Continued fraction algorithm for Sturmian colorings of trees
aa r X i v : . [ m a t h . D S ] A p r CONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OFTREES
DONG HAN KIM AND SEONHEE LIM
Abstract.
Factor complexity b n ( φ ) for a vertex coloring φ of a regular tree is the number ofclasses of n -balls up to color-preserving automorphisms. Sturmian colorings are colorings ofminimal unbounded factor complexity b n ( φ ) = n + 2.In this article, we prove an induction algorithm for Sturmian colorings using colored ballsin a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, wecharacterize Sturmian colorings in terms of the data appearing in the induction algorithm. Introduction
Factor complexity (also called subword complexity or block complexity) b n ( u ) of an (one-sided) infinite word u , which counts the number of distinct n -subwords, has been studied for along time in symbolic dynamics [6]. A classical theorem of Hedlund and Morse says that Sturmiansequences, which are sequences of minimal unbounded complexity b n ( u ) = n + 1, correspond toirrational rotations [7].Factor complexity was generalized from sequences to vertex colorings of a regular tree in[8] and [9]. Fo a graph X , let us denote its vertex set by V X and its set of oriented edges by EX . For a vertex coloring φ : V T → A of a d -regular tree T , let B n ( φ ) the set of classes of n -balls appearing in T colored by φ up to coloring-preserving isomorphisms of n -balls. The factorcomplexity b n ( φ ) (called subword complexity in [8] ) is the cardinality of the set B n ( φ ).A coloring φ is called periodic if b n ( φ ) is bounded. A coloring φ g associated to an automor-phism g of a uniform tree T was constructed in [11] and [2] so that g is a commensurator elementof a cocompact lattice Γ ⊂ G = Aut ( T ) if and only if φ g is a periodic coloring. (See [11], [4], [15],Chapter 6 of [3] for commensurators.) One can associate a vertex coloring to any automorphismof a tree, thus classifying vertex colorings of trees may give us a tool to classify automorphismsof a tree. Date : September 20, 2016.2010
Mathematics Subject Classification.
Key words and phrases. tree, Sturmian, colorings, factor complexity, continued fraction algorithm. B ( φ ) = { ◦ , • } , B ( φ ) = (cid:26) ◦ •◦ ◦ , ◦ •◦ • , • •◦ ◦ (cid:27) , B ( φ ) = ( ◦ •◦ ◦ •◦•◦ • ◦ , ◦ •◦ • •◦•• • ◦ , • •◦ ◦ ◦◦◦◦ • ◦ , • •◦ ◦ ◦◦•◦ • ◦ ) . Figure 1.
An example of a Sturmian treeA coloring φ is called Sturmian if it has minimal unbounded factor complexity b n ( φ ), whichis n + 2. See Figure 1 for an example of a Sturmian coloring and the sets B n ( φ ) for n = 0 , , n , there is a unique class of n -balls, called the special n -ball and denoted by S n , which is the restriction (to the concentric n -ball) of two distinct classes of ( n + 1)-balls denoted by A n +1 , B n +1 . We denote by C n thecentral n -ball of S n +1 . For φ in Figure 1, S is the white vertex, S is the last ball in B ( φ ).Let G be the group of φ -preserving automorphisms of T , which is a subgroup of the auto-morphism group Aut( T ). Define X = X φ := G \ T to be the graph obtained from quotienting T by G . We call X φ the quotient graph of ( T, φ ).The quotient graph X = X φ has a structure of edge-indexed graph ( X, i ), i.e. a graph withindex i : EX → N on the set of oriented edges. It is in fact the edge-indexed graph associatedto the graph of groups ( X, Stab G ( · )) which is the graph X with stabilizers Stab G ( x ) attached toeach x ∈ V X ∪ EX . (See Chapter 2 of [3] for details.)The tree T is the universal cover of the edge-indexed graph ( X, i ) (see Appendix of [3]). Thegraph X is again colored by φ and the original coloring is the lift of φ from X to T . In a previouswork, we defined Stumian colorings of bounded type and characterized the quotient graph X asfollows. ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 3
For v ∈ V T , we define the type set of v as the set of m ∈ Z ≥ such that the colored m -ballcentered at v is special. A Sturmian coloring φ on tree T is called of bounded type if the type setof each vertex is finite. If one vertex is of bounded type, then all vertices are of bounded type([8]). Throughout the paper, by a cycle , we mean a cycle of length >
1. A cycle of length 1 willbe called a loop . Theorem 1.1 ([8] Theorem 3.4) . Any Sturmian coloring φ is a lift of a coloring on a graph X which is an infinite ray (with loops possibly attached) • • • • • • · · · or a biinfinite line (with loops possibly attached) · · · • • • • • • • • • • · · · Moreover, if the coloring φ is of bounded type, then the graph X is the former. If x i denotesthe i -th vertex from the left, then there exists m such that the maximum of the type set of x i is m + i, ∀ i ≥ . The edge-indexed graph (
X, i ) of the first graph in the theorem above can be the edge-indexedgraph associated to lattice of Nagao type (see Chapter 10 of [3] for such lattices).In this article, we show an induction algorithm for Sturmian colorings that characterizesSturmian colorings completely, analogous to a continued fraction algorithm for Sturmian words.1.1.
Induction algorithm.
Sturmian words.
Let us recall the correspondence between Sturmian words and irrationalrotations. Fix 0 < θ < ≤ ρ < . Consider the orbit { ρ + nθ (mod 1) } n ∈ N of a rotationwith irrational slope θ and intercept ρ . Partition [0 ,
1) into [0 , − θ ) ⊔ [1 − θ,
1) and give index0 , s = s s · · · is a Sturmian word if and only if s n = ⌊ ( n + 1) θ + ρ ⌋ − ⌊ nθ + ρ ⌋ or ⌈ ( n + 1) θ + ρ ⌉ − ⌈ nθ + ρ ⌉ , for some θ and ρ . Note that in the first case, s n is the index of the set where the orbit belongsfor the partition above and the second case corresponds to another similar partition. See forexample [10, Section 2.1.2].We choose two sequences of finite words ( u n ) and ( v n ) with alphabets { , } , satisfying u − =0 , v − = 1 and(1.1) ( u n +1 , v n +1 ) = R ( u n , v n ) := ( u n , u n v n ) or ( u n +1 , v n +1 ) = L ( u n , v n ) := ( v n u n , v n ) . DONG HAN KIM AND SEONHEE LIM u k v k Figure 2.
Rauzy graphs of Strumian word, case (i) and case (ii)Rauzy showed that both sequences u n , v n have the same limit which is a characteristic Sturmianword (i.e. Sturmian word with ρ = 0), and conversely any characteristic Sturmian word is thelimit of two such sequences [13].Two Sturmian words have the same factors if and only if they have the same slope [10,Proposition 2.1.18]. For each θ , the characteristic Sturmian word s θ is constructed by the partialquotients a k of the continued fraction expansion of the slope θ = a + a + . . . : s θ = lim n →∞ u n = lim n →∞ v n where(1.2) lim n →∞ ( u n , v n ) = lim k →∞ R a k +1 · · · R a L a R a − ( u − , v − ) . (see e.g. [10, Proposition 2.2.24]).There is an induction algorithm, closely related to continued fraction algorithm, for Sturmianwords using Rauzy graphs (also called factor graphs) [12] (see also [5]): for a given infinite word u , the Rauzy graph G n is a finite oriented graph whose vertices are distinct n -words. There isan oriented edge from u to v if there are two letters x, y such that uy = xv is an ( n + 1)-word.For Sturmian words, the Rauzy graph G n is always a union of two cycles with an intersectionwhich is either a vertex (case (i)) or a line segment (case (ii)). The G n belongs to the case (i)infinitely often. (See Arnoux and Rauzy’s work [1] for transitions between case (i) and case (ii).)Let n k be the sequence of positive integers such that G n k belongs to case (i). The two cycles in G n k correspond to the finite words ( u k , v k ). Thus, the Rauzy graph G n k evolves by the formulaof (1.2).1.1.2. Sturmian colorings.
Now let us describe our induction algorithm for Sturmian colorings.For a given Sturmian coloring φ , let G n be the graph whose vertices are elements of B n ( φ ). Thereis an edge between two classes D, E of n -balls if there exist n -balls of center x, y in the class D, E , respectively, such that d ( x, y ) = 1. The graph G n is an analogue of Rauzy graphs G n . ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 5
Except at the vertex S n , the graph G n is locally isomorphic as colored graphs to the quotientgraph X φ since for any class D = S n of n -balls, there is a unique class of ( n + 1)-ball containing D concentrically. For the special class S n , there are two classes of ( n + 1)-balls A n +1 , B n +1 containing S n .We define graphs G An , G Bn as follows: the vertices of G An , G Bn are vertices of G n that are connectedto S n with edges defined by adjacency in A n +1 , B n +1 , respectively. The graphs G An , G Bn bothcontain C n (Lemma 2.3).For n such that C n is the end vertex of both G An and G Bn , we define the concatenation G An ⊲⊳ G Bn of two graphs G An , G Bn . (See Definition 3.2 for details.)If C n = S n , we define the vertex set to be V ( G An ⊲⊳ G Bn ) = V G An ∪ V G Bn where the end vertices C n in G An and C n in G Bn are identified. The edge set is E ( G An ⊲⊳ G Bn ) = E G An ∪ E G Bn , where theloops at C n in G An and G Bn are identified in G An ⊲⊳ G Bn . If C n = S n , then define V ( G An ⊲⊳ G Bn ) = V G An ⊔ V G Bn where C n = S n in G An and G Bn remaintwo distinct vertices in G An ⊲⊳ G Bn . Define E ( G An ⊲⊳ G Bn ) = E G An ⊔ E G Bn ∪ { e } , where e is the edgebetween two end vertices C n in G An and G Bn .Our first main theorem is the following induction algorithm, an analog of continued fractionalgorithm described in (1.1) and (1.2). Theorem 1.2.
Let φ be a Sturmian coloring.(1) If φ is such that G n does not have any cycle for all n , then there exists K ∈ [0 , ∞ ] and asequence ( n k ) k ≥ such that n k = k for ≤ k ≤ K and G An ∼ = G An − , G Bn ∼ = G An − ⊲⊳ G Bn − , if ≤ n < K, G An ∼ = G An − ⊲⊳ G Bn − , G Bn ∼ = G An − ⊲⊳ G Bn − or G An ∼ = G An − ⊲⊳ G Bn − , G Bn ∼ = G Bn − , if n = K, G An ∼ = G An − , G Bn ∼ = G Bn − , if n = n k , n > K, G An ∼ = G An − ⊲⊳ G Bn − , G Bn ∼ = G Bn − or G An ∼ = G An − , G Bn ∼ = G An − ⊲⊳ G Bn − , if n = n k , n > K. (2) If φ is such that G n has a cycle, for some n , then φ is of bounded type. The coloring φ isof bounded type if and only if either G An or G Bn eventually stabilizes. Part (1) is proved in Theorem 3.5 and part (2) is proved in Proposition 5.4 and Theorem 5.7.Note that the graphs G An , G Bn has canonical structure of edge-indexed graphs (see Definition 2.9).Theorem 3.5 is more precise than part (1) of the above theorem in the sense that the concatenationwe define in Section 3 include the vertices V, V ′ where the concatenation occurs and edge-indices i k of edges around V, V ′ . DONG HAN KIM AND SEONHEE LIM
Direct limit as the inverse process.
In the second part of the article, we construct aninverse process of induction algorithm described in Theorem 1.2.We first define the α k -admissible sequences of indices i k : α k is an arbitrary sequence of A and B and i k is defined so that the universal cover of the concatenated edge-indexed graph hasdegree d (see Definition 4.1). We then define α k -admissible β k in (4.1), which is a sequence of A, B closely related to α k . Two sequences β k , β ′ k are equivalent if they are eventually equal. Ourmain result in the second part of the article is that there is a one-to-one correspondence betweenSturmian colorings and equivalence classes of admissible sequences: { ( X, φ ) Sturmian colorings } Φ ⇄ Ψ { ( α k , i k , [ β k ]) : i k , β k are α k -admissible } , where [ β k ] is the equivalence class of β k .1.2.1. Ψ : a direct limit. For any given sequence α k and α k -admissible sequence of indices i k , wedefine F Ak , F Bk recursively using concatenations with edge-indices i k . For an α k -admissible β k ,we first show that the direct limit of F β k k is the quotient graph ( X, φ ) of a Sturmian coloring φ (Theorem 4.14), i.e. Ψ is well-defined. If φ is of bounded type, the sequence β k has a furtherrestriction: β k is eventually constant (see Proposition 5.4).1.2.2. Φ : the data of vertices and edge-indices in the induction algorithm. For a given Sturmiancoloring φ , we define α k , i k and β k = β k ( t ) for each t ∈ V X φ : ( α k ) is the sequence of lettersdefined by α k = A if | V G An k | ≥ | V G Bn k | and α k = B otherwise. The vector i k is the new indicesthat appear in the concatenation: for example, i k = i and ( i, j ) respectively, in the two figuresbelow (see Definition 4.8). · · · ⊚ • S n +1 G An ⊲⊳ G Bn ⊛ · · · i m − iℓ · · · ⊚ • A n +1 G An ⊲⊳ G Bn • B n +1 ⊛ · · · i jm m ℓ − i ℓ − j We show that Φ is well-defined i.e. i k , β k are α k -admissible and distinct t, t ′ ∈ V X φ giveseventually equal sequences β k ( t ) , β k ( t ′ ). The next theorem shows that Ψ ◦ Φ = Id . Theorem 1.3.
Let φ be a Sturmian coloring.(1) For a sequence α k ∈ { A, B } , if i k and β k are α k -admissible, then the direct limit of F β k k is a Sturmian coloring. Furthermore, the sequence induced from the direct limit F β k k is ( α k , i k , β ′ k ) such that β k and β ′ k are eventually equal. ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 7 (2) If φ is such that G n does not have any cycle, for all n , then there are ( α k ) , ( α k ) -admissibleindices ( i k ) and an α k -admissible sequence β k such that the direct limit lim k →∞ G β k n k of G β k n k is the original Sturmian coloring ( X, φ ) .(3) If φ is such that G n has a cycle, then it is a coloring of bounded type. If φ is of boundedtype, then ( X, φ ) = lim −→ G ∗ n where ∗ = A or B . Part (1) is proved in Theorem 4.14 and part (2) is proved in Theorem ?? . Part (3) is provedin Theorem 5.7 (2). The colorings have the same classes of balls if and only if the sequences( α k , i k ) are the same. In this regard, the sequences ( α k , i k ) correspond to partial quotients of theslope of a Sturmian word and the sequence β k corresponds to the intercept of a Sturmian word.The article is organized as follows. In Section 2, we gather preliminary facts about coloredballs, define the graph G n and the edge-indexed graphs G An , G Bn . In Section 3, we study coloringsfor which G n does not have a cycle for all n and show induction algorithm (Theorem 1.2). InSection 4, we define α k -admissible sequences i k and β k and show Theorem 1.3 (1), (2). InSection 5, cyclic Sturmian colorings are treated and we investigate Sturmian colorings of boundedtype further to show Theorem 1.3 (3).2. Graphs of colored balls
Let T be a d -regular tree, i.e. the degree of each vertex is d . Let V T, ET be the set ofvertices and the set of oriented edges of T , respectively. The group Aut ( T ) of automorphismsof T is a locally compact topological group with compact-open topology. Consider the pathmetric d on T with edge length all equal to 1. The (closed) n -ball around x is defined by B n ( x ) = { y ∈ T : d ( x, y ) ≤ n } .Throughout the paper, φ : V T → A is a Sturmian coloring, i.e., a coloring of factor complexity b n ( φ ) = n + 2. Since b (0) = 2, A has two elements. Set A = { a, b } .2.1. Preliminary : basic properties of Sturmian colorings.
In this subsection, we recallpreliminary facts from [8] and prove basic properties of Sturmian colorings, mostly about variousadjacencies of n -balls in ( T, φ ). Definition 2.1.
For a Sturmian coloring φ on T , denote the colored tree by T φ .(1) Two vertices x and y are called congruent if there exists a color-preserving automorphismof T φ sending x to y .(2) Two n -balls B n ( x ) , B n ( y ) are called equivalent if there exists a color-preserving isomor-phism of n -balls between them. Such an equivalence class is called a class of n -balls and DONG HAN KIM AND SEONHEE LIM is denoted it either by an n -ball with brackets, e.g. [ B n ( x )], or by capital alphabets, e.g. A n , B n , C n , D, E .(3) For two classes D, E of balls, D is (always) adjacent to E if for any n -balls B n ( x ) in theclass D , there exists an n -ball B m ( y ) in the class E such that d ( x, y ) = 1. Two classes D, E are (always) adjacent if D is adjacent to E and vice versa.(4) Two classes D, E of balls can be adjacent if D = [ B n ( x )] , E = [ B m ( y )] for some balls B n ( x ) , B m ( y ) in T φ such that d ( x, y ) = 1.(5) A class of n -balls is called admissible if it appears in T φ . Let B n ( φ ) be the set of admissibleclasses of n -balls. As defined in the introduction, b n ( φ ) = | B n ( φ ) | .We will omit the word “always” in part (3) when there is no confusion. Remark 2.2.
Remark that if two classes
C, D of n -balls are both not special, then various kindsof adjacencies in part (3) and (4) above are equivalent. The only subtle situation is when one ofthem is S n : for a class D , it is possible that S n , D can be adjacent, D is always adjacent to S n and S n is not always adjacent to D .Recall that S n is the unique ball contained in two distinct classes of ( n + 1)-balls A n +1 , B n +1 ,and C n is the central n -ball of S n +1 . For a class of n -balls B = [ B n ( x )], denote the class of[ B n − ( x )] by B and call it the restriction of B . One of the most basic properties of a Sturmiancoloring is that for any non-special class of n -balls B , there is a unique class of ( n + 1)-ballscontaining B concentrically, which we denote by B and call the extension of B . For the notationalsimplicity, we denote the empty ball by S − = A − = C − . Note that B − is not defined. Lemma 2.3.
For a Sturmian coloring φ , without loss of generality, we assume that S = A = [ a ] and B = [ b ] .(1) We can choose { A n } , { B n } so that A n +1 , B n +1 are always adjacent to A n , B n , respectively.Moreover, A n +1 , B n +1 are uniquely determined if we impose the condition that A n +1 contains balls of class A n more than B n +1 does.(2) For each n -ball B n ( x ) , the n -balls adjacent to B n ( x ) belong to at most two classes of n -balls apart from [ B n ( x )] . Thus for any class of n -balls D = S n , there are at most twoclasses of n -balls adjacent to D .(3) If A n = S n ( B n = S n ), then A n ( B n , respectively) is always adjacent to S n .(4) The classes S n , C n are always adjacent.Proof. (1) Lemma 2.11 of [8] for m = 1 says that there are two balls C, D such that A n +1 , B n +1 are always adjacent to C, D , respectively and { C, D } = { A n , B n } . Thus we ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 9 may choose A n , B n so that A n +1 , B n +1 are always adjacent to A n , B n , respectively. Thenumbers of adjacent balls of class A n and B n to each A n +1 or B n +1 are constants since theballs of A n , B n are contained in A n +1 or B n +1 . To show the uniqueness, denote the num-ber of A n , B n adjacent to A n +1 , B n +1 by i ( A n +1 , A n ) , i ( A n +1 , B n ) , i ( B n +1 , A n ) , i ( B n +1 , B n ) , respectively. Remark that i ( A n +1 , A n ) + i ( A n +1 , B n ) = i ( B n +1 , A n ) + i ( B n +1 , A n ) , since they are both equal to the number of S n − adjacent to S n . If i A ( A n , A n +1 ) = i B ( A n , A n +1 ), then A n +1 = B n +1 since all the n -balls distinct from S n − adjacent to S n have unique extensions to ( n + 1)-balls.(2) It is clear that the centers of distinct classes of n -balls are not congruent, and there areat most two congruence classes of vertices adjacent to any given vertex by Theorem 1.1.Any D = S n has the same set of classes of colored balls in its 1-neighborhood, thus thereare at most two classes that can be adjacent to D .(3) By part (1), A n can be adjacent to A n +1 = S n . Thus if A n = S n , A n is always adjacentto S n .(4) If S n = C n , then A n +1 = S n +1 and B n +1 = S n +1 , thus by part (3), both A n +1 and B n +1 are always adjacent to S n +1 , thus S n is always adjacent to C n .If S n = C n , we may assume that A n +1 = S n +1 = B n +1 . By part (1), A n +2 and B n +2 are always adjacent to A n +1 and B n +1 respectively, thus S n +1 = A n +1 is always adjacentto either A n +1 or B n +1 . On the other hand, by part (3) B n +1 is always adjacent to S n +1 = A n +1 . Therefore, S n is always adjacent to A n +1 = B n +1 = S n = C n . (cid:3) Lemma 2.4.
Let φ be a Sturmian coloring.(1) Suppose that D is not equal to any of A n , B n , S n . If S n , D can be adjacent, then S n , D are always adjacent.(2) If D in part (1) satisfies D = C n , then S n = C n .(3) Apart from S n itself, there are at most three classes of n -balls which can be adjacent to S n .Proof. (1) We only need to show that S n is always adjacent to D . Since S n can be adjacent to D , it follows that S n contains D , i.e. S n is always adjacent to D . Note that D = S n − since D = A n , B n . Therefore D is uniquely extended to D and S n is always adjacent to D . (2) Suppose D = C n . If S n = C n , then either A n +1 = S n +1 or B n +1 = S n +1 . By Lemma 2.3(3), it follows that B n +1 ( = S n +1 ) is adjacent to S n +1 = A n +1 or A n +1 ( = S n +1 ) is adjacent to S n +1 = B n +1 . Therefore, A n +1 , B n +1 can be adjacent. Since A n +1 , B n +1 are always adjacent to D by part (1) and to A n , B n , respectively, by Lemma 2.3 (1), it follows that A n +1 and B n +1 canbe adjacent to B n +1 , A n , D and to A n +1 , B n , D respectively. Since either A n = S n or B n = S n ,either B n +1 , A n , D or A n +1 , B n , D are distinct, which contradicts Lemma 2.3 (2).(3) By Lemma 2.3 (2), apart from S n , there are at most four classes of n -balls adjacent to S n . If there exists D distinct from A n , B n , S n and adjacent to S n , by part (1), D, S n are alwaysadjacent, thus two of the four classes are D . (cid:3) Graph G n of colored balls. In this subsection, we establish basic properties of the graph G n . Recall that V G n = B φ ( n ) and two classes of n -balls are adjacent in G n , i.e. they are connectedby an edge in E G n if they can be adjacent. Recall also that a loop is an edge whose initial andterminal vertices are the same, and a cycle is of length larger than 1. Lemma 2.5. (1) Suppose that D is distinct from A n , B n , C n , S n . If S n , D are adjacent in G n , then D is a vertex of degree 1 in G n and there is a cycle in G n +1 .(2) If B n +1 (resp. A n +1 ) is adjacent to A n (resp. B n ) in G n , and A n = S n , C n , then thereis a cycle in G n +1 .Proof. (1) By Lemma 2.4 (2), we have S n = C n . Since S n = D as well, both A n +1 and B n +1 aredistinct from S n +1 and D . Lemma 2.3 (3) implies that both A n +1 and B n +1 are adjacent to S n +1 .By Lemma 2.4 (1), S n is always adjacent to D , thus both A n +1 and B n +1 can be adjacent to D .Therefore, the path with vertices [ S n +1 A n +1 DB n +1 S n +1 ] is a cycle in G n +1 . The restriction ofthe path to n -balls is a line segment [ C n S n D ] with D a vertex of degree 1.(2) By assumption, A n is adjacent to B n +1 in G n . By Lemma 2.3 (1), A n is adjacent to A n +1 in G n . Since A n = S n , A n differs from both A n +1 and B n +1 . If S n = C n , then S n +1 differs fromboth A n +1 and B n +1 . Thus, by Lemma 2.3 (3), the path with vertices [ S n +1 A n +1 A n B n +1 S n +1 ]is a cycle in G n +1 . If S n = C n , then S n +1 is one of A n +1 , B n +1 , By Lemma 2.3 (3), the path[ S n +1 A n B n +1 S n +1 ] or [ S n +1 A n A n +1 S n +1 ] is a cycle in G n +1 . (cid:3) Recall that S − = C − = A − = ∅ and S = A . Lemma 2.6.
There exists ≤ K ≤ ∞ such that S n = A n = C n if and only if − ≤ n < K .Therefore, either A K = S K or B K = S K . ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 11
Proof.
Suppose that S n +1 = A n +1 = C n +1 and n ≥
0. Then by Lemma 2.4 (2), S n +1 is adjacentto only B n +1 and itself. By Lemma 2.3 (2), B n +1 has exactly one more adjacent vertex D in G n +1 since G n +1 is connected and | V G n +1 | ≥
3. Then S n = B n +1 = A n +1 = S n +1 = C n and S n is adjacent to only D and itself. Since S n can be adjacent to A n and B n by Lemma 2.3 (1), wehave A n = S n = C n and B n = D . The case B n = S n = C n and A n = D does not occur sinceotherwise A n +1 would be adjacent to D which is a contradiction to the fact that S n +1 is adjacentto only B n +1 and itself. If S = A = C , then put K = 0.Since S K − = C K − , either A K = S K or B K = S K . (cid:3) Edge-indexed Graphs G An , G Bn of colored balls. Although the vertices of G n are equiva-lence classes of vertices of X , G n do not resemble the graph X even locally, because of the specialball S n , which can be extended to two ways in T φ . In this section, we define two edge-indexedgraphs G An , G Bn so that X locally looks like either G An or G Bn . Definition 2.7.
Define the indices i, i A , i B : S n ( V G n × V G n ) → N as follows.(1) If an n -ball X n is not special, let i ( X n , Y n ) be the number of n -balls colored by Y n adjacentto X n . It is independent of the position of X n in the tree.(2) Define i A ( S n , Y n ), i B ( S n , Y n ) to be the number of Y n adjacent to S n in A n +1 , B n +1 ,respectively. For simplicity, for X n = S n denote i A ( X n , Y n ) = i B ( X n , Y n ) = i ( X n , Y n ).In particular, i ( X n , Y n ) = 0 if there is no edge between X n and Y n . Remark 2.8.
Remark that i, i A , i B are not reflexive in general, however the positivity of i isreflexive in the following sense:(1) If X, Y are not special, then i ( X, Y ) > i ( Y, X ) > X = S n , then i ( X, S n ) > i A ( S n , X ) + i B ( S n , X ) > Definition 2.9.
Define G An ( G Bn ) to be the edge-indexed oriented graph whose vertices are thosewhich are connected by a path from S n with edges of positive index for i A ( i B , respectively).Their oriented edge set is the set of oriented edges between vertices in G An ( G Bn , respectively) withpositive index for i A ( i B , respectively), endowed with the index i A ( i B , respectively).For n = −
1, define G A − = G B − to be the edge-indexed graph consisting of the vertex of emptyball S − = C − and a loop on it indexed by d , the degree of T .It is clear that the set of vertices V G An ∪ V G Bn is the set of classes of n -balls B φ ( n ).Note that the 1-neighborhood of a given vertex in G An and G Bn are identical except at S n , sinceany non-special n -ball has a unique extension to ( n + 1)-ball. The following lemma is immediate by definition. The index i B enjoys properties similar tothose of i A . Lemma 2.10.
Let V = S n . Then for each n ≥ − (1) For an n -ball U = S n , C n we have i ( U, V ) = i ( U , V ) , i ( U, S n ) = i ( U , A n +1 ) + i ( U , B n +1 ) . (2) If C n = S n , then i ( C n , V ) = i A ( S n +1 , V ) , i ( C n , S n ) = i A ( S n +1 , A n +1 ) + i A ( S n +1 , B n +1 ) ,i A ( S n , V ) = i ( A n +1 , V ) , i A ( S n , S n ) = i ( A n +1 , A n +1 ) + i ( A n +1 , B n +1 ) . (3) If C n = S n , say S n +1 = A n +1 , then i A ( S n , V ) = i A ( A n +1 , V ) i A ( S n , S n ) = i A ( A n +1 , A n +1 ) + i A ( A n +1 , B n +1 )= i B ( A n +1 , V ) = i B ( A n +1 , A n +1 ) + i B ( A n +1 , B n +1 ) i B ( S n , V ) = i ( B n +1 , V ) i B ( S n , S n ) = i ( B n +1 , A n +1 ) + i ( B n +1 , B n +1 ) . Definition 2.11.
We say that a Sturmian coloring is cyclic if there is a cycle in G n for some n .We say that a Sturmian coloring is acyclic if it is not cyclic. Example 2.12.
For the coloring φ in Figure 1, we have the sequence of G n as follows. A = S = ◦ , B = C = • ; ◦G : S • B ◦G A : S • B ◦G B : S • B A = ◦ •◦ ◦ , B = C = ◦ •◦ • , S = B = • •◦ ◦ ; ◦G : A • S ◦ B ◦G A : A • S ◦ B •G B : S ◦ B ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 13 A = ◦ •◦ ◦ •◦•◦ • ◦ , B = S = ◦ •◦ • •◦•• • ◦ , A = • •◦ ◦ ◦◦◦◦ • ◦ , B = • •◦ ◦ ◦◦•◦ • ◦ . ◦G : A • A • B ◦ S ◦G A : A • A ◦ S •G B : B ◦ S S = C and A = S . Let us omit the balls from n = 4. •G : B ◦ B ◦ A • C ◦ A ◦G A : A • C ◦ A •G B : B ◦ B ◦ A • C ◦ A •G : B ◦ B ◦ B • S ◦ C ◦ A ◦G A : A • S ◦ C •G B : B ◦ B ◦ B • S ◦ C •G : B ◦ B ◦ B • B ◦ S ◦ A • A ◦G A : A • A ◦ S •G B : B ◦ B ◦ B • B ◦ S Induction algorithm for acyclic Sturmian colorings
Throughout Section 3 and Section 4, we assume that φ is acyclic, i.e. G n does not have anycycle of length larger than 1 for all n . Concatenation of G An , G Bn . The following lemma gives some special condition which ensuresthat the index i for G n +1 is zero for some balls. Lemma 3.1.
The following properties hold. (Similar properties hold for i B .)(1) For D = A n , B n , S n , C n , i ( D, S n ) = 0 . If A n = S n , C n , then i ( A n , S n ) = i ( A n , A n +1 ) .(2) If C n = S n , then i A ( S n , S n ) = i ( A n +1 , A n +1 ) .If furthermore A n +1 = C n +1 , then i ( C n , S n ) = i B ( S n +1 , B n +1 ) . (3) If C n = S n , say A n +1 = S n +1 and B n +1 = C n +1 , then i A ( S n , S n ) = i A ( A n +1 , A n +1 ) . Proof. (1) It follows from Lemma 2.5 (1) that i ( D, S n ) = 0 for D = A n , B n , C n , S n . If A n = S n , C n , then by Lemma 2.5 (2) i ( A n , B n +1 ) = 0. Lemma 2.10 (1) implies that i ( A n , S n ) = i ( A n , A n +1 ).(2) C n = S n implies that S n +1 = A n +1 , S n +1 = B n +1 . If A n +1 and B n +1 are adjacent, then[ S n +1 A n +1 B n +1 S n +1 ] is a cycle in G n +1 , thus i ( A n +1 , B n +1 ) = i ( B n +1 , A n +1 ) = 0. Moreover, if A n +1 = C n +1 , then by Lemma 2.5 (2) we deduce that i B ( S n +1 , A n +1 ) = 0. By Lemma 2.10, thesecond assertion follows.(3) If A n +1 = S n +1 , B n +1 = C n +1 , then since B n +1 = A n +1 = S n +1 it follows from Lemma 2.5(2) that i A ( S n +1 , B n +1 ) = 0. Using Lemma 2.10, we complete the proof. (cid:3) Definition 3.2. [Concatenation] Let G , G be two edge-indexed graphs. Let m i , ℓ i ( i = 1 ,
2) bethe indices of the edge and the loop coming out of V in G i . When m = m and ℓ = ℓ , we writethem as m and ℓ .(1) For 1 ≤ i < m , the ( i )-concatenation G V,V ′ ⊲⊳ i G of G and G at ( V, V ′ ) is the edge-indexedgraph defined as follows : the vertex set is V G ∪ V G , where only the vertices V ∈ G and V ′ ∈ G are identified (all the other vertices of G and G are distinct in G V,V ′ ⊲⊳ i G ).The oriented edge set is E G ∪ E G , where the loop at V in G and the loop at V in G areidentified. The edge-index is i, m − i for the non-loop edge at V in G , G , respectively,and ℓ for the loop at V . Note that ℓ can be zero. · · · ⊚ G • Vm ℓ • V ′ ⊛ G · · · mℓ · · · ⊚ • V = V ′ G V,V ′ ⊲⊳ i G ⊛ · · · i m − iℓ (2) For 1 ≤ i ≤ ℓ , ≤ j ≤ ℓ , the ( i, j )-concatenation G ⊲⊳ i,j G of G and G at ( V, V ′ ) is theedge-indexed graph with vertex set V G ⊔ V G (the vertices V in G and G are distinctin the concatenation) and the oriented edge set E G ⊔ E G ⊔ { e, e } with one new pair of ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 15 oriented edges e, e between V in G and V in G . The edge-index is ℓ − i , ℓ − j for loopsfrom G and G , respectively, and i ( V, V ′ ) = i, i ( V ′ , V ) = j for e . · · · ⊚ G • Vm ℓ • V ′ ⊛ G · · · m ℓ · · · ⊚ • V G V,V ′ ⊲⊳ i,j G • V ′ ⊛ · · · i jm m ℓ − i ℓ − j Let us call
V, V ′ the joining vertices of the concatenation. We will denote by G ⊲⊳ G eitheran i -concatenation or an ( i, j )-concatenation of G and G . In this section, when we omit thevertices V, V ′ , all the concatenations of G An , G Bn are at V = C n , V ′ = C n . Theorem 3.3.
Acyclic Sturmian colorings enjoy the following properties.(1) If A n +1 = S n +1 and A n +1 = C n +1 , then G Bn +1 ∼ = G Bn . (2) If S n = C n and A n +1 = C n +1 , then for some i , G Bn +1 ∼ = G An ⊲⊳ i G Bn . · · · ⊚ S n G An • C n m ℓ • C n ⊛ S n G Bn · · · mℓ · · · ⊚ A n +1 • S n +1 G Bn +1 ⊛ B n +1 · · · i m − iℓ (3) If (i) A n +1 = S n +1 or (ii) S n = C n and A n +1 = C n +1 , then for some i, j , G Bn +1 ∼ = G An ⊲⊳ i,j G Bn . · · · ⊚ G An • C n = S n m ℓ • C n = S n ⊛ G Bn · · · m ℓ · · · ⊚ • A n +1 G Bn +1 • B n +1 ⊛ · · · i jm m ℓ − i ℓ − j By symmetry, similar properties hold for G An +1 .The isomorphism of edge-indexed graphs from G Bn +1 to G Bn or G An ⊲⊳ G Bn is given by therestriction of ( n + 1) -balls to n -balls. D D .Proof. We show the edge-indexed graph isomorphism between G Bn +1 and G Bn or G An ⊲⊳ G Bn by theextension of n -ball or the restriction of ( n + 1)-balls.(1) If S n = C n , then by Lemma 3.1 (2), i ( C n , S n ) = i B ( S n +1 , B n +1 ) and i B ( S n , S n ) = i ( B n +1 , B n +1 ). Also, by Lemma 3.1 (1), i ( B n , S n ) = i ( B n , B n +1 ) if B n = S n , C n . Since there isno other class of n -balls adjacent to S n by Lemma 2.3 (2), we get G Bn +1 ∼ = G Bn . If S n = C n , then B n +1 = S n +1 and A n +1 = C n +1 . Thus by Lemma 3.1 (3) i B ( S n , S n ) = i B ( B n +1 , B n +1 ). Also, by Lemma 3.1 (i), i ( B n , S n ) = i ( B n , B n +1 ) if B n = S n = C n . Since thereis no other adjacent n -ball to S n , we get G Bn +1 ∼ = G Bn . (2) If S n = C n and A n +1 = C n +1 , then by Lemma 3.1 (2) i B ( S n , S n ) = i ( B n +1 , B n +1 ) . ByLemma 2.10 (2), i ( C n , S n ) = i B ( S n +1 , A n +1 ) + i B ( S n +1 , B n +1 ) , where i B ( S n +1 , A n +1 ) > i B ( S n +1 , B n +1 ) > G Bn +1 ∼ = G An ⊲⊳ i G Bn , where i = i B ( S n +1 , A n +1 ) . (3) If A n +1 = S n +1 (thus S n = C n ), by Lemma 2.10 (3), we have i A ( S n , S n ) = i B ( A n +1 , A n +1 ) + i B ( A n +1 , B n +1 ) ,i B ( S n , S n ) = i ( B n +1 , A n +1 ) + i ( B n +1 , B n +1 ) . where i B ( A n +1 , B n +1 ) > i ( B n +1 , A n +1 ) > G Bn +1 ∼ = G An ⊲⊳ i,j G Bn , where i = i B ( A n +1 , B n +1 ) and j = i ( B n +1 , A n +1 ) . Suppose that A n +1 = S n +1 , S n = C n and A n +1 = C n +1 . By Lemma 2.10 (3) we have i A ( S n , S n ) = i ( A n +1 , A n +1 ) + i ( A n +1 , B n +1 ) ,i B ( S n , S n ) = i B ( B n +1 , A n +1 ) + i B ( B n +1 , B n +1 ) . where i ( A n +1 , B n +1 ) > i B ( B n +1 , A n +1 ) > G Bn +1 ∼ = G An ⊲⊳ i,j G Bn , where i = i ( A n +1 , B n +1 ) , j = i B ( B n +1 , A n +1 ) . (cid:3) Induction algorithm.
In this section, we prove Theorem 1.2.
Definition 3.4.
We define n k to be the sequence of integers n k ≥ A n k or B n k is identical to S n k or C n k . By Lemma 2.6, we have n k = k for 0 ≤ k ≤ K .The constant K and the sequence ( n k ) indicate for which n ’s either G An or G Bn becomes strictlylarger than G An − , G Bn − . Theorem 3.5.
Let φ be an acyclic Sturmian coloring. ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 17 (1) For ≤ n < K , we have A n = C n = S n , thus for some i, j G An ∼ = G An − , G Bn ∼ = G An − ⊲⊳ i,j G Bn − . (2) For n = K , we have the following cases:(a) If A n = S n , B n = C n , then for some i < i ′ , j G An ∼ = G An − ⊲⊳ i,j G Bn − , G Bn ∼ = G An − ⊲⊳ i ′ ,j G Bn − . (b) If B n = S n , then C n = B n − and G An ∼ = G An − ⊲⊳ i,j G Bn − , G Bn ∼ = G Bn − . (3) For every n > K , G An ∼ = G An − , G Bn ∼ = G Bn − , if n = n k , G An ∼ = G An − ⊲⊳ G Bn − , G Bn ∼ = G Bn − or G An ∼ = G An − , G Bn ∼ = G An − ⊲⊳ G Bn − , if n = n k . Here, ⊲⊳ is either i -concatenation or ( i, j ) -concatenation.Proof. (1) For 0 ≤ n < K , by Lemma 2.6, A n = C n = S n = B n . Theorem 3.3 (1) and (3) impliesthat G An ∼ = G An − and G Bn ∼ = G An − ⊲⊳ i,j G Bn − for some i, j .(2) Let n = K . Then C n − = S n − which implies either (a) S n = A n or (b) S n = B n .(a) S n = A n : we have C n = A n = S n by the definition of K . By Lemma 2.3 (4) andLemma 2.5 (1) we have B n = C n . Thus, Theorem 3.3 (3) implies that G An ∼ = G An − ⊲⊳ i ′ ,j G Bn − and G Bn ∼ = G An − ⊲⊳ i,j G Bn − for some i ′ , i, j .(b) S n = B n = A n : by the choice of A and B , n = K ≥ S n is adjacent to distint balls A n and B n − . By Lemma 2.5 (1), we have C n = B n − = A n , thus Theorem 3.3 (1) and (3) impliesthat G An ∼ = G An − ⊲⊳ i,j G Bn − for some i, j and G Bn ∼ = G Bn − .(3) Assume that n > K . If S n − = C n − , then we have either A n = S n or B n = S n .Suppose that B n = A n = S n . If B n = C n , then by Lemma 2.3 (4) and Lemma 2.5 (1), A n isadjancent to A n and B n only. By Lemma 2.3 (1), A n is adjacent to A n − , which implies that A n − = S n − = C n − , which contradicts n ≥ K + 1. Thus B n = C n and Theorem 3.3 (1) and(3) implies that G An ∼ = G An − and G Bn ∼ = G An − ⊲⊳ i,j G Bn − for some i, j . For the case A n = B n = S n ,we apply the same argument.If S n − = C n − , then we have A n , B n = S n Suppose that B n = A n = C n or A n = B n = C n .Then by Theorem 3.3 (1), (2) for some i , G An ∼ = G An − , G Bn ∼ = G An − ⊲⊳ i G Bn − or G An ∼ = G An − ⊲⊳ i G Bn − , G Bn ∼ = G Bn − . In case of A n , B n = C n , by Theorem 3.3 (1), G An ∼ = G An − and G Bn ∼ = G Bn − . This is thecase of n = n k . (cid:3) For α ∈ { A, B } , define α = B if α = A and vice versa. Recall from Section 1.2.2 that α k satisfies | V G α k n k | > | V G α k n k | if k = K and α K = A . The following proposition shows how n k isrelated to α k and i k , j k . It will be used in Theorem 5.7. Proposition 3.6.
The special ball S n k is a vertex of degree 1 in G α k n k . Put m = | V G α k n k | . For i = 0 , , · · · , m − , the vertex of G α k n k of distance i from S n k , which we denote by [ i ] , is the central n k -ball of S n k + i . Therefore n k +1 = n k + m − , if there is a ( i ) -concatenation at n k +1 ,n k + m, if there is a ( i, j ) -concatenation at n k +1 . Moreover, none of the vertices of G α k n k − is the center of a special n -ball if n k + 1 ≤ n ≤ n k +1 for ( i ) -concatenation at n k and if n k ≤ n ≤ n k +1 for ( i, j ) -concatenation at n k .Proof. We may assume that α k = A . From the definition of concatenation, it is immediate that C n k − is a vertex of degree 1 in G Bn k . By Theorem 3.5, S n k = C n k . By Lemma 2.3 (4), the onlyadjacent vertex of [0] = S n k is [1], which is equal to C n k . By the canonical projection from G Bn k to G Bn k +1 , the vertices [0] , [1] are mapped to B n k +1 , S n k +1 respectively, thus again Lemma 2.3 (4)implies that [2] is mapped to C n k +1 . Inductively, for 1 ≤ i ≤ m − i ] is mapped to S n k + i by the canonical projection from G Bn k to G Bn k + i . By Theorem 3.5, we continue this procedureuntil n k +1 . (cid:3) Example 3.7.
Let s i , t i , i = 1 , , t i ≥ s i ≥ s i + 2 t i = d for each i = 1 , , s = s . Example 7 in [8] is the following: Y : · · · • c • d • c • c • d • c • d · · · X : · · · ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ · · · t t t t t t t t t t t t t t t t t t t t t t t t t t s s s s s s s s s s s s s G A : ◦ • t t s s G B : ◦ • t t s s G A : ◦ • t t s s G B : ◦ • ◦ t t t t s s G A : • ◦ • ◦ t t t t t t s s G B : ◦ • ◦ t t t t s s G A : • ◦ • ◦ t t t t t t s s G B : ◦ • ◦ t t t t s s G A : • ◦ • ◦ t t t t t t s s G B : ◦ • ◦ • ◦ • t t t t t t t t t t s s In this example, K = 0, and moreover, we have n k = f k +2 − f k is the Fibonacci sequencedefined by f k = f k − + f k − , f = 1 , f = 1, i.e., n = 0 , n = 1 , n = 2 , n = 4 and so on. Notethat α k = A if k is even and α k = B if k is odd. For the sequence of i k we have v = (2 t , t , t )and v k + i = ( t i ) for each i = − , , Inverse process for acyclic Sturmian coloring
In this section, we characterize acyclic Sturmian colorings completely by showing that theconverse of Theorem 3.5 also holds : we define admissible sequences which determine sequencesof edge-indexed graphs F Ak , F Bk constructed by ( i )-concatenations or ( i, j )-concatenations recur-sively, so that the appropriate direct limit F is a linear graph canonically colored by a, b . Weshow that these colorings are Sturmian.4.1. Admissible sequences.
Let us first define the admissible sequences of indices which wewill use in concatenations. Set D = { , , . . . , d − , d } , where d is the degree of the tree. Let { ( α k , i k ) } ∞ k =0 be a sequence of pairs of α k ∈ { A, B } and i k is one of i k ∈ D , ( i k , j k ) ∈ D or( i k , i ′ k , j k ) ∈ D . Put K = min { k ≥ α k = A } and i − = 0. To define admissibility, we needan extra notation for the edge-indices at the end vertices: define the sequence (ı Ak , ı Bk , ı Ck ) k ≥ K recursively by ı AK = i ′ K , ı BK = i K − , ı CK = j for i K = 0 , ı AK = i K , ı BK = i ′ K , ı CK = j for i K > k > K ı Ck = ı α k k − , ı α k k = i α k k − , ı α k k = ı Ck − . Definition 4.1 (Admissible sequence) . We call a sequence of indices i k an α k -admissible sequence if it satisfies the following conditions.(1) If 0 ≤ k < K , then i k = ( i k , j k ) ∈ D and satisfies1 ≤ i k < d, ≤ j k ≤ d − i k − . (2) If k = K , then i K = ( i K , i ′ K , j K ) ∈ D and satisfies1 ≤ i K < i ′ K ≤ d, ≤ j K ≤ d − i K − , or i K = ( i K , j K ) ∈ D and satisfies1 ≤ i K ≤ d, ≤ j K ≤ d − i K − . Furthermore i K ∈ D if K = 0.(3) If k > K , then i k = i k or ( i k , j k ) and satisfies1 ≤ i k < ı Ck − , if i k = i k , ≤ i k ≤ d − ı Ck − , ≤ j k ≤ d − ı Ck − , if i k = ( i k , j k ) . Definition 4.2 (Definition of edge-index graphs F Ak , F Bk ) . To an α k -admissible sequence i k , weassociate the edge-indexed graphs F Ak , F Bk defined as follows.(1) For k = − F A − , F B − are both the graph with one vertex and one loop of index d . ◦F A − d •F B − d (2) For k = 0 , · · · , K −
1, define F Ak = F Ak − , F Bk = F Ak − V,V ′ ⊲⊳ i k F Bk − . Here, V is the unique vertex of F Ak − . The vertex V ′ is the unique vertex of F Bk − comingfrom F Ak − for k ≥ F B − for k = 0. ◦ V F Ak d ◦ V ′ ◦ · · ·F Bk ◦ • i k − j i j i k j k d − j d − i k (3) For k = K , define F AK = F AK − V,V ′ ⊲⊳ i K ,j K F BK − , F BK = F AK − V,V ′ ⊲⊳ i ′ K ,j K F BK − , if i K = ( i K , i ′ K , j K ) , F AK = F AK − V,V ′ ⊲⊳ i K ,j K F BK − , F BK = F BK − , if i K = ( i K , j K ) , where V, V ′ are as in part (2). By the common end vertices of F AK , F BK , we mean theimages of the end vertex of F BK − different from V ′ in F AK , F BK , respectively.(4) For k > K , define F α k k = F Ak − V,V ′ ⊲⊳ i k F Bk − , F α k k = F α k k − where V, V ′ are the common end vertices of F Ak − , F Bk − . By the common end verticesof F Ak , F Bk , we mean the images of the non-common end vertex of F α k k − in F Ak , F Bk ,respectively. ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 21
The graphs F Ak , F Bk have canonical colorings ϕ Ak , ϕ Bk , resp.: the color of each vertex y is a or b , according to whether y comes from F A − or F B − , respectively. Remark 4.3.
The indices in the definition above can be interpreted as follows. For k > K , d − i Ck is the index of the loop at the common end vertex and d − i Ak , d − i Bk are the indices of theloops at the non-common end vertices of F Ak , F Bk , respectively.4.2. Ψ is a direct limit. For a given sequence α k , a sequence β k ∈ { A, B } is called α k -admissible if(4.1) β k = α k or β k − . For an α k -admissible β k , we have a natural inclusions ( F β k k , ϕ β k k ) ֒ → ( F β k +1 k +1 , ϕ β k +1 k +1 ) of coloredgraphs. Consider the colored graph ( Y, ϕ ) which is the direct limit determined by such inclusions:(
Y, ϕ ) = lim −→ ( F β k k , ϕ β k k ) . The coloring ϕ is the color of each vertex y is a or b , according to whether y comes from F A − or F B − , respectively. By construction, the universal cover of the edge-indexed graph Y is a d -regulartree T , thus ( Y, ϕ ) defines a coloring on the tree T , which is denoted by ( T, ϕ ). Similarly, theedge-indexed graph F Ak , F Bk colored by ϕ Ak , ϕ Bk have a d -regular colored tree as their universalcover, which we denote by ( T, ϕ Ak ) , ( T, ϕ Bk ), respectively.Let n k = | V F α k k | −
2. Fix a vertex t ∈ V F Ak ( t ∈ V F Bk ). For n ≤ n k , denote by B n ( t ) theclass of n -balls with the center a lift of t in ( T, ϕ Ak ) (( T, ϕ Bk ), respectively). Lemma 4.4. (1) For any vertices v, w in F Ak − ⊲⊳ F Bk − , we have [ B n k ( v )] = [ B n k ( w )] .(2) Let v be a vertex of F Ak − ⊲⊳ F Bk − corresponding to a vertex v ′ of F Ak − or F Bk − . Then [ B n k ( v )] = [ B n k ( v ′ )] .Proof. Denote by v ′ , · · · , v ′ n and w ′ , · · · , w ′ m the vertices of F Ak − and F Bk − in the linear orderso that v ′ , w ′ are the common end vertices with index i Ck − and v ′ n , w ′ m are the end vertices withindex ı Ak − , ı Bk − .If F · k is ( i, j )-concatenation, then there are ( n + m ) vertices in F · k denoted by v , · · · , v n , w , · · · , w m corresponding to vertices v ′ , · · · , v ′ n of F Ak − and w ′ , · · · , w ′ m of F Bk − , respectively. If F · k is ( i )-concatenation, then there are ( n + m −
1) vertices in F · k denoted by v , v , · · · , v n , w , · · · , w m where v is a vertex of F · k corresponding to two vertices v ′ of F Ak − and w ′ of F Bk − .We use induction on k on part (1) and part (2) at the same time. (a) For k ≤ K −
1, recall that α k = B , n k = k , n = 1, m = k + 1 and the figures of F Ak and F Bk are as follows: ◦ v ′ F Ak = F Ak − d ◦ v ◦ w · · ·F Bk ◦ w k • w k +1 i k − j i j i k j k d − j d − i k ◦ w ′ · · ·F Bk − ◦ w ′ k • w ′ k +1 i k − j i j d − j d − i k − Obviously, there is exactly one class of n k -balls in ( T, φ α k k ), namely the ball colored by a (colored white in the figure) only. There are n k + 2 = k + 2 distinct classes of n k -balls in( T, φ α k k ), namely the balls [ B n k ( v )] and [ B n k ( w i )] for i = 1 , · · · , k + 1. They are distinctclasses since the vertex in B n k ( w i ) closest to the center v i and colored by b (colored blackin the figure) is of distance n k + 2 − i for i = 1 , · · · , k + 1 from the center. For part (2),we have [ B n k ( v )] = [ B n k ( v ′ )] since all the vertices are colored by a in both balls. Let w ′ i be the vertices of F Bk − . Then for each 1 ≤ i ≤ n k + 1 we have [ B n k ( w i )] = [ B n k ( w ′ i )] sincethe number of neighboring n k − k = K and i K = ( i K , j K ), then F Ak = F Ak − ⊲⊳ i K ,j K F Bk − . An argument similar to (1)shows that [ B n K ( w i )] = [ B n K ( w ′ i )] for 1 ≤ i ≤ m + K + 1 and [ B n K ( v )] = [ B n K ( v ′ )].(c) If k = K and i K = ( i K , i ′ K , j K ), then F Ak = F Ak − ⊲⊳ i K ,j K F Bk − , F Bk = F Ak − ⊲⊳ i ′ K ,j K F Bk − .Let us denote by v A , w Ai and v B , w Bi the vertices of F AK and F BK respectively. Then[ B n K ( w Ai )] = [ B n K ( w Bi )] = [ B n K ( w ′ i )] for all 1 ≤ i ≤ m = K + 1 and [ B n K ( v A )] =[ B n K ( v B )] = [ B n K ( v ′ )].(d) Now let us prove the case k ≥ K + 1 assuming induction hypothesis for up to k − α k = A . • v ′ • v ′ · · ·F Ak − • v ′ n • v n · · · • v • v • w F Ak = F Ak − ⊲⊳ F Bk − · · · • w m • w ′ · · ·F Bk = F Bk − • w ′ m If α k − = A , then F Ak − = F Ak − ⊲⊳ F Bk − and n ≥ m . Moreover the vertices v ′ , . . . v ′ m corresponds to w ′ , . . . , w ′ m by the definition of the common end vertices. By the inductionargument, [ B n k − ( v ′ i )] = [ B n k − ( w ′ i )] for all 1 ≤ i ≤ m . Since [ B n k − ( v ′ i )] = [ B n k − ( v ′ j )] for i = j and [ B n k − ( w ′ i )] = [ B n k − ( w ′ j )] for i = j , to show part (1) it remains to show that[ B n k ( v ′ i )] = [ B n k ( w ′ i )] for all 1 ≤ i ≤ m for ( i, j )-concatenation and for all 2 ≤ i ≤ m for( i )-concatenation.Note that d ( v ′ i , v ′ m +1 ) ≤ m = n k − n k − for 1 ≤ i ≤ m in the case of ( i, j )-concatenation.Similarly, d ( v ′ i , v ′ m +1 ) ≤ m − n k − n k − , and for 2 ≤ i ≤ m in the case of ( i )-concatenation. It follows that B n k ( v ′ i ) contains B n k − ( v ′ m +1 ). In contrast, B n k ( w ′ i ) does ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 23 not contain B n k − ( v ′ m +1 ) since all n k − -balls in F Bk − are of the class [ B n k − ( w ′ i )] =[ B n k − ( v ′ i )], 1 ≤ i ≤ m . Therefore, [ B n k ( w ′ i )] = [ B n k ( v ′ i )] in ( T, φ Ak ), for i ≥ i, j )-concatenation and for i ≥ i )-concatenation. Thus part (1) for k follows.If n = m , then k = K + 1 and n = m = K + 2. In this case, i K = ( i K , i ′ K , j K ),i.e., F AK = F AK − ⊲⊳ i K ,j K F BK − , F BK = F AK − ⊲⊳ i ′ K ,j K F BK − . Note that [ B K ( v ′ i )] = [ B K ( w ′ i )]for all 1 ≤ i ≤ n = m . Since i K < i ′ K , the number [ B K ( v ′ n )] adjacent to v ′ n and w ′ n are different. Thus [ B K +1 ( w ′ n )] = [ B K +1 ( v ′ n )]. By the similar argument, we have[ B n K +1 ( w ′ i )] = [ B n K +1 ( v ′ i )] for each 1 ≤ i ≤ n .For part (2), we consider B n k +1 ( v i ) as a colored m = ( n k +1 − n k )-ball by coloring of B n k ( t ) on each vertex. Then v i and w i are colored by the same color [ B n k ( v i )] = [ B n k ( w i )].Note that for 1 < i ≤ m k −
1, since they both come from the same F k − , we have i ( w i , w j ) = i ( v i , v j )for j = i, i − , i + 1 . For i = 1, i ( w , w ) + i ( w , v ) = i ( v , w ) + i ( v , v ) . In other words, the edge-indexed graph colored by n k -balls in F Ak +1 with vertices ofdistance m k from the vertex w i is isomorphic to the edge-indexed graph colored by n k -balls in F Bk +1 with vertices of distance m k from the vertex w ′ i .Therefore [ B m ( v i )] = [ B m ( v ′ i )] and [ B m ( w i )] = [ B m ( w ′ i )]. The proof for the case α k = B is similar. (cid:3) Proposition 4.5.
Two admissible sequences β k , β ′ k are eventually equal if and only if they havethe same direct limit lim −→ ( F β k k , ϕ β k k ) = lim −→ ( F β ′ k k , ϕ β ′ k k ) .Proof. It is clear that if β k , β ′ k are eventual equal, the direct limits are the same. Suppose that β k = β ′ k for infinitely many k .Let t, t ′ be the vertices of lim −→ F β k k , lim −→ F β ′ k k . Then there exist k , k such that t ∈ V F β k k for k ≥ k and t ′ ∈ V F β ′ k k for k ≥ k . Choose k ≥ max( k , k ) as β k = β ′ k . Then t ∈ V F β k k , t ′ ∈ V F β ′ k k , thus t , t ′ are different vertices in V F α k +1 k +1 unless they are the common ends of F β k k , F β ′ k k . By Lemma 4.4 (1), we have [ B n k +1 ( t )] = [ B n k +1 ( t ′ )]. If t , t ′ are the common end vertices of F β k k , F β ′ k k , then they are the joining vertex of theconcatenation. Therefore, one of t , t ′ is not end vertex of F β k + ℓ k + ℓ or F β ′ k + ℓ k + ℓ for all ℓ ≥
1. Chooseanother k ′ > k such that β k ′ = β ′ k ′ and apply the same argument. (cid:3) Lemma 4.6.
For any vertex v in F Ak + ℓ or F Bk + ℓ with ℓ ≥ , there exists v ′ ∈ V F α k k such that [ B n k ( v )] = [ B n k ( v ′ )] .Proof. We may assume that v is a vertex of F Ak + ℓ . If α k + i = B for all 0 ≤ i ≤ ℓ , then F Ak + ℓ = F Ak − , thus v is also a vertex of F Ak − . By Lemma 4.4 (2) there exists v ′ ∈ V F α k k such that[ B n k ( v )] = [ B n k ( v ′ )].Let m be the largest integer with k ≤ m ≤ k + ℓ satisfying α m = A . Then F Ak + ℓ = F Am , thus v is also a vertex of F Am . By Lemma 4.4 (2), there exists v m − ∈ V F α m − m − such that [ B n m ( v )] =[ B n m ( v m − )]. Inductively, we can choose v i ∈ V F α i i such that [ B n i +1 ( v i +1 )] = [ B n i +1 ( v i )] for k ≤ i ≤ m −
1. Therefore, [ B n k +1 ( v k )] = [ B n k +1 ( v k +1 )] = · · · = [ B n k +1 ( v m − )] = [ B n k +1 ( v )]. (cid:3) Proposition 4.7.
Assume that β k = α k for infinitely many k . Then the direct limit ( Y, ϕ ) is aSturmian coloring.Proof. By Lemma 4.6 for k such that β k = α k , we have B n k ( ϕ ) ⊂ B n k ( ϕ α k k ). On the other hand, B n k ( ϕ ) ⊃ B n k ( ϕ α k k ) is implied by Lemma 4.4 (2). Using Lemma 4.4 (1), we deduce that | B n k ( ϕ ) | = | B n k ( ϕ α k k ) | = | V F α k k | = n k + 2 . Since there are infinitely many such k ’s, ( Y, φ ) is Sturmian. (cid:3)
We remark that if α k is not stabilized i.e. there are infinitely many k such that α k +1 = α k ,then any admissible sequence β k satisfies Proposition 4.7, thus the direct limit is a Sturmiancoloring.4.3. Sturmian coloring and direct limits.
In this section we show that Ψ ◦ Φ = Id. Let usfirst define Φ. For a given Sturmian coloring (
T, φ ), we need to define ( α k , i k , [ β k ]). The sequence α k and G An , G Bn are defined in Section 3. They determine i k as follows. Definition 4.8 (Definition of Φ) . Let (
X, φ ) be an acyclic Sturmian coloring.
ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 25 (1) Define i k by i k = ( i, j ) , for case (1) , ( i, i ′ , j ) , for case (2)(a) , ( i, j ) , for case (2)(b) , ( i ) or ( i, j ) , for case (3) , where the cases are as in Theorem 3.5.(2) For a fixed vertex t ∈ T , define β k = β k ( t ) as follows:(i) if [ B n k ( t )] ∈ V G α k n k then set β k − = α k .(ii) if [ B n k ( t )] / ∈ V G α k n k then set β k − = α k .One can easily check that i k is α k -admissible. Lemma 4.9.
For any k ≥ , we have [ B n k ( t )] ∈ V G β k ( t ) n k . Proof.
By Theorem 3.5 (3), G α k +1 n k +1 ∼ = G α k +1 n k , where the isomorphism is given by the restrictionand extension, i.e. [ B n k +1 ( t )] ∈ V G α k +1 n k +1 if and only if [ B n k ( t )] ∈ V G α k +1 n k . If β k = α k +1 , then by Definition 4.8 (2) (i), [ B n k +1 ( t )] ∈ V G α k +1 n k +1 , thus [ B n k ( t )] ∈ V G β k n k . If β k = α k +1 , then by Definition 4.8 (2) (ii), [ B n k +1 ( t )] / ∈ V G α k +1 n k +1 , thus [ B n k ( t )] / ∈ V G α k +1 n k , which implies that [ B n k ( t )] ∈ V G α k +1 n k = V G β k n k . (cid:3) Lemma 4.10.
The sequence β k defined above is α k -admissible, i.e. it satisfies β k = α k or β k − .Proof. Suppose that β k = β k − = α k . By Definition 4.8 (2) (ii), we have[ B n k ( t )] / ∈ V G α k n k . However, by Lemma 4.9, we have [ B n k ( t )] ∈ V G β k n k = V G α k n k , which is a contradiction. (cid:3) Remark 4.11. If φ is unbounded, then β k = α k for arbitrary large k . If φ is bounded, thenthere exists k such that β k = α k for all k ≥ k . Lemma 4.12.
Fix t ∈ V T and let β k = β k ( t ) . Let t ′ be the vertex adjacent to t . Then thereexists k such that β k ( t ′ ) = β k ( t ) for all k ≥ k .Proof. It is enough to show that if there is some k > K such that[ B n k ( t )] ∈ V G α k n k and [ B n k ( t ′ )] / ∈ V G α k n k , then for β l = β l ( t ′ ) for all l > k . Let k be such an integer. Without loss of generality, let us assume that α k = A . Then bythe definition of G An , G Bn , we have [ B n k ( t )] = S n k and [ B n k ( t ′ )] = A n k . Since A n k +1 is adjacent to A n k and not B n k , we have [ B n k +1 ( t )] = A n k +1 . Let t ′′ ∈ V T be an adjacent vertex of t such that[ B n k ( t ′′ )] = B n k = C n k . From | V G Bn k | >
1, we check B n k = S n k . Therefore t has two adjacentvertices t ′ , t ′′ with distinct n k -balls.It follows that for any n ≥ n k + 1, [ B n ( t )] always adjacent to two different n -balls in both G An and G Bn , whenever [ B n ( t )] is a vertex of them. We conclude that that [ B n ( t )] is not an end vertexof G An nor G Bn for n ≥ n k + 1. In other words, β l ( t ) = β l ( t ′ ) for all l > k . (cid:3) Repeating the above lemma for vertices between any pair of vertices t, t ′ , the sequence β k ( t ′ )is eventually equal to β k ( t ), thus Φ is well-defined. Remark 4.13.
The sequence ( α k ) has a role corresponding to the slope for the irrational rotationassociated to a Sturmian word (or the ratio of alphabets a and b appearing in the word). Thefreedom coming from the intercept (starting point of the irrational rotation) of Sturmian wordsare replaced by a sequence β k satisfying Lemma 4.10.Define ( Y, ϕ ) to be the direct limit lim −→ F β k k in Section 4.2 with F Ak = G An k , F Bk = G Bn k . Theorem 4.14.
We have Ψ ◦ Φ = 1 , i.e. for given Sturmian coloring, let ( X, φ ) be the quotientgraph defined in the introduction. Then the direct limit ( Y, ϕ ) is equal to ( X, φ ) .Proof. By Lemmas 4.9 and 4.12, for any vertex t ∈ V X , there exists k ( t ) such that B n k ( t ) ∈ V G β k n k for k ≥ k ( t ). Moreover, for k > k ( t ), the n k -ball centered at t is not an end vertex of G β k n k , thusthe injection preserves the adjacencent n k -balls. Thus there is a injection from V X to the set ofvertices of lim −→ G β k n k ( t ) preserving the edge indices. This map is clearly surjective, since any vertexof lim −→ G β k n k ( t ) determines a sequence of equivalence classes of n -balls, which determines a vertexof X . (cid:3) Sturmian colorings of bounded type
Cyclic Sturmian colorings.
In this subsection, we investigate cyclic Sturmian colorings.The main result in this section is Proposition 5.4.
Lemma 5.1.
Suppose that G n has a cycle. Then(1) the special ball S n is in the cycle.(2) the special ball S n is adjacent to A n , B n , C n only, apart from S n itself. ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 27
Proof. (1) Suppose that G n has a vertex v which is not in the cycle. Since G n is connected, thereexists a vertex w in the cycle connected to v . Since S n is the unique vertex of G n with degree 3by Lemma 2.4 (2), it follows that w = S n and S n is in the cycle.(2) Suppose that there exists D = S n distinct from A n , B n , C n and adjacent to S n . ByLemma 2.4 (1), S n = C n and by Lemma 2.5 (1), D is of degree 1. Since S n is adjacent to C n byLemma 2.3 (4), the classes of distance 1 from S n in A n +1 , B n +1 are { C n , S n , D } (with possiblydistinct indices). Using Lemma 2.3 (4) again, C n and D belong to the cycle, contradicting that D is of degree 1. (cid:3) Lemma 5.2. If G n has a cycle not containing C n , then G n + ℓ has a cycle containing C n + ℓ forsome ℓ ≥ .Proof. By Lemma 5.1 (2), S n is adjacent to A n , B n , C n , S n only. Since S n is the unique vertex ofdegree at least 3 and it is in the cycle, and since C n is adjacent to S n , the graph G n is the unionof a cycle containing S n , A n , B n and a line segment containing S n , C n . •G n : E ℓ · · · . . . • E • C n • S n • A n • B n • D • D m •• ••G n +1 : E ℓ · · · . . . • C n +1 • S n +1 • A n +1 • B n +1 • A n • B n • D • D m •• • Denote the cycle by [ S n A n D D . . . D m B n S n ] and the line segment by [ S n C n E · · · E ℓ ]. Since( n + 1)-ball extension of S n which is adjacent to A n (resp. B n ) is A n +1 (resp. B n +1 ), it followsthat [ S n +1 A n +1 A n D . . . D m B n S n +1 ] is a cycle in G n +1 .If there are no vertices E i , C n is the unique vertex not in the cycle in G n , thus all the verticesin G n +1 belong to the cycle.If ℓ ≥
2, then E is adjacent to every C n = S n +1 , which implies that E = C n +1 . Only thepath [ E E . . . E ℓ ] in G n +1 is not in the cycle. Repeating this procedure j times, it follows thatonly the path [ E j +1 j E j +2 j · · · E ℓj ] is not in the cycle. (Here we denote the extension of an n -ball E to the ( n + j )-ball by E j .) Thus G n + ℓ contains all the vertices, in particular C n + ℓ . (cid:3) Lemma 5.3. If G n has a cycle containing C n = S n , then G n +1 has a cycle containing C n +1 = S n +1 .Proof. Suppose that G n has a cycle which contains C n = S n . By definition of C n and S n , itfollows that either A n +1 = S n +1 or B n +1 = S n +1 . As C n = S n , by Lemma 2.4 (1), the cycle in G n is of the form [ S n A n C . . . C m B n S n ]. Since we have by Lemma 2.3 (3), either B n +1 is adjacentto S n +1 = A n +1 or A n +1 is adjacent to S n +1 = B n +1 . Therefore, we have a cycle either [ S n +1 (= A n +1 ) A n C . . . C m B n B n +1 S n +1 (= A n +1 )] or [ S n +1 (= B n +1 ) A n +1 A n C . . . C m B n S n +1 (= B n +1 )].Since S n +1 is adjacent to A n +1 , B n +1 or C n +1 , either A n = C n +1 or B n = C n +1 . Therefore G n +1 has a cycle which contains C n +1 = S n +1 . (cid:3) Proposition 5.4.
Suppose that for some n , G n has a cycle. Let m be the smallest integer suchthat S m = C m are in the cycle of G m .(1) The coloring φ is of bounded type.(2) Then the quotient graph X φ of ( T, φ ) is isomorphic to lim −→ G Bn (resp. X = lim −→ G An ). Allthe graphs G An (or G Bn ) are isomorphic for n ≥ m .(3) By part (1), the quotient graph X φ is an infinite ray by Theorem 1.1. As in Theorem 1.1,let x i be the ( i − -th vertex from the left. The ball B m + i ( x i ) is special for all i ≥ .Proof. By Lemmas 5.2 and 5.3, such m exists.Let n ≥ m . Denote the cycle in G n by [ S n C n C . . . C k S n ], where C n , C , . . . , C k are distinct.It follows from Lemma 5.1 that C k is A n or B n , say C k = A n . Thus the cycle is the underlyinggraph of G An . Choose an n -ball colored by S n and adjacent to A n . Its extension to ( n + 1)-ballis A n +1 which is adjacent to S n +1 = C n and A n = C k . Therefore, [ A n +1 S n +1 C . . . A n A n +1 ] isthe cycle in G n +1 and C = C n +1 . By the the same argument for G An , we deduce that G An +1 is theedge indexed graph with underlying graph the cycle. By Lemma 3.1, we have G An +1 ∼ = G An .By repeating this procedure, it follows that G Am + i ∼ = G Am and B m + i is not a vertex of G Am + i for any i ≥
1. Let x i be a vertex of T such that B m + i ( x i ) = B m + i for i ≥
1. Then B m + j ( x i ) ∈ V G Bm + j − V G Am + j for any j ≥ i which implies that B m + j ( x i ) is not special for any j ≥ i . Sincethe type set at x i is finite, φ is of bounded type. It follows that each vertex x in T satisfies B m + i ( x ) = B m + i for some i ≥
1. Moreover, the number of neighboring vertices is determined bythe edge index of G Bn and the quotient graph X = lim −→ G Bn .We now prove that G Am = G Am − . If G m − does not have a cycle, then G Am − is not isomorphicto G Am which is cyclic. If G m − has a cycle which does not contain C m − , then by the proof ofLemma 5.2, the cycle in G Am is smaller than the cycle of G m − . If G m − has a cycle containing S m − = C m − , then by the proof of Lemma 5.3, the cycle in G Am is smaller than the cycle of G m − .As for part (3), it is easy to see that the integer m in Theorem 1.1 and the m here coincide. (cid:3) ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 29
Example 5.5.
An example of a cyclic coloring: X : • ◦ ◦ • ◦ ◦ • ◦ ◦ · · · G A : ◦ • G B : ◦ • G A : ◦◦ •
21 1 32 0 G B : ◦◦ •
21 1 22 1 G A : • ◦ •◦ G B : ◦ •◦
21 12 12 G A : • ◦ ◦ ◦• G B : ◦ ◦•
21 12 12 G A : • ◦ ◦ • ◦◦ G B : • ◦◦
21 12 12
Example 5.6.
An example with an n -ball C adjacent to S n and which is not one of S n , A n , B n , C n . Note that this case can happen only for bounded type Sturmian colorings. X : ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • · · · G A : ◦ G B : ◦ • G A : ◦ ◦ • G B : ◦ ◦ • G A : ◦ ◦ •◦
12 21 1 11 2 G B : ◦ ◦ •◦
13 20 1 11 2 G A : ◦ ◦ •◦
12 21 1 11 2 G B : ◦ ◦ •◦◦ n = 1. Then any special 1-ball is adjacent to the 1-ball with center b = • which is not thecentral 1-ball of special j - balls, j = 0 , , • x • x • x • x • x • x · · · Figure 3.
Quotient graph of bounded type Sturmian coloring5.2.
Sturmian colorings of bounded type.
In this subsection, we show that Proposition 5.4 isa more general phenomenon: the second statement of part (2) is a characterization of a Sturmiancoloring of a bounded type. Let φ be a Sturmian coloring of bounded type, i.e. the type setof each vertex is finite. By Theorem 1.1, we know that the quotient graph X is an infinite ray.Denote the vertices of X from the left by ( x i ) i ≥ as in Figure 3. Theorem 5.7.
Let φ be a Sturmian coloring.(1) The coloring φ is of bounded type if and only if either all the G An ’s or all the G Bn ’s areisomorphic for sufficiently large n .(2) Moreover, if G An (resp. G Bn ) are all isomorphic for sufficiently large n , then the quotientgraph X φ = lim −→ G Bn (resp. X = lim −→ G An ).(3) Let m be the smallest integer such that G An +1 ∼ = G An for all n ≥ m . Then B m + i ( x i ) isspecial for all i ≥ . In particular, B m ( x ) is the m -special ball.Proof. We showed all the statements in Proposition 5.4 for cyclic Sturmian colorings.Now we prove part (1) for acyclic Sturmian colorings. If φ is of bounded type, then thereexists a vertex t ∈ V T and some integer m such that t is not the center of the special n -ball forall n > m . For any n k > m + 1,(5.1) B n k ( t ) ∈ V G α k n k − V G α k n k by the first statement of Lemma 3.6. We claim that α k + l = α k for all l >
0. Indeed, otherwise,for the minimal l such that α k + l = α k + l − , B n k + l − ( t ) is a vertex in G α k + l n k + l − thus B n k + l − ( t ) is avertex of G α k + l − n k + l − , which is a contradiction to (5.1).Conversely, if G Bn are all isomorphic for sufficiently large n , then by Proposition 3.6, for any B n ( t ) ∈ V G An − V G Bn , B m ( t ) is not special for all m ≥ n . Thus φ is of bounded type.For part (2), suppose that there exists m such that G Am + ℓ is isomorphic for all ℓ ≥
0. Choose t ′ , t ′′ ∈ V X of distance larger than | V G Am | . By Lemma 4.12, for n k > m , the n k -balls aroundvertices between t ′ , t ′′ are in G β k n k , thus β k = B for n k > m .Part (3) for acyclic colorings follows from Proposition 3.6. (cid:3) ONTINUED FRACTION ALGORITHM FOR STURMIAN COLORINGS OF TREES 31
Acknowledgement
We thank the anonymous referee for valuable comments. We would also like to thank thehospitality of KIAS, of which both authors are associate members and where part of this workwas done. The first author was supported by the National Research Foundation of Korea (NRF-2015R1A2A2A01007090) and the second author is supported by Samsung Science and TechnologyFoundation under Project No. SSTF-BA1601-03.
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Department of Mathematics Education, Dongguk University–Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul, 04620
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