p (cid:48) ) − l ( p )) , v β (cid:105) = 1 β (cid:104) x + l ( p (cid:48) ) − l ( p ) , v β (cid:105) . If [ γ, j ] ∈ Γ srs , then 0 ≤ (cid:104) x , v β (cid:105) < (cid:104) e j , v β (cid:105) implies that − β − (cid:104) l ( p ) , v β (cid:105) ≤ (cid:104) x , v β (cid:105) < β − (cid:104) e j + l ( p (cid:48) ) , v β (cid:105) . Using the fact that σ ( i ) = pis and σ ( j ) = p (cid:48) js (cid:48) for some s, s (cid:48) ∈ A ∗ , we obtain −(cid:104) e i , v β (cid:105) < (cid:104) x , v β (cid:105) < (cid:104) e j , v β (cid:105) , hence [ γ , j ] or [ − γ , i ] belongs to Γ srs . A similar computation holds if [ − γ, i ] ∈ Γ srs .See also [38, Proof of Theorem 5.6]. (cid:3) Remark . In [38, 44], all the vertices [ i, γ, j ] of the boundary graph satisfy [ γ, j ] ∈ Γ srs ,but two types of edges are used. In the present article, we do not introduce two types ofedges. In this way, the labels of infinite walks in G are sequences of prefixes that alsooccur as labels of infinite walks in the prefix-suffix graph. In other words, the language ofthe boundary of T is directly visualized as a sublanguage of T . This will be important forthe proof of our main results, that requires to find out the infinite sequences of prefixes( p k ) k ≥ , ( p (cid:48) k ) k ≥ satisfying (cid:80) k ≥ h k π l ( p k ) = (cid:80) k ≥ h k π l ( p (cid:48) k ). We explain in the core ofthe proof of Proposition 3.10 how to get rid off the two types of edges from the boundarygraphs of [38, 44] in order to derive our boundary graph G .We call S = { [ i, γ, j ] ∈ S ; γ (cid:54) = 0 , [ γ, j ] ∈ Γ srs } the set of neighbors of T in the tiling (2.6).This gives us the first boundary GIFS. Proposition 3.5.
Let B [ i, γ, j ] the non-empty compact sets solutions of the GIFS (3.2) ∀ [ i, γ, j ] ∈ S , B [ i, γ, j ] = (cid:91) [ i,γ,j ] p | p (cid:48) −−→ [ i ,γ ,j ] ∈G h B [ i , γ , j ] + π l ( p ) . Then B [ i, γ, j ] = T ( i ) ∩ ( T ( j ) + γ ) and ∂ T = (cid:83) [ i,γ,j ] ∈S B [ i, γ, j ] . Proof.
The proof follows [38, Proof of Theorem 5.7]. The set { x (cid:55)→ h x + π l ( p ) } [ i,γ,j ] p | p (cid:48) −−→ [ i ,γ , j ] ∈G is a graph iterated function system, since h is a contraction. By a result of Mauldin andWilliams [28], there is a unique sequence of non-empty compact sets ( B [ i, γ, j ]) [ i,γ,j ] ∈S which is the attractor of this GIFS.We now show that the sequence of sets ( T ( i ) ∩ ( T ( j ) + γ )) [ i,γ,j ] ∈S also satisfies theset equations of the above GIFS and then use the unicity of the attractor.Let [ i, γ, j ] be a vertex of G . Using (2.4), we can subdivide each intersection of tilesas follows:(3.3) T ( i ) ∩ ( T ( j ) + γ )= (cid:83) i p −→ i ∈ Γ ,j p (cid:48) −→ j ∈ Γ π l ( p ) + h T ( i ) ∩ T ( j ) + h − π ( l ( p (cid:48) ) − l ( p ) + h − γ (cid:124) (cid:123)(cid:122) (cid:125) =: γ . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 11
Let [ i , γ , j ] be as in the above union. If it is a vertex of G , then by a similarcomputation as in the first part of the proof of Lemma 3.2, one obtains a point in T ( i ) ∩ ( T ( j ) + γ ), thus this intersection is non-empty.On the contrary, suppose T ( i ) ∩ ( T ( j ) + γ ) (cid:54) = ∅ . We wish to show that [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ] ∈ G . First, since [ i, γ, j ] is a vertex of G , we can write γ = π ( x ) for some x ∈ Z and γ = π ( M − ( x + l ( p (cid:48) ) − l ( p ))) ∈ π ( Z ). Also, since T ( i ) ∩ ( T ( j ) + γ ) (cid:54) = ∅ , thereare ( p k ) k ≥ and ( p (cid:48) k ) k ≥ labels of infinite walks of Γ starting from i and j respectivelysuch that (cid:88) k ≥ h k π l ( p k ) = γ + (cid:88) k ≥ h k π l ( p (cid:48) k ) . Consequently, γ is bounded as in (3.1). Hence the edge [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ] satisfies( i ), as well as ( ii ) of Definition 3.1. Moreover, from the above equality of expansions,one can construct as in the proof of Lemma 3.2 an infinite sequence of edges startingfrom [ i , γ , j ] and satisfying ( i ) and ( ii ) of Definition 3.1. Lastly, by assumption on[ i, γ, j ], one can find a walk [ i , γ , j ] q | q (cid:48) −−−→ · · · q l | q (cid:48) l −−−→ [ i, γ, j ] in G with [ γ , j ] ∈ Γ srs and ( γ (cid:54) = 0 or i < j ). Altogether, we have found an infinite sequence of edges satisfying( i ) and ( ii ) and including the edge [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ]. Therefore, [ i , γ , j ] fulfills( iii ) of Definition 3.1 and [ i, γ, j ] p | p (cid:48) −−→ [ i , γ , j ] belongs to G .It follows that (3.3) can be re-written as(3.4) T ( i ) ∩ ( T ( j ) + γ )= (cid:83) [ i,γ,j ] p | p (cid:48) −−→ [ i ,γ ,j ] ∈G π l ( p ) + h T ( i ) ∩ T ( j ) + h − π ( l ( p (cid:48) ) − l ( p ) + h − γ (cid:124) (cid:123)(cid:122) (cid:125) =: γ . By unicity of the GIFS-attractor, we conclude that B [ i, γ, j ] = T ( i ) ∩ ( T ( j ) + γ ) for all[ i, γ, j ] ∈ S .The second equality is a consequence of the tiling property and the definition of S : ∂ T = (cid:91) i =1 (cid:91) [ γ,j ] ∈ Γ srs ,γ (cid:54) =0 T ( i ) ∩ ( T ( j ) + γ ) = (cid:91) [ i,γ,j ] ∈S T ( i ) ∩ ( T ( j ) + γ ) . (cid:3) Therefore, ∂T is the attractor of a graph directed self-affine system. To proceed to theboundary parametrization, the natural idea would be to order the vertices and edges ofthe graph and use the induced Dumont-Thomas numeration system [19]. Geometrically,this corresponds to an ordering of the boundary parts and their subdivisions clockwise orcounterclockwise along the boundary. This method requires the strongly connectednessof the graph, or at least the existence of a positive dominant eigenvector for its incidencematrix. However, in general, the above boundary graph does not have this property.Roughly speaking, there may be many redundances in the boundary language given bythe boundary graph: the mapping[ i, γ, j ] p | p (cid:48) −−−→ [ i , γ , j ] p | p (cid:48) −−−→ . . . ∈ G (cid:55)→ (cid:88) k ≥ h k π l ( p k ) ∈ ∂ T sending an infinite walk in the boundary graph to a boundary point may be highly notinjective. The level of non-injectivity reflects the complexity of the topology of T . Forexample, many neighbors (that is, many states in the automaton) suggest an intricatetopological structure. In fact, if an intersection T ( i ) ∩ ( T ( j ) + γ ) is a point, or has a Hausdorff dimensionsmaller than that of the boundary, it shall be redundant (contained in other intersections),thus not essential. In the next subsection, we introduce a subgraph of the boundary graphthat will be more appropriate.3.2. The graph G . In 2006, J¨org Thuswaldner defined a graph which is in generalsmaller than the boundary graph but always contains enough information to describethe whole boundary [44]. As an example, he computed this graph for our class of sub-stitutions.
Definition . Let a ≥ b ≥
1. Let G = G ,a,b be the graph with • Vertices: R = R ,a,b = { A, B, C, C − , D, D − , E, E − , F, F − , G, G − , H, H − , I, I − , J, K, N, N − , O, O − , P, P − }∪ { M } \ { I, I − } , if a ≥ , b = 1 ∪ { L } \ { G, G − , N − } , if a = b ≥ ∪ { L, M } \ {
G, G − , I, I − , N − } , if a = b = 1as in Table 1. Here, if S = [ i, γ, j ], then S − := [ j, − γ, i ]. • Edges : in addition to the edges of Table 1, we have S − p | p −−−→ T − ∈ G ⇐⇒ S p | p −−−→ T ∈ G , and S − p | p −−−→ T ∈ G ⇐⇒ S p | p −−−→ T − ∈ G (as long as S − , T − belong to R defined above). Remark . The states
A, B, C, D, . . . , P correspond to the vertices [ i, γ, j ] with [ γ, j ] ∈ Γ srs .One can check that G satisfies the conditions ( i ) , ( ii ) and ( iii ) of the definition of theboundary graph (Definition 3.1). Therefore, the following lemma holds. Lemma 3.8.
For all a ≥ b ≥ , G ,a,b ⊂ G ,a,b . Remark . The graph G is related to the contact graph defined in [44] or [38]. Thisnotion of contact graph was first introduced by Gr¨ochenig and Haas [20] in the contextof self-affine tiles (see also [37]). For substitution tiles, the contact graph is obtainedfrom a sequence of polygonal approximations of the Rauzy fractal constructed via thedual substitutions on the stepped surface (see [6]). Each approximation gives rise to apolygonal tiling of the stepped surface. In these tilings, the structure of the adjacentneighbors (neighbors whose intersection with the approximating central tile has non-zero1-dimensional Lebesgue measure) stabilizes after finitely many steps. The collection ofadjacent neighbors of a good enough polygonal approximation of the Rauzy fractal resultsin the set R of Definition 3.6. As the prefixes p , p belong to { (cid:15), , , . . . , · · · (cid:124) (cid:123)(cid:122) (cid:125) a } , the labels just indicate the number of 1’s in p , p . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 13Vertex Edge(s) p | p Condition A [1 , π (0 , , , C k | b − k, ≤ k ≤ a − bD | b − O | b − N k | b + k, ≤ k ≤ a − b − a (cid:54) = bB [1 , π (0 , , , N a − b | aC a − b + 1 | a b ≥ C [1 , π (0 , , − , P k | a − b + k, ≤ k ≤ b − H k | a − b + 1 + k, ≤ k ≤ b − b ≥ I k | a − b + 1 + k, ≤ k ≤ b − b ≥ D [1 , π (0 , , − , H b − | aI b − | a b ≥ E [2 , π (1 , , − , C − a | a − bN − a | a − b − a (cid:54) = bF [3 , π (1 , , − , D − b | O − b | G [1 , π (1 , , − , a (cid:54) = b C − a − − k | a − b − − k, ≤ k ≤ a − b − a (cid:54) = bN − a − − k | a − b − − k, ≤ k ≤ a − b − a ≥ b + 2 H [2 , π (1 , − , , P − a | b − H − a | b − b ≥ I − a | b − b ≥ I [1 , π (1 , − , , b ≥ P − a − − k | b − − k, ≤ k ≤ b − b ≥ H − a − − k | b − − k, ≤ k ≤ b − b ≥ I − a − − k | b − − k, ≤ k ≤ b − b ≥ J [1 , π (0 , , , A a − | aK [1 , π (0 , , , B b − | bJ b | b a (cid:54) = bM b − | b b = 1 L [2 , π (0 , , , a = b J a | a a = bM [2 , π (0 , , , b = 1 C a | a b = 1 N [1 , π (0 , , , E | a − F | a − G | a − a (cid:54) = bO [3 , π (0 , , − , P b | aP [2 , π (1 , − , , E − a | F − a | G − a | a (cid:54) = b Table 1.
The subgraph G of the self-replicating boundary graph. Proposition 3.10 ([44, Theorem 4.3]) . Let a ≥ b ≥ and σ = σ a,b the substitutiondefined in 2.1. Consider the graph G = G ,a,b of Definition 3.6. We denote by R a,b = R ⊂ R the set R = { A, B, C, D, E, F, G, H, I, N, O, P }∪ { M } \ { I } , if a ≥ , b = 1 \{ G } , if a = b ≥ ∪ { M } \ { G, I } , if a = b = 1 . Then ∂ T = (cid:83) [ i,γ,j ] ∈ R C [ i, γ, j ] , where the sets C [ i, γ, j ] ([ i, γ, j ] ∈ R ) are the solutions of the GIFS directed by G , i.e. , (3.5) ∀ [ i, γ, j ] ∈ R , C [ i, γ, j ] = (cid:83) [ i,γ,j ] p | p (cid:48) −−→ [ i ,γ ,j ] ∈ G h C [ i , γ , j ] + π l ( p ) ⊂ T ( i ) ∩ ( T ( j ) + γ ) Proof.
The last inclusion is an easy consequence of Lemma 3.8 and Proposition 3.5.The lengthy proof is given in [44, Section 6]. However, in that article, two types ofedges are used and the Rauzy fractals are defined in terms of suffixes instead of prefixes.We refer to Remark 3.4 and to [44, Section 4.3] as well as [6]. The correspondence withour setting reads as follows.
Let C = C a,b be the graph as in [44, Theorem 6.2], depicted in Figures 9 and 10 withinthis reference, and C ∂ the subgraph obtained from C after successively deleting the stateshaving no outgoing edges (as in [44, Definition 4.5]). For a state S = [(0 , , , i ] , [ γ, j ]occurring in [44, Figures 9-10], we shall simply write S = [ i, γ, j ]. Step 1 . The aim is to remove the two types of edges. By [44, Definition 3.6], an edge[ i, γ, j ] ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ [ i (cid:48) , γ (cid:48) , j (cid:48) ] ∈ C ∂ is – of type 1 if(3.6) σ ( i (cid:48) ) = p is , σ ( j (cid:48) ) = p js and h γ (cid:48) = γ + π l ( s ) − π l ( s ) – of type 2 if(3.7) σ ( j (cid:48) ) = p is , σ ( i (cid:48) ) = p js , and − h γ (cid:48) = γ + π l ( s ) − π l ( s ) . Replace each edge S ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ T ∈ C ∂ of type 1 by two edges S ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ T and S − ( p ,j,s ) | ( p ,i,s ) −−−−−−−−−−−→ T − , and each edge S ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ T ∈ C ∂ of type 2 by two edges S ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ T − and S − ( p ,j,s ) | ( p ,i,s ) −−−−−−−−−−−→ T. Here, for S = [ i, γ, j ] state of C ∂ , we wrote S − := [ j, − γ, i ]. See also [38, Section7, Proof of Theorem 5.6]. This procedure results in a graph whose number ofstates has doubled. Delete successively the states S − having no incoming edges.We denote by C ∂ the remaining graph. Note that all edges in this graph nowsatisfy the relation (3.6). Step 2 . The aim is to use prefixes instead of suffixes. Note that if X i is defined as in [44]by X i = (cid:91) σ ( j )= pis h X j + π l ( s ) , then we have T ( i ) = − X i − π l ( i ) . This uses the unicity of the attractor solution of (2.4) and the relation (2.2): for σ ( j ) = pis in the above union, we have π l ( s ) = π l ( σ ( j )) − π l ( p ) − π l ( i ) = h π l ( j ) − π l ( p ) − π l ( i )Replace each edge[ i, γ, j ] ( p ,i,s ) | ( p ,j,s ) −−−−−−−−−−−→ [ i (cid:48) , γ (cid:48) , j (cid:48) ] ∈ C ∂ by an edge[ j, γ − π l ( j ) + π l ( i ) , i ] p | p −−−→ [ j (cid:48) , γ (cid:48) − π l ( j (cid:48) ) + π l ( i (cid:48) ) , i (cid:48) ] . This change relies on the following computation. For an edge in C ∂ as above, wehave the relation (3.6). In particular, h γ (cid:48) = γ + π l ( s ) − π l ( s ) , which is equivalent to h ( γ (cid:48) − π l ( j (cid:48) ) + π l ( i (cid:48) )) = γ − π l ( j ) + π l ( i ) + π l ( p ) − π l ( p ) , again by using (2.2). The resulting graph is G . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 15
We write X := (cid:83) i =1 X i . By [44, Theorem 4.3],(3.8) ∂X = (cid:91) [ i,γ,j ] ∈ R C [ i, γ, j ]and(3.9) ∀ i = 1 , , , ∂X i = (cid:91) [ i,γ,j ] ∈ R C [ i, γ, j ] ∪ (cid:91) [ i, ,j ] ∈ R C [ i, , j ] , where the sets C [ i, γ, j ] ([ i, γ, j ] ∈ R ) are the solutions of the GIFS directed by C ∂ , i.e. , ∀ [ i, γ, j ] ∈ R , C [ i, γ, j ] = (cid:83) [ i,γ,j ] ( p ,i,s | ( p ,j,s −−−−−−−−−−−→ [ i ,γ ,j ] ∈C ∂ h C [ i , γ , j ] + π l ( s ) ⊂ X i ∩ ( X j + γ ) . Here, the sets R , R are defined for the graph C ∂ analogously to R, R . In particular, R = { [ j, γ − π l ( j ) + π l ( i ) , i ]; [ i, γ, j ] ∈ R } , and a similar relation holds between R and R .By unicity of the attractor of the GIFS (3.5) directed by G , one can check that, forall [ i, γ, j ] ∈ R , − C [ i, γ, j ] − π l ( i ) = C [ j, γ − π l ( j ) + π l ( i ) , i ] . Using (3.9), this leads to ∂ T ( i ) = (cid:91) [ i,γ,j ] ∈ R C [ i, γ, j ] ∪ (cid:91) [ i, ,j ] ∈ R ∩{ J,K,L } C [ i, , j ]for all i = 1 , ,
3. As T = (cid:83) i =1 T ( i ) and C [ i, , j ] ⊂ T ( i ) ∩ T ( j ), we finally obtain that ∂ T = (cid:91) [ i,γ,j ] ∈ R C [ i, γ, j ] . (cid:3) The following lemma is essential for the construction of the boundary parametrizationin the next section.
Lemma 3.11.
Let a ≥ b ≥ and G = G ,a,b as in Definition 3.6. We denote by G = G a,b the graph obtained from G after deleting the states J, K, L and all their in-and outcoming edges. Let r = r a,b be the number of states in R \ { J, K, L } and L = L a,b the incidence matrix of G : L = ( l m,n ) ≤ m,n ≤ r with l m,n = { S n p | p −−−→ S m ∈ G } , where { S , . . . , S r } = R \ { J, K, L } . Then there exists a strictly positive vector u = u a,b satisfying Lu = λ u , where λ = λ a,b is the largest root of the characteristic polynomial of L . In particular, λ is the largest root of p a,b ( x ) = x + (1 − b ) x + ( b − a ) x − ( a + 1) x − . We normalize u = ( u (1) , . . . , u ( r ) ) to have u (1) + · · · + u ( r ) = 1 .Proof. We refer to Tables 2, 3, 4, 5 and the corresponding Figures 5, 6, 7, 8. Note thatthe restriction of the graph G to the set of states • R \ { A, B, J, K } if a ≥ b + 1 , b ≥ • R \ { A, B, M, J, K } if a ≥ , b = 1, • R \ { A, B, N, J, K, L } if a = b ≥ • R \ { A, B, M, N, J, K, L } if a = b = 1, is strongly connected. Moreover, every walk in G starting from any of the remain-ing states A, B, M or N reaches this strongly connected part after at most two edges.This justifies the existence of a strictly positive eigenvector corresponding to the Perron-Frobenius eigenvalue λ of L , which is easily computed to be the largest root of p a,b forall these cases. (cid:3) Remark . In general, even for the contact graph, the incidence matrix needs not havea positive dominant eigenvector. Our class of substitutions is therefore a special case.4.
Boundary parametrization
Throughout this section, we fix a ≥ b ≥
1. We will prove Theorem 2.2, that includesa parametrization of the boundary of T = T a,b based on the graph G = G a.b . InDefinition 4.1, we order the states and edges of the graph G . This ordering seemsto be arbitrary, but it has a geometrical interpretation: it corresponds to an orderingof the boundary pieces and subpieces in the GIFS (3.10) counterclockwise around theboundary of the Rauzy fractal T . This choice of ordering will insure the left continuityof our parametrization (see the proof of Proposition 4.6). Definition . We call G + a,b = G + the ordered graph obtained from G a,b = G afterordering the states and edges as listed in Tables 2, 3, 4, 5, according to the values of a, b .Moreover, we set S −− := S and call o max S or just o max the number of edges startingfrom the state S . We define the following edges in G + . Suppose • S / ∈ { , } if a ≥ b + 1 , b ≥ • S / ∈ { , , } if a ≥ , b = 1; • S / ∈ { , , } if a = b ≥ • S / ∈ { , , , } if a = b = 1.Then S − p | p || o −−−−−→ T − ∈ G + : ⇐⇒ S p | p || o max + − o −−−−−−−−−−−→ T ∈ G + . Finally, we call the states { , , . . . , } (case a > b ) or { , , . . . , } (case a = b ) the starting states of G + . A finite or infinite walk in G + is admissible if it starts from astarting state.The corresponding graphs are depicted in Figures 5, 6, 7, 8. The starting states arecolored. For simplicity, we write S ( m ) −−→ T whenever there are m edges from S to T . If m = 1, we write the complete edge S p || o −−→ T . If m = 0, there is simply no edge betweenthe states. Remark . All the edges of G + are of the form S p | p || o −−−−−→ T . Since there is no ambiguity,we may simply write S p || o −−−→ T or S o −→ T or ( S ; o ). Note that if the condition of thelast column in the tables is not fulfilled, then the edges of the corresponding line do notexist and there is exactly one edge starting from the associated state. Remark . We write ( S ; o , . . . , o n ) for the walk of G + starting from the state S with the edges successively labelled by o , . . . , o n . By the above ordering of states andedges in G + , the set of admissible walks of length n ( n ≥
0) is lexicographically or-dered, from the walk (1; , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n times ) to the walk (12; o max , o max , . . . , o max (cid:124) (cid:123)(cid:122) (cid:125) n times ) ( a > b ) or(11; o max , o max , . . . , o max (cid:124) (cid:123)(cid:122) (cid:125) n times ) ( a = b ). This holds also for the infinite admissible walks. OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 17Vertex Edge(s) p | p Order Condition C k | a − b + k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − k | a − b + 1 + k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − k | a − b + 1 + k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − N | a − | a −
10 0 | a − A | b −
12 0 | b − k | b − k, ≤ k ≤ a − b + , ≤ k ≤ a − b k | b + k, ≤ k ≤ a − b − + , ≤ k ≤ a − b − B a − b | a a − b + 1 | a I − a − − k | b − − k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − − a − − k | b − − k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − b ≥ − a − − k | b − − k, ≤ k ≤ b − + ( b − − k ) , ≤ k ≤ b − b ≥ H − a | b − − a | b − − a | b − P − a | − a | − a | E − a | a − b − a | a − b − G − a − − k | a − b − − k, ≤ k ≤ a − b − + , ≤ k ≤ a − b − − a − − k | a − b − − k, ≤ k ≤ a − b − + , ≤ k ≤ a − b − a ≥ b + 2 F
10 12 − b | − b | O
11 7 b | a D
12 5 b − | a b − | a J a − | aK b − | bJ b | b Table 2. G ,a,b for a ≥ b + 1 , b ≥ Vertex Edge(s) p | p Order Condition C | a − N | a − | a −
10 0 | a − A |
12 0 | k | k, ≤ k ≤ a − + , ≤ k ≤ a − k | k, ≤ k ≤ a − + , ≤ k ≤ a − B a − | a M a | a H − a | P − a | − a | − a | E − a | a − − a | a − G − a − − k | a − − k, ≤ k ≤ a − + , ≤ k ≤ a − − a − − k | a − − k, ≤ k ≤ a − + , ≤ k ≤ a − a ≥ F
10 12 − | − | O
11 7 1 | a D
12 6 0 | a J a − | aK | J |
15 0 | Table 3. G ,a,b for a ≥ , b = 1.The parametrization procedure now runs along the same lines as in [5, Section 3].Let L be the incidence matrix of G + (or G ) and u = ( u (1) , . . . , u ( r ) ) a strictly positive p | p Order Condition C k | k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − k | k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − k | k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − N | a − | a − A | a −
11 0 | a − | a − B | a | a I − a − − k | a − − k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − − a − − k | a − − k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − a ≥ − a − − k | a − − k, ≤ k ≤ a − + ( a − − k ) , ≤ k ≤ a − a ≥ H − a | a − − a | a − − a | a − P − a | − a | E − a | F − a | − a | O
10 7 a | a D
11 5 a − | a a − | a J a − | aK a − | aL J a | a Table 4. G ,a,b for a = b ≥ Vertex Edge(s) p | p Order C | N | | A |
11 0 | | B | M | H − | P − | − | E − | F − | − | O
10 7 1 | D
11 6 0 | J | K |
15 0 | L J | Table 5. G ,a =1 ,b =1 (Tribonacci substitution).eigenvector for the dominant eigenvalue λ as in Lemma 3.11, normalized to have u (1) + · · · + u ( r ) = 1. The automaton G + induces a number system, also known as Dumont-Thomas numeration system [19]. We map each admissible infinite walk of G + to a pointin the unit interval [0 , ,
1] in subintervals of lengths given by the coordinates of u ; we then subdivide eachsubinterval, for example the subinterval [ u (1) , u (1) + u (2) ], by using Lu = λ u ; and weiterate this procedure. More precisely, we define a function OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 19 − − − − − − − − − −
241 867 5 93 11 1012 0 || || || a − || a − || a − || ( b ) ( b − b −
1) ( b )( b − b − a − b + 1)( a − b )0 || || a − b || a − b + 1 || ( b − b −
2) ( b − b − || ( b − b − a || ( b − b − || b − || a || a || a || a || a || || || || a || a || a − b || a − b || ( a − b )( a − b −
1) ( a − b ) ( a − b − b || b || || || b || a || b − || b − || a || a || Figure 5. G + a,b for a ≥ b + 1 , b ≥ φ ( S ; o ) = , if o = ; (cid:88) ≤ k < o ,S k −→ S (cid:48) u ( S (cid:48) ) , if o (cid:54) = . Thus φ ( S ; o ) < (cid:88) ≤ k ≤ o max ,S k −→ S (cid:48) u ( S (cid:48) ) = λ u ( S ) for each edge ( S ; o ). − − − − − − − − −
241 865 7 93 11 10120 || || || a − || a − || a − || || a − || ( a )( a − || || a − || a || || a || a || a || a || || || || a || a || a − || a − || ( a − a −
2) ( a −
1) ( a − || || || || || a || || a || Figure 6. G + a,b for a ≥ , b = 1.We set u (0) := 0 and map the admissible infinite walks in G + to [0 ,
1] via(4.1) φ : G + −→ [0 , w (cid:55)→ lim n →∞ ( u (0) + u (1) + . . . + u ( S − + λ φ ( S ; o ) + λ φ ( S ; o ) + . . . + λ n φ ( S n − ; o n ) )whenever w is the admissible infinite walk: w : S o −→ S o −→ . . . o n −−→ S n o n + −−−→ . . . This mapping φ is well-defined, increasing, onto, and it is almost 1 to 1, as identificationsoccur exactly on pairs of lexicographically consecutive infinite walks. Indeed, let w (cid:54) = w (cid:48) OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 21 − − − − − − − −
241 867 53 10 911 0 || || ( a ) ( a − a −
1) ( a )( a − a − || || || || || ( a − a −
2) ( a − a − || ( a − a − a || ( a − a − || a − || a || a || a || a || || || a || || a || a || || || a || a || a − || a − || a || a || Figure 7. G + a,b for a = b ≥ G + , say for example w > lex w (cid:48) . Then φ ( w ) = φ ( w (cid:48) ) iff(4.2) 1 . (cid:26) w = ( S + 1; ) w (cid:48) = ( S ; o max ) or 2 . (cid:26) w = ( S ; o , . . . , o m , o + , ) w (cid:48) = ( S ; o , . . . , o m , o , o max )holds for some state S = 1 , . . . , S max or some prefix ( S ; o , . . . , o m ) and an order o . Here, S max = 12 (case a > b ) or 11 (case a = b ). By o , we mean the infinite repetition of theorder o . We omit the proofs of these facts here, since they are similar to proofs givenin [5].Consequently, if t ∈ [0 , φ − ( t ) consists of either one or two elements. Hencean inverse of φ can be defined as φ (1) : [0 , −→ G + t (cid:55)→ max lex φ − ( t ) , where max lex maps a finite set of walks to its lexicographically maximal walk. − − − − − − −
241 865 73 10 911 0 || || || || || || || || || || || || || || || || || || || || || || || || || Figure 8. G + a,b for a = b = 1 (Tribonacci substitution). u (1) u (2) u (3) u (4) . . . (1; . . . ) (2; . . . ) (3; . . . ) (4; . . . ) . . . u (8) λ u (9) λ u (10) λ (2; , . . . ) (2; , . . . ) (2; , . . . ) u (2) = ! u (8) + u (9) + u (10) " /λ Figure 9.
Parametrization procedure: interval subdivisions (case a > b ).We finally denote by P the natural bijection:(4.3) P : G + → G ( S ; o , o , . . . ) (cid:55)→ w : S p | p (cid:48) −−−→ S p | p (cid:48) −−−→ . . . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 23 whenever ( S ; o , o , . . . ) = S p | p (cid:48) || o −−−−−−→ S p | p (cid:48) || o −−−−−−→ . . . is an admissible walk of G + , andby ψ the mapping(4.4) ψ : G → ∂ T w = [ i, γ, j ] p | p (cid:48) −−−→ [ i , γ , j ] p | p (cid:48) −−−→ . . . (cid:55)→ (cid:80) k ≥ h k π l ( p k )This allows us to define our parametrization mapping C . Proposition 4.4.
The mapping C : [0 , φ (1) −−→ G + P −→ G ψ −→ ∂ T is well-defined andsurjective. Furthermore, let A := { t ∈ [0 ,
1] ; t = u (0) + u (1) + . . . + u ( S − + λ φ ( S ; o ) + λ φ ( S ; o ) + . . . + λ n φ ( S n − ; o n ) for some admissible finite walk S p || o −−−−→ S p || o −−−−→ . . . p n − || o n −−−−−−→ S n ∈ G + } . Then C is continuous on [0 , \ A , and right continuous on A . Also, if t ∈ A , then lim t − C exists. The proof relies on arguments of Hata [22] and is given in [5, Proposition 3.4]. Notethat here the contractions, associated with the prefixes p ∈ { (cid:15), . . . , · · · (cid:124) (cid:123)(cid:122) (cid:125) a } , are affinemappings of the form f p ( x ) = hx + π l ( p ).The above proposition means that discontinuities of C may occur if ψ does not identifywalks that are “trivially” identified by the numeration system φ as in (4.2). The followingproposition insures the identifications for finitely many such pairs of walks. Because ofthe graph directed self-similarity of the boundary, this will be sufficient to infer thecontinuity of C on the whole interval [0 ,
1] (see Proposition 4.6).
Proposition 4.5.
The following equalities hold. ψ ( P ( S ; o max )) = ψ ( P ( S + 1; )) (1 ≤ S ≤ S max − ψ ( P ( S max ; o max )) = ψ ( P (1; ))(4.6) ψ ( P ( S ; o , o max )) = ψ ( P ( S ; o + , )) ( ∀ S, ≤ o < o max ) , (4.7) where S max = 12 (case a > b ) or (case a = b ).Proof. We check (4.5) in the following way. Let us consider the case a ≥ b + 1 , b ≥
2. Werefer to Table 2 and Figure 5. We read for S = 1 P (1; o max ) = 1 −→ a −→ − a − b −−→ P (2; ) = 2 −→ a −→ − a − b −−→ − −→ a ( a − b ), it follows from the definition of ψ (see (4.4)) that the equality ψ ( P (1; o max )) = ψ ( P (2; )) is trivially satisfied.The same happens for the other values of S , apart from S = 5 , , ,
11. For example,we read for S = 5 P (5; o max ) = 5 a − −−→ − −→ b −→ − a −→ − and P (6; ) = 6 a −→ − b − −−→ a −→ − −→ b −→ . Note that this does not exclude the case b = 2 (we then simply have o max = for thestate 5). Therefore we read the infinite sequence of prefixes( a − ba and a ( b − a b. To prove that these sequences lead to the same boundary point, we use Lemma 3.2.Indeed, there exists the pair of following infinite walks in G ⊂ G : p = a − | p (cid:48) = b − −−−−−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p = b | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = b −−−−−−−→ − p (cid:48)(cid:48) = a | p (cid:48) = b − −−−−−−−−−→ − p (cid:48)(cid:48) = b − | p (cid:48) = a −−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = b −−−−−−−→ p (cid:48)(cid:48) = b | p (cid:48) = a −−−−−−−→ , and 5 = [1 , π (1 , − , , , , π (1 , − , , (cid:88) k ≥ h k π l ( p k ) = π (1 , − ,
1) + (cid:88) k ≥ h k π l ( p (cid:48) k ) = (cid:88) k ≥ h k π l ( p (cid:48)(cid:48) k ) . Conditions (4.6) and (4.7) are checked similarly. The computations are carried out inAppendix A. See also this appendix for the remaining values of a, b . (cid:3) In the appendix and in the rest of this section, we use the following notation for theconcatenation of walks:(4.8) ( S ; o , . . . , o n )&( S (cid:48) ; o n + , . . . ) := ( S, o , . . . , o n , o n + , . . . ) , where S (cid:48) is the ending state of the walk ( S ; o , . . . , o n ) of G + . Proposition 4.6.
The mapping C satisfies C (0) = C (1) and it is H¨older continuouswith exponent s = − log | α | log λ , where | α | = max {| α | , | α |} ( α , α are the conjugates of thePisot number β ).Proof. The proof mainly relies on an argument of Hata [22]. By Proposition 4.4, wejust need to check the left-continuity of C on the countable set A . This will result fromProposition 4.5. First note that (4.7) means that C (0) = C (1). Also, (4.5) and (4.7)mean that C is left continuous at the points associated to walks of length n = 1 in thedefinition of A . We now prove that this is sufficient for C to be continuous on the wholeset A . This follows from the definition of ψ . Indeed, let t ∈ A associated to a walk oflength n ≥ t = φ ( S ; o , . . . , o n , (cid:124) (cid:123)(cid:122) (cid:125) w )with o n (cid:54) = . We write ( p , p , . . . ) for the labeling sequence of P ( w ). Then, C ( t ) = ψ ( P ( w )) = (cid:80) n − k =0 h k π l ( p k ) + h n ψ ( P ( S (cid:48) ; o n , ))= (cid:80) n − k =0 h k π l ( p k ) + h n ψ ( P ( S (cid:48) ; o n − , o max )) (by Condition (4.7))= C ( t − )(here S (cid:48) is the ending state of the finite admissible walk w | n − in the automaton G +0 ).Thus C is left continuous in t .More details as well as the proof of the H¨older continuity can be found in [5, Proposi-tion 3.5]. The exponent is − log | δ | log λ , where δ is the maximal contraction factor among allthe contractions in the GIFS, i.e. , the maximal contraction factor of the mapping h (seeSubsection 2.2). This is exactly | α | . (cid:3) Remark . If | α | = | α | = | α | , i.e. , if the contraction h is a similarity, then β | α | =1, therefore the above H¨older exponent is related to the Hausdorff dimension of theboundary: s = 1dim H ( ∂ T )(see [44, Theorem 6.7] or [38, Theorem 4.4]). As mentioned in [44, Section 6.4], this isthe case as soon as D = (cid:0) − b + 18 ab − a b + 4 a (cid:1) ≥ OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 25
We now give the sequence of polygonal closed curves ∆ n converging to ∂ T with respectto the Hausdorff metric. For N points M , . . . , M N of R , we denote by [ M , . . . , M N ]the curve joining M , . . . , M N in this order by straight lines. Definition . Let w ( n )1 , . . . , w ( n ) N n be the walks of length n in the graph G + , written inthe lexicographical order:(1; , . . . , ) = w ( n )1 ≤ lex w ( n )2 ≤ lex . . . ≤ lex w ( n ) N n = ( S max ; o max , . . . , o max ) , where N n is the number of these walks. For n = 0, these are just the states 1 , . . . , S max .Let C ( n ) j := C ( φ ( w ( n ) j & )) ∈ ∂T (1 ≤ j ≤ N n ) . Then we call ∆ n := (cid:104) C ( n )1 , C ( n )2 , . . . , C ( n ) N n , C ( n )1 (cid:105) , the n -th approximation of ∂ T n .The first terms of the sequences (∆ n ) n ≥ are depicted for a = b = 1 in Figure 10 andfor a = 10 , b = 7 in Figure 11. Proposition 4.9. ∆ n is a polygonal closed curve and its vertices have Q ( λ ) -addressesin the parametrization C . Moreover, (∆ n ) n ≥ converges to ∂ T in Hausdorff metric.Proof. By definition, ∆ n is a polygonal closed curve with vertices on ∂ T . The ver-tices have Q ( λ )-addresses in the parametrization, since they correspond to parameters t ∈ A , where A is the countable set defined in Proposition 4.4. Note that u , definedin Lemma 3.11, is the dominant eigenvector of a non-negative matrix with dominanteigenvalue λ , hence its components belong to Q ( λ ). Finally, one can check that ∆ n +1 is obtained from ∆ n after applying the GIFS (3.5). This is due to the fact that thecontractions are affine mappings. Therefore, (∆ n ) n ≥ converges in Hausdorff metric tothe unique attractor, which is ∂ T . Details can be found in [5, Section 3]. (cid:3) Proof of Theorem 2.2.
Theorem 2.2 is now a consequence of Lemma 3.11, Proposition 4.6and Proposition 4.9. (cid:3)
Remark . The way of generation of the approximations ∆ n is analogous to Dekking’srecurrent set method [17, 18]. Consider for example the Tribonacci case. The orderedautomaton G + on Figure 8 gives rise to a free group endomorphismΘ : 1 → , → , →
10 11 1 , . . . on the free group generated by the letters 1 , , . . . ,
11. An edge is associated to eachletter, the word W = 1 2 3 · · ·
11 is mapped to the 11-gon ∆ depicted on Figure 10.The iterations Θ n ( W ) map to ∆ n after renormalization.5. Proof of Theorem 2.1
We recall the statement concerning non disk-like tiles.
Let T a,b be the tile associatedto the substitution σ a,b ( a ≥ b ≥ ). If b − a > , then T a,b is not homeomorphic to aclosed disk.Proof of Theorem 2.1 (non disk-like tiles). One proof can be found in [27], but neededthe additional computation of a subgraph of the lattice boundary graph for all parameters a, b - a graph that describes the boundary in the periodic tiling induced by T . Here, wemake no use of this periodic tiling. The proof below uses the parametrization derivedfrom the graph G , already obtained by Thuswaldner in [44], or, more precisely, from ourordered version G + . Figure 10.
Boundary approximations for the Rauzy fractal T , Figure 11.
Boundary approximations for T , In our assumptions a ≥ b ≥ b − a >
3, we just need to consider two cases:( i ) b ≥ a ≥ b + 1;( ii ) a = b > i ) we find infinite walks associated to distinct parameters 0 < t < t (cid:48) < t : 5 b − || + ( − a − ) −−−−−−−−−−−−−→ − b − || −−−−→ b − || + ( − a − ) ←−−−−−−−−−−−−− t (cid:48) : 12 b − || −−−−→ b − || + ( − a − ) −−−−−−−−−−−−−→ b − || ←−−−− − in G . We refer to Table 2 and the corresponding Figure 5. OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 27
Similarly, in case ( ii ) we find the following infinite walks associated to distinct param-eters 0 < t < t (cid:48) < t : 5 a − || + ( a − ) −−−−−−−−−−→ − a − || −−−−→ a − || + ( a − ) ←−−−−−−−−−− t (cid:48) : 11 a − || −−−−→ a − || + ( a − ) −−−−−−−−−−→ a − || ←−−−− − in G (see Table 4 and the corresponding Figure 7).Therefore, in both cases, we have 0 < t < t (cid:48) < C ( t ) = C ( t (cid:48) ), since theassociated infinite walks in G carry the same labels. Hence ∂ T is not a simple closedcurve. (cid:3) We now come to the characterization of the disk-like tiles.
Let T a,b be the tile associatedto the substitution σ a,b ( a ≥ b ≥ ). If b − a ≤ , then ∂ T a,b is a simple closed curve.Therefore, T a,b is homeomorphic to a closed disk. We wish to show that all pairs ( t, t (cid:48) ) ∈ [0 ,
1[ with C ( t ) = C ( t (cid:48) ) satisfy t = t (cid:48) . In otherwords, we shall show that all pairs of walks ( w, w (cid:48) ) ∈ G + with ψ ( P ( w )) = ψ ( P ( w (cid:48) ))satisfy φ ( w ) = φ ( w (cid:48) ), where φ, P, ψ are defined in (4.1), (4.3) and (4.4), respectively.We first characterize the infinite sequences of prefixes ( p k ) k ≥ , ( p (cid:48) k ) k ≥ leading to thesame boundary point (cid:80) k ≥ h k π l ( p k ) = (cid:80) k ≥ h k π l ( p (cid:48) k ). Lemma 5.1.
Let ( p k ) k ≥ and ( p (cid:48) k ) k ≥ be the labels of infinite walks in the prefix-suffixgraph Γ starting from i ∈ A and i (cid:48) ∈ A , respectively. Then (cid:88) k ≥ h k π l ( p k ) = (cid:88) k ≥ h k π l ( p (cid:48) k ) =: x ∈ ∂ T if and only if there exist j ∈ A , γ ∈ π ( Z ) \ { } with [ γ, j ] ∈ Γ srs and ( p (cid:48)(cid:48) k ) k ≥ sequenceof prefixes such that (5.1) [ i, γ, j ] p | p (cid:48)(cid:48) −−−→ · · · p | p (cid:48)(cid:48) −−−→ · · · ∈ G [ i (cid:48) , γ, j ] p (cid:48) | p (cid:48)(cid:48) −−−→ · · · p (cid:48) | p (cid:48)(cid:48) −−−→ · · · ∈ G . Proof.
By the tiling property, a boundary point x can also be written x = γ + (cid:88) k ≥ h k π l ( p (cid:48)(cid:48) k )for some γ ∈ π ( Z ) \ { } , an infinite walk ( p (cid:48)(cid:48) k ) k ≥ in Γ, starting from a j ∈ A , with[ γ, j ] ∈ Γ srs . Thus the lemma follows from Lemma 3.2. (cid:3) The above characterization requires the knowledge of the boundary graph G - thesubgraph G would not be sufficient to obtain all the identifications. G is not known forour whole class of substitutions σ a,b . However, in the case 2 b − a ≤
3, it was computedin our joint work [27].
Proposition 5.2 ([27, Theorem 3.2]) . Let σ a,b be the substitution (2.1), G ,a,b the bound-ary graph as in Definition 3.1 and G ,a,b the graph of Definition 3.6. Suppose b − a ≤ .Then G ,a,b = G ,a,b . We can now characterize the disk-like tiles of our class.
Proof of Theorem 2.1 (disk-like tiles).
As mentioned above, we need to check that allidentifications are trivial in the parametrization, i.e. , that infinite sequences of prefixes( p k ) k ≥ and ( p (cid:48) k ) k ≥ in the prefix-suffix graph Γ satisfying (cid:88) k ≥ h k π l ( p k ) = (cid:88) k ≥ h k π l ( p (cid:48) k ) =: x ∈ ∂ T correspond only to labels of admissible infinite walks w and w (cid:48) in G + satisfying φ ( w ) = φ ( w (cid:48) ) defined in (4.1). The pairs of walks identified by φ are given in (4.2).To this effect, we first look for all pairs of infinite sequences of prefixes ( p k ) (cid:54) = ( p (cid:48) k )such that (cid:80) k ≥ h k π l ( p k ) = (cid:80) k ≥ h k π l ( p (cid:48) k ) ∈ ∂ T . This amounts to finding all the pairsof infinite admissible walks in G = G satisfying (5.1). This can be done algorithmicallyby constructing an automaton A ψ , with the following states and edges.( i ) S | S (cid:48) p || o | p || o (cid:48) −−−−−−→ T | T (cid:48) if and only if there is a prefix p (cid:48)(cid:48) satisfying S p | p (cid:48)(cid:48) || o −−−−−→ T ∈ G + S (cid:48) p | p (cid:48)(cid:48) || o (cid:48) −−−−−−→ T (cid:48) ∈ G + . ( ii ) S | S (cid:48) p || o | p (cid:48) || o (cid:48) −−−−−−−→ T || T (cid:48) if and only if p (cid:54) = p (cid:48) and there is a prefix p (cid:48)(cid:48) satisfying S p | p (cid:48)(cid:48) || o −−−−−→ T ∈ G + S (cid:48) p (cid:48) | p (cid:48)(cid:48) || o (cid:48) −−−−−−→ T (cid:48) ∈ G + . ( iii ) S || S (cid:48) p || o | p (cid:48) || o (cid:48) −−−−−−−→ T || T (cid:48) if and only if there is a prefix p (cid:48)(cid:48) satisfying S p | p (cid:48)(cid:48) || o −−−−−→ T ∈ G + S (cid:48) p (cid:48) | p (cid:48)(cid:48) || o (cid:48) −−−−−−→ T (cid:48) ∈ G + . We call an infinite walk in A ψ admissible if it starts from a state S | S (cid:48) with S = [ i, γ, j ], S (cid:48) = [ i (cid:48) , γ, j ] and [ γ, j ] ∈ Γ srs (possibly S = S (cid:48) ), and if it goes through at least one state ofthe shape T || T (cid:48) . Now, two sequences of the prefix-suffix automaton Γ, ( p k ) k ≥ (cid:54) = ( p (cid:48) k ) k ≥ ,satisfy (cid:80) k ≥ h k π l ( p k ) = (cid:80) k ≥ h k π l ( p (cid:48) k ) ∈ ∂ T if and only if there is an admissible walkin A ψ labelled by ( p k || o k | p (cid:48) k || o (cid:48) k ) k ≥ .After deleting the states and edges that do not belong to an admissible walk, we getthe automaton of Figure 12 for the case a ≥ b + 1 , b ≥
2. Note that for a = b + 1 or b = 2, the automaton becomes lighter, as several edges disappear. The starting statesfor admissible walks are colored. For the sake of simplicity, we did not depict the edgesof the form S | S p || o | p || o −−−−−−→ T | T (particular case of ( i )). Therefore, the states S | S in thesefigures may be preceded by a finite walk made of such edges and ending in S | S . Theremaining cases can be found in the Appendix B.Second, we look for all pairs of infinite admissible walks w (cid:54) = w (cid:48) of G + such that P ( w )and P ( w (cid:48) ) carry the same infinite sequence of prefixes ( p k ) k ≥ : the parameters t, t (cid:48) ∈ [0 , C .Again, these pairs of walks can be obtained algorithmically via an automaton A sl withthe following states and edges.( iv ) S | S p | o || o −−−−→ T | T if and only if S p || o −−−→ T ∈ G + . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 29 ( v ) S | S p | o || o (cid:48) −−−−−→ T || T (cid:48) if and only if o (cid:54) = o (cid:48) and (cid:40) S p || o −−→ T ∈ G + S p || o (cid:48) −−−→ T (cid:48) ∈ G + . ( vi ) S || S (cid:48) p | o || o (cid:48) −−−−−→ T || T (cid:48) if and only if and (cid:40) S p || o −−→ T ∈ G + S (cid:48) p || o (cid:48) −−−→ T (cid:48) ∈ G + . We call an infinite walk in A sl admissible if it starts from a state S | S or S || S (cid:48) with S = [ i, γ, j ], S (cid:48) = [ i (cid:48) , γ, j ] and [ γ, j ] ∈ Γ srs ( S (cid:54) = S (cid:48) ), and if it goes through at least onestate of the shape T || T (cid:48) . Now, for two admissible infinite walks of G + : w = ( S ; o , o , . . . ) (cid:54) = w (cid:48) = ( S (cid:48) ; o (cid:48) , o (cid:48) , . . . ) ,P ( w ) and P ( w (cid:48) ) carry the same sequence of prefixes ( p k ) k ≥ if and only if there is anadmissible walk in A sl labelled by ( p k | o k || o (cid:48) k ) k ≥ .After deleting the states and edges that do not belong to an admissible walk, weget the automaton of Figure 13 for the case a ≥ b + 1 , b ≥
2. The starting states foradmissible walks are colored. For the sake of simplicity, we did not depict the edges ofItem ( iv ). For the remaining cases, see Appendix B.Note that in A ψ as well as in A sl , no more pairs ( w, w (cid:48) ) than the pairs given in (4.2)are found. Therefore, we conclude that for all w, w (cid:48) ∈ G + ψ ( P ( w )) = ψ ( P ( w (cid:48) )) ⇒ φ ( w ) = φ ( w (cid:48) ) . Consequently, the parametrization C : [0 , → ∂ T is injective, apart from C (0) = C (1).Hence ∂T is a simple closed curve and, by a theorem of Sch¨onflies [45], T is homeomorphicto a closed disk. (cid:3) Concluding remarks
Other projects using the parametrization method may concern the topological studyof further classes of substitutions, for example families of Arnoux-Rauzy substitutions.These substitutions are of the form σ = τ · · · τ r , where r ≥ { τ , . . . , τ r } = { σ , σ , σ } ( σ i are the Arnoux-Rauzy substitutions). For the moment, the connect-edness of the associated Rauzy fractals could be obtained (see [9]), but the classificationdisk-like/non-disk-like is still outstanding.Another challenge is the study of non-disk-like tiles, which happens to be rather dif-ficult. A criterion [38] allows to decide whether the fundamental group is trivial oruncountable, but more precise descriptions are not known. For given examples of self-affine tiles, the description of cut points and of connected components could be achieved(see [31, 8]). We can understand the degree of difficulty of these studies via the followingconsiderations. In our framework, non-disk-likeness implies non-trivial identifications inthe parametrization and requires to find out non-injective points of the parametrization.To speak roughly, we need the computation of the language of G \ G . Therefore, thisis related to the complementation of B¨uchi automata, which is known to be a difficultproblem ([42, 32]). Note that we have here the tools to complete such a study. Similarlyto [5, Section 4] and as in the above proof of Theorem 2.1 (disk-like tiles), we can definethree automata whose edges take the form S | S (cid:48) p || o | p (cid:48) || o (cid:48) −−−−−−−→ T | T (cid:48) , − ||
11 11 − || − − | − || − | − || − || − − | − − | − − | − − | − || − | − − | − − || − − || − || || − | | |
12 5 | | | | | |
12 8 | | a − b − || | a − b || || o m | b || b || o m | a || a || o m | || a − b − − k || + ( a − b − − k ) | a − b − − k || + ( a − b − − k )0 ≤ k ≤ a − b − a − b + k || + | a − b + 1 + k || + ≤ k ≤ b − a − b || | a − b + 1 || || o m | b − || || o m | b − || a − || o m | a || b || o m | b || b − || | b − || a − || o m | a || a − − k || + ( b − − k ) | a − − k || + ( b − − k )0 ≤ k ≤ b − k || + | k + 1 || + ( k + )0 ≤ k ≤ a − b − || | || b − || | b − || b − || | b − || k || + ( b − − k ) | k || + ( b − − k )0 ≤ k ≤ b − k + 1 || + ( b − − k ) | k || + ( b − − k )0 ≤ k ≤ b − b − − k || + | b − − k || + ≤ k ≤ b − b − − k || + | b − − k || + ≤ k ≤ b − || | || || | || || | || || | || b || o m | b − || a || o m | a − b + 1 || || | || a − b || o m | || || o m | a || a || o m | a − b || a − || o m | a − b || a || o m | a − || a − − k || + | a − − k || + ( k + )0 ≤ k ≤ a − b − Figure 12. A ψ for a ≥ b + 1 , b ≥ o m stands for o max ).where S p || o −−→ T and S (cid:48) p (cid:48) || o (cid:48) −−−→ T (cid:48) are edges of G . A first automaton A φ gives the walksidentified by the Dumont-Thomas numeration system φ , i.e. , the pairs ( w, w (cid:48) ) ∈ G + given in (4.2). In the disk-like case, no other walks are identified. The second automaton A ψ gives pairs of walks ( w, w (cid:48) ) identified by ψ and is computed via Lemma 5.1. The thirdautomaton A sl gives the pairs of walks ( w, w (cid:48) ) carrying the same sequence of prefixes.Topological information might be read off from the automaton A ψ ∪ A sl \ A φ . Appendix A. Details for the proof of Proposition 4.5.
We check Conditions (4.5), (4.6) and (4.7) of Proposition 4.5. The ideas were givenin the proof after the statement of this proposition. We sum up the computations inTables 6 to 13. As mentioned, some computations require the use of Lemma 3.2. Thelast column refers to the items below for these special cases. In the cases below, we givethe pairs of infinite walks in G ⊂ G : S = [ i, γ, j ] p | p (cid:48) −−−→ S p | p (cid:48) −−−→ S p | p (cid:48) −−−→ · · · and S (cid:48) = [ i (cid:48) , γ, j (cid:48) ] p (cid:48)(cid:48) | p (cid:48) −−−→ S (cid:48) p (cid:48)(cid:48) | p (cid:48) −−−→ S (cid:48) p (cid:48)(cid:48) | p (cid:48) −−−→ · · · , OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 31 − || − || − − || − − || − − || − − || − − || − − || − − || − || − | || | || | || | | || | | || || || | || a | || a | o m || b | o m || a − − k | + || + ≤ k ≤ a − b − a − | o m || | o m || | o m || k | + ( b − − k ) || + ( b − − k )0 ≤ k ≤ b − a | o m || a − − k | + ( b − − k ) || + ( b − − k )0 ≤ k ≤ b − a − − k | + ( b − − k ) || + ( b − − k )0 ≤ k ≤ b − k | + || + ≤ k ≤ a − b − | || a | || a | || b − | o m || b − | o m || a − b − | o m || | o m || a | || a | || b | || a − b | o m || | o m || a | o m || | o m || a | o m || a − b | o m || | o m || a − b + 1 | o m || Figure 13. A sl for a ≥ b + 1 , b ≥ o m stands for o max ).for which holds (cid:88) k ≥ h k π l ( p k ) = γ + (cid:88) k ≥ h k π l ( p (cid:48) k ) = (cid:88) k ≥ h k π l ( p (cid:48)(cid:48) k )by Lemma 3.2. The concatenation of walks, using the symbol &, was defined in (4.8). Case a ≥ b + 1 , b ≥
2. Note that the states S = 11 , − have only one outgoing edge,thus it does not show up in the checking of Conditon (4.7). This happens also with thestates S = 5 , − whenever b = 2, and S = 9 , − whenever a = b + 1.(1) See proof of Proposition 4.5.(2) p = a | p (cid:48) = a − b − −−−−−−−−−−−→ − p = a − | p (cid:48) =0 −−−−−−−−−→ − p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) = a − | p (cid:48) = a − b − −−−−−−−−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) =0 −−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − . (3) p = b | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ p (cid:48)(cid:48) = b − | p (cid:48) = a −−−−−−−−−→ p = a − b +1 | p (cid:48) =0 −−−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) =0 −−−−−−−−−→ − . (4) For 0 ≤ k ≤ b − p = b − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =( a − b ) | p (cid:48) = a −−−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ p (cid:48)(cid:48) = b − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a − b +1 | p (cid:48) =0 −−−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) =0 −−−−−−−−−→ − . (5) For 0 ≤ k ≤ a − b − p (cid:48)(cid:48) = k | p (cid:48) = b + k −−−−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p (cid:48)(cid:48) = b | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) = a | p (cid:48) = b −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p = k +1 | p (cid:48) = b + k −−−−−−−−−−−→ p = b − | p (cid:48) = a − −−−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) = b −−−−−−−→ p = b | p (cid:48) = a −−−−−−−→ . (6) p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p = b | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = b −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a − b +1 | p (cid:48) = a −−−−−−−−−−−→ p = b − | p (cid:48) = a − −−−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = b −−−−−−−→ p (cid:48)(cid:48) = b | p (cid:48) = a −−−−−−−→ . (7) For 0 ≤ k ≤ a − b − p = a − − k | p (cid:48) = a − b − − k −−−−−−−−−−−−−−−−→ − p = a − | p (cid:48) =0 −−−−−−−−−→ − p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) = a − − k | p (cid:48) = a − b − − k −−−−−−−−−−−−−−−−→ − p = a − b | p (cid:48) =0 −−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − . (8) For 0 ≤ k ≤ b − − p = a − b + k | p (cid:48) = k −−−−−−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p = b | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = b −−−−−−−→ − − p (cid:48)(cid:48) = a − b + k +1 | p (cid:48) = k −−−−−−−−−−−−−→ − p = b − | p (cid:48) = a −−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = b −−−−−−−→ p (cid:48)(cid:48) = b | p (cid:48) = a −−−−−−−→ . (9) For 0 ≤ k ≤ b − − p = b − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = b − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a − b +1 | p (cid:48) =0 −−−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) =0 −−−−−−−−−→ − . (10) − p = b − | p (cid:48) = a −−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a − b | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = b − | p (cid:48) = a −−−−−−−−−→ p = a − b +1 | p (cid:48) =0 −−−−−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − b −−−−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) =0 −−−−−−−−−→ − . (11) − p = a − b − | p (cid:48) = a −−−−−−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p = b | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = b −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ − p (cid:48)(cid:48) = a − b | p (cid:48) = a −−−−−−−−−→ p = b − | p (cid:48) = a − −−−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = b −−−−−−−→ p (cid:48)(cid:48) = b | p (cid:48) = a −−−−−−−→ . Case a ≥ , b = 1. We take advantage of the similarities with the preceding case(compare the graph of Figure 6, Table 3 with the graph of Figure 5, Table 2 when taking b = 1).Conditions (4.5) is checked as in Table 6 for S = 1 , , , , , ,
10 and the referreditems by taking b = 1. OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 33
Condition (4.7) is checked as in Tables 7 and 8 for( S ; o ) ∈ { (2; ) , (3; + ) , (3; + ) , (7; ) , (7; ) , (8; ) , (9; ) , (9; ) , (2 − ; ) , (2 − ; ) , (7 − ; ) , (9 − ; + ) , (9 − ; + ) } . by taking b = 1. Note that the states S = 1 , , , , , , − , − , − , − have only oneoutgoing edge, that is why they do not show up in the checking of Conditon (4.7). Thisalso happens for S = 9 , − , whenever a = 2.The remaining cases and Condition (4.6) are presented in Table 9, with references tothe items below when the use of Lemma 3.2 is required.(12) p = a − | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) =1 −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =1 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) = a −−−−−−−→ . (13) p =1 | p (cid:48) =0 −−−−−−−→ − p = a − | p (cid:48) =0 −−−−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) =1 −−−−−−−→ − p (cid:48)(cid:48) =1 | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =1 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) = a −−−−−−−→ . (14) p =1 | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a − | p (cid:48) = a −−−−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a − −−−−−−−−−→ − p (cid:48)(cid:48) = a − | p (cid:48) =0 −−−−−−−−−→ − . Case a = b ≥
2. This case has similarities with the case a ≥ b + 1 , b ≥
2. However,the number of starting states reduces to 11. We present the results in Tables 10 to 12.Note that the states S = 8 , , − , − have only one outgoing edge, thus they do notshow up in the checking of Conditon (4.7). This also happens for S = 5 , − , whenever a = 2.(15) p = a − | p (cid:48) = a − −−−−−−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = a −−−−−−−→ − p (cid:48)(cid:48) = a | p (cid:48) = a − −−−−−−−−−→ − p (cid:48)(cid:48) = a − | p (cid:48) = a −−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a −−−−−−−→ , (16) p = a | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ p (cid:48)(cid:48) = a − | p (cid:48) = a −−−−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =0 −−−−−−−→ − . (17) For 0 ≤ k ≤ a − p = a − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ p (cid:48)(cid:48) = a − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =0 −−−−−−−→ − . (18) p =0 | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) = a − −−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = a −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) = a −−−−−−−→ p = a − | p (cid:48) = a − −−−−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a −−−−−−−→ . (19) For 0 ≤ k ≤ a − − p = k | p (cid:48) = k −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p = a | p (cid:48) = a −−−−−−−→ − − p (cid:48)(cid:48) = k +1 | p (cid:48) = k −−−−−−−−−→ − p = a − | p (cid:48) = a −−−−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) = a −−−−−−−→ . (20) For 0 ≤ k ≤ a − − p = a − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) = a − − k | p (cid:48) = a − − k −−−−−−−−−−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =0 −−−−−−−→ − . (21) − p = a − | p (cid:48) = a −−−−−−−−−→ p = a | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) = a −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) = a − | p (cid:48) = a −−−−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) = a −−−−−−−→ p (cid:48)(cid:48) = a | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =0 −−−−−−−→ − . Case a = b = 1. There are similarities with the preceding case a = b ≥ a = 1).Conditions (4.5) and Condition (4.6) are checked as in Table 10 for S = 1 , , , , , , , a = 1.Condition (4.7) is checked as in Tables 11 and 12 for( S ; o ) ∈ { (2; ) , (3; ) , (7; ) , (7 − ; ) } . by taking a = 1. Note that the states S = 1 , , , , , , , − , − , − , − , − haveonly one outgoing edge, that is why they do not show up in the checking of Conditon (4.7).The remaining cases are presented in Table 13, with references to the items belowwhen the use of Lemma 3.2 is required.(22) p =0 | p (cid:48) =1 −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p =1 | p (cid:48) =1 −−−−−−−→ − p =0 | p (cid:48) =1 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) =1 −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =1 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) =1 −−−−−−−→ . (23) p =1 | p (cid:48) =1 −−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p =0 | p (cid:48) =1 −−−−−−−→ p =0 | p (cid:48) =0 −−−−−−−→ p (cid:48)(cid:48) =0 | p (cid:48) =1 −−−−−−−→ p =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =1 −−−−−−−→ p (cid:48)(cid:48) =1 | p (cid:48) =0 −−−−−−−→ − p (cid:48)(cid:48) =0 | p (cid:48) =0 −−−−−−−→ − . OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 35Walk 1 Walk 2 Sequence 1 Sequence 2 Checking via Lemma 3.2: see Item. . .(1; o max ) (2; ) 0 a ( a − b ) 0 a ( a − b )0(2; o max ) (3; ) 0 ba ba o max ) (4; ) ( a − b )0 a ( a − b ) ( a − b )0 a ( a − b )0(4; o max ) (5; ) ( a − b + 1)0 a ( a − b ) ( a − b + 1)0 a ( a − b )(5; o max ) (6; ) ( a − ba a ( b − a b (1)(6; o max ) (7; ) a ba a b (7; o max ) (8; ) a ( a − b )0 a ( a − b )0(8; o max ) (9; ) a ( a − a − b )0 a ( a − a − b )0 a (2)(9; o max ) (10; ) b ( a − ba ba ( b − a b (1)(10; o max ) (11; ) ba ba o max ) (12; ) ba ( a − b )0 ( b − a − b + 1)0 a ( a − b ) (3)(12; o max ) (1; ) ( b − a ba ( b − a b ) Table 6.
Case a ≥ b + 1 , b ≥
2, Conditions (4.5) and (4.6).
Appendix B. Details for the proof of Theorem 2.1 (disk-like tiles).
We depict the automata A ψ and A sl for the remaining cases: • b = 1 , a ≥
2, Figures 14 and 15; • a = b ≤ b − a ≤ a ∈ { , } , as well asFigures 18 and 19 for a = b = 1.Again, in A ψ as well as in A sl , no more pairs ( w, w (cid:48) ) than the pairs given in (4.2) arefound. Thus the same conclusion as in the core of the proof of Theorem 2.1 (disk-liketiles) applies. References [1] Roy L. Adler and Benjamin Weiss.
Similarity of automorphisms of the torus . Memoirs of the Amer-ican Mathematical Society, No. 98. American Mathematical Society, Providence, R.I., 1970.[2] Shigeki Akiyama. Dynamical norm conjecture and Pisot tiling.
Kyoto University Research Infor-mation Repository, RIMS Lecture Note , in Japanese (1091):241–250, 1999.[3] Shigeki Akiyama. Topological structure of fractal tilings attached to number systems.
Colloquiumof the Japanese Mathematical Society, Algebra Section: Lecture Note , in Japanese, 1999.[4] Shigeki Akiyama, Guy Barat, Val´erie Berth´e, and Anne Siegel. Boundary of central tiles associatedwith Pisot beta-numeration and purely periodic expansions.
Monatsh. Math. , 155(3-4):377–419,2008.[5] Shigeki Akiyama and Benoˆıt Loridant. Boundary parametrization of self-affine tiles.
J. Math. Soc.Japan , 63(2):525–579, 2011.[6] Pierre Arnoux and Shunji Ito. Pisot substitutions and Rauzy fractals.
Bull. Belg. Math. Soc. SimonStevin , 8(2):181–207, 2001. Journ´ees Montoises d’Informatique Th´eorique (Marne-la-Vall´ee, 2000).[7] Marcy Barge and Jaroslaw Kwapisz. Geometric theory of unimodular Pisot substitutions.
Amer. J.Math. , 128(5):1219–1282, 2006.[8] Julien Bernat, Benoˆıt Loridant, and J¨org Thuswaldner. Interior components of a tile associated toa quadratic canonical number system—Part II.
Fractals , 18(3):385–397, 2010.[9] Val´erie Berth´e, Timo Jolivet, and Anne Siegel. Connectedness of the fractals associated with arnoux-rauzy substitutions.
RAIRO Theoretical Informatics and Application , to appear, 2013.[10] Val´erie Berth´e and Michel Rigo, editors.
Combinatorics, automata and number theory , volume135 of
Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge,2010. − ||
11 11 − || − − | − || || − || − − | − − | − || − | − − | − − || − − || − || || − | | | | |
12 8 | | a − || | a − || || o m | || || o m | a || a || o m | || a − − k || + ( a − − k ) | a − − k || + ( a − − k )0 ≤ k ≤ a − || o m | || || o m | || a − || o m | a || || o m | || a − || o m | a || k || + | k + 1 || + ( k + )0 ≤ k ≤ a − || | || || | || || | || || | || || | || || o m | || a || o m | a || || | || a − || o m | || || o m | a || a || o m | a − || a − || o m | a − || a || o m | a − || a − − k || + | a − − k || + ( k + )0 ≤ k ≤ a − Figure 14. A ψ for b = 1 , a ≥ [11] Val´erie Berth´e and Anne Siegel. Tilings associated with beta-numeration and substitutions. Integers ,5(3):A2, 46, 2005.[12] Rufus Bowen. Markov partitions for Axiom A diffeomorphisms.
Amer. J. Math. , 92:725–747, 1970.[13] Rufus Bowen. Markov partitions are not smooth.
Proc. Amer. Math. Soc. , 71(1):130–132, 1978.[14] Alfred Brauer. On algebraic equations with all but one root in the interior of the unit circle.
Math.Nachr. , 4:250–257, 1951.[15] Vincent Canterini and Anne Siegel. Automate des pr´efixes-suffixes associ´e `a une substitution prim-itive.
J. Th´eor. Nombres Bordeaux , 13(2):353–369, 2001.[16] Vincent Canterini and Anne Siegel. Geometric representation of substitutions of Pisot type.
Trans.Amer. Math. Soc. , 353(12):5121–5144, 2001.[17] F. M. Dekking. Recurrent sets.
Adv. in Math. , 44(1):78–104, 1982.[18] F. M. Dekking. Replicating superfigures and endomorphisms of free groups.
J. Combin. Theory Ser.A , 32(3):315–320, 1982.[19] Jean-Marie Dumont and Alain Thomas. Systemes de numeration et fonctions fractales relatifs auxsubstitutions.
Theoret. Comput. Sci. , 65(2):153–169, 1989.[20] Karlheinz Gr¨ochenig and Andrew Haas. Self-similar lattice tilings.
J. Fourier Anal. Appl. , 1(2):131–170, 1994.[21] Branko Gr¨unbaum and G. C. Shephard.
Tilings and patterns . A Series of Books in the MathematicalSciences. W. H. Freeman and Company, New York, 1989. An introduction.[22] Masayoshi Hata. On the structure of self-similar sets.
Japan J. Appl. Math. , 2(2):381–414, 1985.[23] Pascal Hubert and Ali Messaoudi. Best simultaneous Diophantine approximations of Pisot numbersand Rauzy fractals.
Acta Arith. , 124(1):1–15, 2006.
OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 37 − || − || − − || − − || − − || − − || − − || − − || − || − | || | || || | || | || || || | || a | || a | o m || | o m || a − − k | + || + ≤ k ≤ a − a − | o m || | o m || | o m || a | o m || k | + || + ≤ k ≤ a − | || a | || a | || | o m || a − | o m || | o m || | || a − | o m || | o m || a | o m || | o m || a | o m || a − | o m || | o m || a | o m || Figure 15. A sl for b = 1 , a ≥ [24] Shunji Ito and Minako Kimura. On Rauzy fractal. Japan J. Indust. Appl. Math. , 8(3):461–486,1991.[25] Shunji Ito and Makoto Ohtsuki. Modified Jacobi-Perron algorithm and generating Markov partitionsfor special hyperbolic toral automorphisms.
Tokyo J. Math. , 16(2):441–472, 1993.[26] Shunji Ito and Hui Rao. Atomic surfaces, tilings and coincidence. I. Irreducible case.
Israel J. Math. ,153:129–155, 2006.[27] Benoˆıt Loridant, Ali Messaoudi, Paul Surer, and J¨org M. Thuswaldner. Tilings induced by a classof cubic Rauzy fractals.
Theoret. Comput. Sci. , 477:6–31, 2013.[28] R. Daniel Mauldin and S. C. Williams. Hausdorff dimension in graph directed constructions.
Trans.Amer. Math. Soc. , 309(2):811–829, 1988.[29] Ali Messaoudi. Fronti`ere du fractal de Rauzy et syst`eme de num´eration complexe.
Acta Arith. ,95(3):195–224, 2000.[30] Ali Messaoudi. Propri´et´es arithm´etiques et topologiques d’une classe d’ensembles fractales.
ActaArith. , 121(4):341–366, 2006.[31] Sze-Man Ngai and Nhu Nguyen. The Heighway dragon revisited.
Discrete Comput. Geom. ,29(4):603–623, 2003.[32] Dominique Perrin and Jean-Eric Pin.
Infinite words - Automata, Semigroups, Logic and Games ,volume 141 of
Pure and applied Mathematics . Elsevier, 2004.[33] Brenda Praggastis. Numeration systems and Markov partitions from self-similar tilings.
Trans.Amer. Math. Soc. , 351(8):3315–3349, 1999.[34] Brenda L. Praggastis.
Markov partitions for hyperbolic toral automorphisms . ProQuest LLC, AnnArbor, MI, 1992. Thesis (Ph.D.)–University of Washington.[35] G´erard Rauzy. Nombres alg´ebriques et substitutions.
Bull. Soc. Math. France , 110(2):147–178, 1982.[36] Yuki Sano, Pierre Arnoux, and Shunji Ito. Higher dimensional extensions of substitutions and theirdual maps.
J. Anal. Math. , 83:183–206, 2001.[37] Klaus Scheicher and J¨org M. Thuswaldner. Neighbours of self-affine tiles in lattice tilings. In
Fractalsin Graz 2001 , Trends Math., pages 241–262. Birkh¨auser, Basel, 2003. − ||
10 10 − || − || − | − || − || − − | − − | − − | − − | − || − | − − || − || || − | | |
11 5 | | | | | | || o m | a || a || o m | a || a || o m | || k || + | k || + ≤ k ≤ a − || | || || o m | a − || || o m | a − || a − || o m | a || a || o m | a || a − || | a − || a − || o m | a || || | || || | || a − || | a − || a − || | a − || k || + ( a − − k ) | k || + ( a − − k )0 ≤ k ≤ a − k + 1 || + ( a − − k ) | k || + ( a − − k )0 ≤ k ≤ a − || | || || | || || | || || | || a || o m | a − || a || o m | || || | || || o m | || || o m | a || a || o m | || Figure 16. A ψ for 2 ≤ a = b ≤ a = 3). [38] Anne Siegel and J¨org M. Thuswaldner. Topological properties of Rauzy fractals. M´em. Soc. Math.Fr. (N.S.) , 118:140, 2009.[39] Ja. G. Sina˘ı. Markov partitions and U-diffeomorphisms.
Funkcional. Anal. i Priloˇzen , 2(1):64–89,1968.[40] V´ıctor F. Sirvent and Yang Wang. Self-affine tiling via substitution dynamical systems and Rauzyfractals.
Pacific J. Math. , 206(2):465–485, 2002.[41] Boris Solomyak. Substitutions, adic transformations, and beta-expansions. In
Symbolic dynamicsand its applications (New Haven, CT, 1991) , volume 135 of
Contemp. Math. , pages 361–372. Amer.Math. Soc., Providence, RI, 1992.[42] Wolfgang Thomas. Automata on infinite objects. In
Handbook of theoretical computer science, Vol.B , pages 133–191. Elsevier, Amsterdam, 1990.[43] William Thurston. Groups, tilings, and finite state automata. AMS Colloquium lecture notes, 1989.[44] J¨org M. Thuswaldner. Unimodular Pisot substitutions and their associated tiles.
J. Th´eor. NombresBordeaux , 18(2):487–536, 2006.[45] Gordon Whyburn and Edwin Duda.
Dynamic topology . Springer-Verlag, New York, 1979. Under-graduate Texts in Mathematics, With a foreword by John L. Kelley.
Montanuniversit¨at Leoben, Lehrstuhl Mathematik & Statistik, Franz Josef Strasse 18,8700 Leoben Austria
E-mail address : [email protected] OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 39 − || − || − − || − − || − − || − − || − − || − || − ||
10 2 || | || | | || | | || || || | || a | o m || a | o m || | o m || | o m || k | + ( a − − k ) || + ( a − − k )0 ≤ k ≤ a − a | o m || | || | || | || a | || a − | o m || a − | o m || | o m || a | || a | || a | || | o m || | o m || a | o m || | o m || a | o m || | o m || | o m || | o m || Figure 17. A sl for 2 ≤ a = b ≤ a = 3). − | − || − || − || || − − | − − || − − || − || − || − − || | | | | || | || || o m | || || o m | || || o m | || || o m | || || o m | || || | || || o m | || | o m | || || o m | || || o m | || || o m | || || o m | || || | || || | || Figure 18. A ψ for a = b = 1. + , o max )= (1; + )&(7; o max ) ,k = 0 , . . . , b − + , )= (1; + )&(5; ) ( b − − k ) a ( a − b )0 ( b − − k )( a − b + 1)0 a ( a − b ) (4)(1; + , o max )= (1; + )&(5; o max ) ,k = 0 , . . . , b − + , )= (1; + )&(6; ) ( b − − k )( b − ba ( b − − k ) a ( b − a b (1)(1; + , o max )= (1; + )&(6; o max ) ,k = 0 , . . . , b − + ( k + ) , )= (1; + ( k + ))&(7; ) ( b − − k ) a b ( b − − k ) a b (2; , o max )= (2; )&(8; o max ) (2; , )= (2; )&(9; ) 0 a ( a − a − b )0 a a − a − b )0 a (2)(2; , o max )= (2; )&(9; o max ) (2; , )= (2; )&(10; ) 0 b ( a − ba ba ( b − a b (1)(3; , o max )= (3; )&(11; o max ) (3; , )= (3; )&(12; ) 0 ba ( a − b )0 0( b − a − b + 1)0 a ( a − b ) (3)(3; , o max )= (3; )&(12; o max ) (3; , )= (3; )&(1; ) 0( b − a ba b − a b (3; + , o max )= (3; + )&(1; o max ) ,k = 0 , . . . , a − b − + , )= (3; + )&(2; ) k a ( a − b ) k a ( a − b )(3; + , o max )= (3; + )&(2; o max ) ,k = 0 , . . . , a − b − + ( k + ) , )= (3; + ( k + ))&(1; ) k ba k + 1)( b − a b (5)(4; , o max )= (4; )&(2; o max ) (4; , )= (4; )&(1; ) ( a − b )0 ba a − b + 1)( b − a b (6)(5; + , o max )= (5; + )&(7 − ; o max ) ,k = 0 , . . . , b − + , )= (5; + )&(6 − ; ) ( a − b + 1 + k )0 ba ( a − b + 2 + k )( b − a b (6)(5; + , o max )= (5; + )&(6 − ; o max ) ,k = 0 , . . . , b − + , )= (5; + )&(5 − ; ) ( a − b + 2 + k )( b − a b ( a − b + 2 + k )( b − a b (5; + , o max )= (5; + )&(5 − ; o max ) ,k = 0 , . . . , b − + ( k + ) , )= (5; + ( k + ))&(7 − ; ) ( a − b + 2 + k )0 a ( a − b ) ( a − b + 2 + k )0 a ( a − b )(6; , o max )= (6; )&(6 − ; o max ) (6; , )= (6; )&(5 − ; ) a ( b − a b a ( b − a b (6; , o max )= (6; )&(5 − ; o max ) (6; , )= (6; )&(7 − ; ) a a ( a − b ) a a ( a − b )(7; , o max )= (7; )&(10 − ; o max ) (7; , )= (7; )&(9 − ; ) a b − a b a b − a b (7; , o max )= (7; )&(9 − ; o max ) (7; , )= (7; )&(8 − ; ) a ( a − b − a ( a − b ) a ( a − b − a ( a − b )(8; , o max )= (8; )&(1 − ; o max ) (8; , )= (8; )&(2 − ; ) a ( a − ba a ( a − ba (9; + , o max )= (9; + )&(1 − ; o max ) ,k = 0 , . . . , a − b − + , )= (9; + )&(2 − ; ) ( a − − k )( a − ba ( a − − k )( a − ba (9; + , o max )= (9; + ))&(2 − ; o max ) ,k = 0 , . . . , a − b − + ( k + ) , )= (9; + ( k + ))&(1 − ; ) ( a − − k )( a − a − b )0 a ( a − − k )( a − b )0 a (7)(10; , o max )= (10; )&(12 − ; o max ) (10; , )= (10; )&(11 − ; ) ba a ( a − b )0 ba a ( a − b )0(12; , o max )= (12; )&(5; o max ) (12; , )= (12; )&(6; ) ( b − a − ba ( b − a ( b − a b (1) Table 7.
Case a ≥ b + 1 , b ≥
2, Condition (4.7), S = 1 , . . . , OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 41Walk 1 Walk 2 Sequence 1 Sequence 2 Item(1 − ; + , o max )= (1 − ; + )&(7 − ; o max ) ,k = 0 , . . . , b − − ; + , )= (1 − ; + )&(6 − ; ) ( a − b + k )0 ba ( a − b + 1 + k )( b − a b (8)(1 − ; + , o max )= (1 − ; + )&(6 − ; o max ) ,k = 0 , . . . , b − − ; + , )= (1 − ; + )&(5 − ; ) ( a − b + 1 + k )( b − a ba ( a − b + 1 + k )( b − a b (1 − ; + , o max )= (1 − ; + )&(5 − ; o max ) ,k = 0 , . . . , b − − ; + ( k + ) , )= (1 − ; + ( k + ))&(7 − ; ) ( a − b + 1 + k )0 a ( a − b )0 ( a − b + k + 1)0 a ( a − b )(2 − ; , o max )= (2 − ; )&(10 − ; o max ) (2 − ; , )= (2 − ; )&(9 − ; ) ( a − b − a ba ( a − b − a b (2 − ; , o max )= (2 − ; )&(9 − ; o max ) (2 − ; , )= (2 − ; )&(10 − ; ) ( a − a − b − a ( a − b ) ( a − a − b − a ( a − b )(5 − ; + , o max )= (5 − ; + )&(7; o max ) ,k = 0 , . . . , b − − ; + , )= (5 − ; + )&(5; ) ( b − − k ) a ( a − b )0 ( b − − k )( a − b + 1)0 a ( a − b ) (9)(5 − ; + , o max )= (5 − ; + )&(5; o max ) ,k = 0 , . . . , b − − ; + , )= (5 − ; + )&(6; ) ( b − − k )( a − ba ( b − − k ) a ( b − a b (1)(5 − ; + , o max )= (5 − ; + )&(6; o max ) ,k = 0 , . . . , b − − ; + ( k + ) , )= (5 − ; + ( k + ))&(7; ) ( b − − k ) a ba ( b − − k ) a b (6 − ; , o max )= (6 − ; )&(7; o max ) (6 − ; , )= (6 − ; )&(5; ) ( b − a ( a − b )0 ( b − a − b + 1)0 a ( a − b ) (10)(6 − ; , o max )= (6 − ; )&(5; o max ) (6 − ; , )= (6 − ; )&(6; ) ( b − a − ba ( b − a ( b − a b (1)(7 − ; , o max )= (7 − ; )&(8; o max ) (7 − ; , )= (7 − ; )&(9; ) 0 a ( a − a − b )0 a a − a − b )0 a (2)(7 − ; , o max )= (7 − ; )&(9; o max ) (7 − ; , )= (7 − ; )&(10; ) 0 b ( a − ba ba ( b − a b (1)(8 − ; , o max )= (8 − ; )&(2; o max ) (8 − ; , )= (8 − ; )&(1; ) ( a − b − ba a − b )( b − a b (11)(9 − ; + , o max )= (9 − ; + )&(1; o max ) ,k = 0 , . . . , a − b − − ; + , )= (9 − ; + )&(2; ) k a ( a − b ) k a ( a − b )(9 − ; + , o max )= (9 − ; + ))&(2; o max ) ,k = 0 , . . . , a − b − − ; + ( k + ) , )= (9 − ; + ( k + ))&(1; ) k ba k + 1)( b − a b (5)(10 − ; , o max )= (10 − ; )&(11; o max ) (10 − ; , )= (10 − ; )&(12; ) 0 ba ( a − b )0 0( b − a − b + 1)0 a ( a − b )(12 − ; , o max )= (12 − ; )&(6 − ; o max ) (12 − ; , )= (12 − ; )&(5 − ; ) a ( b − a b a ( b − a b Table 8.
Case a ≥ b + 1 , b ≥
2, Condition (4.7), S = 1 − , . . . , − . o max ) (5; ) ( a − a a a
01 (12)(5; o max ) (6; ) a a ( a − a a ( a − o max ) (10; ) 1( a − a a a
01 (13)(11; o max ) (12; ) 1 a ( a − a a ( a −
1) (14)(12; o max ) (1; ) 0 a a a , o max )= (2; )&(9; o max ) (2; , )= (2; )&(10; ) 01( a − a a a
01 (13)(3; , o max )= (3; )&(11; o max ) (3; , )= (3; )&(12; ) 01 a ( a − a a ( a −
1) (14)(3; , o max )= (3; )&(12; o max ) (3; , )= (3; )&(1; ) 00 a a a , o max )= (10; )&(12 − ; o max ) (10; , )= (10; )&(11 − ; ) 1 a a ( a − a a ( a − − ; , o max )= (7 − ; )&(9; o max ) (7 − ; , )= (7 − ; )&(10; ) 01( a − a a a
01 (13)
Table 9.
Case a ≥ , b = 1. See also Tables 6 to 8 with b = 1. Walk 1 Walk 2 Sequence 1 Sequence 2 Checking via Lemma 3.2: see Item. . .(1; o max ) (2; ) 0 a a o max ) (3; ) 0 aa aa o max ) (4; ) 00 a a o max ) (5; ) 10 a a o max ) (6; ) ( a − aa a ( a − a a (15)(6; o max ) (7; ) a aa a a (7; o max ) (8; ) a a o max ) (9; ) a ( a − aa aa ( a − a a (15)(9; o max ) (10; ) aa aa o max ) (11; ) aa
00 ( a − a o max ) (12; ) ( a − a aa ( a − a a Table 10.
Case a = b ≥
2, Conditions (4.5) and (4.6).
OPOLOGICAL PROPERTIES OF A CLASS OF CUBIC RAUZY FRACTALS 43Walk 1 Walk 2 Sequence 1 Sequence 2 Item(1; + , o max )= (1; + )&(7; o max ) ,k = 0 , . . . , a − + , )= (1; + )&(5; ) ( a − − k ) a
00 ( a − − k )10 a + , o max )= (1; + )&(5; o max ) ,k = 0 , . . . , a − + , )= (1; + )&(6; ) ( a − − k )( a − aa ( a − − k ) a ( a − a a (15)(1; + , o max )= (1; + )&(6; o max ) ,k = 0 , . . . , a − + ( k + ) , )= (1; + ( k + ))&(7; ) ( a − − k ) a a ( a − − k ) a a (2; , o max )= (2; )&(8; o max ) (2; , )= (2; )&(9; ) 0 a ( a − aa aa ( a − a a (15)(3; , o max )= (3; )&(10; o max ) (3; , )= (3; )&(11; ) 0 aa
00 0( a − a , o max )= (3; )&(11; o max ) (3; , )= (3; )&(1; ) 0( a − a aa a − a a (4; , o max )= (4; )&(2; o max ) (4; , )= (4; )&(1; ) 00 aa a − a a (17)(5; + , o max )= (5; + )&(7 − ; o max ) ,k = 0 , . . . , a − + , )= (5; + )&(6 − ; ) ( k + 1)0 aa ( k + 2)( a − a a (17)(5; + , o max )= (5; + )&(6 − ; o max ) ,k = 0 , . . . , a − + , )= (5; + )&(5 − ; ) ( k + 2)( a − a a ( k + 2)( a − a a (5; + , o max )= (5; + )&(5 − ; o max ) ,k = 0 , . . . , a − + ( k + ) , )= (5; + ( k + ))&(7 − ; ) ( k + 2)0 a k + 2)0 a , o max )= (6; )&(6 − ; o max ) (6; , )= (6; )&(5 − ; ) a ( a − a a a ( a − a a (6; , o max )= (6; )&(5 − ; o max ) (6; , )= (6; )&(7 − ; ) a a a a , o max )= (7; )&(9 − ; o max ) (7; , )= (7; )&(8 − ; ) a a − a a a a − a a (9; , o max )= (9; )&(11 − ; o max ) (9; , )= (9; )&(10 − ; ) aa a aa a , o max )= (11; )&(5; o max ) (11; , )= (11; )&(6; ) ( a − a − aa ( a − a ( a − a a (15) Table 11.
Case a = b ≥
2, Condition (4.7), S = 1 , . . . , − || − || − − || − − || − − || − − || || || || | || || || || | || | − || − | o m || | o m || | o m || | o m || | o m || | o m || | o m || | o m || | o m || | o m || | || | o m || | || | || | o m || | || | o m || | o m || Figure 19. A sl for a = b = 1. − ; + , o max )= (1 − ; + )&(7 − ; o max ) ,k = 0 , . . . , a − − ; + , )= (1 − ; + )&(6 − ; ) k aa ( k + 1)( a − a a (18)(1 − ; + , o max )= (1 − ; + )&(6 − ; o max ) ,k = 0 , . . . , a − − ; + , )= (1 − ; + )&(5 − ; ) ( k + 1)( a − a aa ( k + 1)( a − a a (1 − ; + , o max )= (1 − ; + )&(5 − ; o max ) ,k = 0 , . . . , a − − ; + ( k + ) , )= (1 − ; + ( k + ))&(7 − ; ) ( k + 1)0 a
00 ( k + 1)0 a − ; + , o max )= (5 − ; + )&(7; o max ) ,k = 0 , . . . , a − − ; + , )= (5 − ; + )&(5; ) ( a − − k ) a
00 ( a − − k )10 a − ; + , o max )= (5 − ; + )&(5; o max ) ,k = 0 , . . . , a − − ; + , )= (5 − ; + )&(6; ) ( a − − k )( a − aa ( a − − k ) a ( a − a a (15)(5 − ; + , o max )= (5 − ; + )&(6; o max ) ,k = 0 , . . . , a − − ; + ( k + ) , )= (5 − ; + ( k + ))&(7; ) ( a − − k ) a aa ( a − − k ) a a (6 − ; , o max )= (6 − ; )&(7; o max ) (6 − ; , )= (6 − ; )&(5; ) ( a − a
00 ( a − a − ; , o max )= (6 − ; )&(5; o max ) (6 − ; , )= (6 − ; )&(6; ) ( a − a − aa ( a − a ( a − a a (15)(7 − ; , o max )= (7 − ; )&(8; o max ) (7 − ; , )= (7 − ; )&(9; ) 0 a ( a − aa aa ( a − a a (15)(9 − ; , o max )= (9 − ; )&(10; o max ) (9 − ; , )= (9 − ; )&(11; ) 0 aa
00 0( a − a − ; , o max )= (11 − ; )&(6 − ; o max ) (11 − ; , )= (11 − ; )&(5 − ; ) a ( a − a aa a ( a − a a Table 12.
Case a = b ≥
2, Condition (4.7), S = 1 − , . . . , − . Walk 1 Walk 2 Sequence 1 Sequence 2 Checking via Lemma 3.2: see Item. . .(4; o max ) (5; ) 00110 10101 (22)(5; o max ) (6; ) 1010 1010(10; o max ) (11; ) 1100 01010 (23)(3; , o max )= (3; )&(10; o max ) (3; , )= (3; )&(11; ) 01100 001010 (23)(9; , o max )= (9; )&(11 − ; o max ) (9; , )= (9; )&(10 − ; ) 110100 11010(9 − ; , o max )= (9 − ; )&(10; o max ) (9 − ; , )= (9 − ; )&(11; ) 01100 001010 (23) Table 13.
Case a = bb