Lipschitz classification of Bedford-McMullen carpets with uniform horizontal fibers
aa r X i v : . [ m a t h . M G ] J u l LIPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETSWITH UNIFORM HORIZONTAL FIBERS
YA-MIN YANG AND YUAN ZHANG ∗ Abstract.
Let M t,v,r ( n, m ), 2 ≤ m < n , be the collection of self-affine carpets withexpanding matrix diag( n, m ) which are totally disconnected, possessing vacant rows andwith uniform horizontal fibers. In this paper, we introduce a notion of structure treeof a metric space, and thanks to this new notion, we completely characterize when twocarpets in M t,v,r ( n, m ) are Lipschitz equivalent. Introduction
Let 2 ≤ m < n be two integers and denote by diag( n, m ) the diagonal matrix withentries n and m . Let D ⊂ { , , . . . , n − } × { , , . . . , m − } and we call it the digit set .For d ∈ D , define S d ( z ) = diag( n − , m − )( z + d ) . Then { S d } d ∈D is an iterated function system (IFS), and there exists a unique non-emptycompact set E = K ( n, m, D ) such that E = S d ∈D S d ( E ); we call E a Bedford-McMullen carpet, or a self-affine carpet .Two metric spaces (
X, d X ) and ( Y, d Y ) are said to be Lipschitz equivalent, denoted by( X, d X ) ∼ ( Y, d Y ), if there exists a map f : X → Y which is bi-Lipschitz, that is, there isa constant C > C − d X ( x, y ) ≤ d Y ( f ( x ) , f ( y )) ≤ Cd X ( x, y ) , for all x, y ∈ X. There are many works on Lipschitz equivalence of self-similar sets, see [3, 4, 14, 18, 19, 21,22] . For example, Rao, Ruan, and Xi [19] and Xi and Xiong [22] showed that for fractalcubes which are totally disconnected and have the same expanding matrix, the Hausdorffdimension completely determines the Lipschitz class. However, there are few works on theclassification of self-affine carpets ([11, 17, 20]).
Date : July 23, 2020.The work is supported by NSFS Nos. 11971195 and 11601172.
Key words and phrases:
Self-affine carpet, uniform horizontal fibers, Lipschitz equivalence.* The correspondence author. ∗ The study of the Lipschitz classification of totally disconnected self-affine carpets ismuch more difficult than that about the self-similar sets. In what follows, we use M t ( n, m )to denote the collection of totally disconnected self-affine carpets with expanding matrixdiag( n, m ).Miao, Xi and Xiong[17] and Rao, Yang and Zhang [20] developed several Lipschitz in-variants of self-affine carpets which are very useful. First, let us introduce some notations.Let A be the cardinality of A . Let E = K ( n, m, D ) be a self-affine carpet. We define(1.1) a j = { i ; ( i, j ) ∈ D} , ≤ j ≤ m − , and call a = ( a j ) m − j =0 the distribution sequence of D , or of E . For a digit set D , we say the j -th row is vacant if a j = 0. Miao et al. [17] showed that if E, F ∈ M t ( n, m ) are Lipschitzequivalent, then either both of them possess vacant rows or neither of them do.Denote N = D . To remove the trivial case, we will always assume that N > µ E supported on E satisfying(1.2) µ E ( · ) = 1 N X d ∈D µ E ◦ S − d ( · ) , and it is called the uniform Bernoulli measure of E . Rao et al. [20] found several Lipschitzinvariants related to the uniform Bernoulli measure of self-affine carpets. They prove thatif E, F ∈ M t ( n, m ) and f : E → F is a bi-Lipschitz map, then µ E and µ F ◦ f areequivalent; consequently, µ E and µ F have the same multifractal spectrum, and µ E isdoubling if and only if µ F is doubling. Remark 1.1. (i) The mulitfractal spectrum of self-affine carpets have been completelycharacterized, see [10, 1, 9].(ii) A measure ν on a metric space X is said to be doubling if there is a constant C ≥ < ν ( B ( x, r )) ≤ Cν ( B ( x, r )) < ∞ for all balls B ( x, r ) ⊂ X with center x and radius r >
0. According to Li, Wei and Wen [12], µ E is doubling if and only if either a a m − = 0, or a j a j +1 = 0 for all j = 0 , . . . , m −
2, or a = a m − .We denote σ = log m/ log n . We shall use M t,v,d ( n, m ) to denote the class of self-affinecarpets in M t ( n, m ) which possess vacant rows and whose uniform Bernoulli measures aredoubling. Using a kind of measure preserving property, [20] proved that Proposition 1.2. ( [20] ) Let σ ∈ Q c , and let E, F ∈ M t,v,d ( n, m ) . If E ∼ F , then thedistribution sequence of E is a permutation of that of F . IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 3 M t,v,r M t, ¯ v,r M t,v,d, ¯ r M t, ¯ v,d, ¯ r M t,v, ¯ d M t, ¯ v, ¯ d Figure 1.
The collection M t is divided into six subclasses.We say E = K ( n, m, D ) has uniform horizontal fibers if all non-zero a j ’s in the distri-bution sequence take only one value. It is shown [2, 16] that dim H E = dim B E if andonly if E has uniform horizontal fibers. (In terms of Falconer [5], a set is called regular ifits Hausdorff dimension and box dimension coincide.) It is seen that if E is a self-affinecarpet with uniform horizontal fibers, then the associated uniform Bernoulli measure mustbe doubling (see Remark 1.1(ii)). We shall use M t,v,r to denote the class of the self-affinecarpets in M t,v,d with uniform horizontal fibers.Hence, the vacant row property, the doubling property and the uniform horizontal fibersproperty divide the totally disconnected self-affine carpets into six subclasses, and if twoself-affine carpets are Lipschitz equivalent, then they must belong to the same subclass.(We shall use ¯ v , ¯ d and ¯ r to denote the negation of the corresponding property.) See Figure1. The main goal of the present paper is to characterize the Lipschitz classification of self-affine carpets in M t,v,r , the green part in Figure 1. For this purpose, we use symbolicspaces. For two sequences x , x ′ ∈ Z ∞ , we use x ∧ x ′ to denote their maximal commonprefix. Let ξ ∈ (0 , Z ∞ by(1.3) d ξ ( x , x ′ ) = ξ | x ∧ x ′ | , where | W | denotes the length of a word W .Let us denote α = 1 /σ − ⌊ /σ ⌋ where ⌊ x ⌋ is the greatest integer no larger than x . Let s = { j ; a j > } be the number of non-vacant rows. For k ≥
1, define(1.4) n k = N s ⌊ /σ ⌋ + δ k − where δ k = ⌊ kα ⌋ − ⌊ ( k − α ⌋ , and setΩ = ∞ Y k =1 { , , , . . . , n k − } . YA-MIN YANG AND YUAN ZHANG ∗ Remark 1.3.
Notice that if σ ∈ Q c , then ( δ k ) k ≥ is a sturmian sequence with rotation α (for sturmian sequence, we refer to Chapter 2 of Lothaire [13]); if σ ∈ Q , then the sequence( δ k ) k ≥ is periodic. Theorem 1.1. If E = K ( n, m, D ) ∈ M t,v,r , then E ∼ (Ω , d /n ) where d /n is definedas (1.3) . Especially, if σ = p/q ∈ Q , then E ∼ ( { , , . . . , N ∗ − } ∞ , d /n p ) where N ∗ = N p s q − p . Theorem 1.1 says that if σ ∈ Q , then E ∈ M t,v,r is Lipschitz equivalent to a self-similarset, and if σ ∈ Q c , then E is Lipschitz equivalent to a homogeneous Moran set. (Forhomogeneous Moran set, we refer to Mauldin and Williams [15], and Feng, Wen and Wu[6].)To prove Theorem 1.1, we introduce a notion of the structure tree of a metric space(Section 3). The nested structure of a set is an important tool to study measures anddimensions. However, to study the Lipschitz equivalence, we need to handle the nestedstructures of two sets; to overcome this difficulty, we regard a nested structure as a tree.As a consequence of Proposition 1.2 (necessity) and Theorem 1.1 (sufficiency), we have Theorem 1.2.
Let
E, F ∈ M t,v,r ( n, m ) . Then ( i ) If σ ∈ Q , then E ∼ F if and only if dim H E = dim H F ; ( ii ) If σ ∈ Q c , then E ∼ F if and only if the distribution sequence of E is a permutationof that of F . Our third result concerns another symbolization of self-affine carpets. We equip D ∞ with the metric λ defined as(1.5) λ (( x , y ) , ( x ′ , y ′ )) = max { d /n ( x , x ′ ) , d /m ( y , y ′ ) } . Theorem 1.3. If E = K ( n, m, D ) ∈ M t,v,r , then E ∼ ( D ∞ , λ ) . It is interesting to know when K ( n, m, D ) ∼ ( D ∞ , λ ). Yang and Zhang[23] found severalother classes of self-affine carpets with this property.The paper is organized as follows. In Section 2, we recall some known results aboutapproximate squares of self-affine carpets. We define structure tree in Section 3 anddevelop several techniques to handle structure trees in Section 4. In Section 5, we proveTheorems 1.1 and 1.2. Theorem 1.3 is proved in Section 6. IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 5 Approximate squares of self-affine carpets
In this section, we study the structure of totally disconnected self-affine carpet withvacant rows. Let E = K ( n, m, D ) be a self-affine carpet. Denote(2.1) E = { j ; a j > } . Throughout the paper, we will use the notation(2.2) ℓ ( k ) = ⌊ k/σ ⌋ . For i = d . . . d k ∈ D k , denote S i = S d ◦ · · · ◦ S d k , and we call S i ([0 , ) a basic rectangleof rank k . Clearly, S i ([0 , ) = S i ((0 , (cid:20) , n k (cid:21) × (cid:20) , m k (cid:21) . Set e E k = S i ∈D k S i ([0 , ), then e E k decrease to E .Let q ≥ x . . . x k ∈ { , , . . . , q − } k , we will use the notation0 .x . . . x k | q = k X j =1 x j q − j . Following McMullen [16], we divide a basic rectangle into approximate squares.
Definition 2.1. ( Approximate squares ) Let x = x . . . x k ∈ { , , . . . , n − } k and y = y . . . y ℓ ( k ) ∈ { , , . . . , m − } ℓ ( k ) . We call(2.3) Q ( x , y ) = (0 . x | n , . y | m ) + (cid:20) , n k (cid:21) × (cid:20) , m ℓ ( k ) (cid:21) an approximate square of rank k of E , if ( x j , y j ) ∈ D for j ≤ k and y j ∈ E for j > k .An approximate square Q ′ is called an offspring of Q if Q ′ ⊂ Q , and it is called a directoffspring if the rank of Q ′ equals the rank of Q plus 1. For j ∈ E , we denote D j = { i ; ( i, j ) ∈ D} . The following lemma is obvious, see [20].
Lemma 2.2. ( [20] ) Let E = K ( n, m, D ) , and Q ( x , y ) be an approximate square of E ofrank k .(i) If ℓ ( k ) > k , then the direct offsprings of Q ( x , y ) are n Q ( x ∗ u, y ∗ z ); u ∈ D y k +1 and z ∈ E ℓ ( k +1) − ℓ ( k ) o and Q ( x , y ) has a y k +1 s ℓ ( k +1) − ℓ ( k ) direct offsprings; YA-MIN YANG AND YUAN ZHANG ∗ (ii) If ℓ ( k ) = k , then the direct offsprings of Q ( x , y ) are n Q ( x ∗ u, y ∗ v ∗ z ); ( u, v ) ∈ D and z ∈ E ℓ ( k +1) − k − o , and Q ( x , y ) has N s ℓ ( k +1) − ( k +1) direct offsprings. Remark 2.3.
As a consequence of the above lemma, it is easy to see that(i) The set of the direct offsprings of an approximate square of E = K ( n, m, D ) of rank k can be written as(2.4) A × B + (cid:20) , n k +1 (cid:21) × (cid:20) , m ℓ ( k +1) (cid:21) , where A ⊂ Z /n k +1 and B ⊂ Z /m ℓ ( k +1) .(ii) If E = K ( n, m, D ) has uniform horizontal fibers, then the number of direct offspringsof an approximate square of E of rank k is always N s ℓ ( k +1) − ℓ ( k ) − no matter ℓ ( k ) > k ornot. Lemma 2.4.
Let E be a self-affine carpet with uniform horizontal fibers. Let W be anapproximate square of rank k , then µ E ( W ) = ( N k s ℓ ( k ) − k ) − .Proof. Let W = Q ( x , y ) be an approximate square of rank k . Then the number of basicrectangles of rank ℓ ( k ) contained in Q ( x , y ) is ( N/s ) ℓ ( k ) − k , and hence its measure in µ E is ( N/s ) ℓ ( k ) − k /N ℓ ( k ) , the lemma is proved. (cid:3) Let E k be the union of all approximate squares of rank k . It is seen that ( E k ) k ≥ is adecrease sequence and E = T ∞ k =1 E k . Let U be a connected component of E k . Hereafter,we will call U a component of E k for simplicity. An approximate square of rank k containedin U will be called a member of U . Denote by k ( U ) the number of members of U . It isshown that k ( U ) has an upper bound which is independent of k . Lemma 2.5. ( [20] ) Let E = K ( n, m, D ) be totally disconnected and possess vacant rows.Then there exists L > such that for every k ≥ and every component U of E k , it holdsthat k ( U ) ≤ L . We shall denote by C E,k the collection of components of E k , and set C E = S k ≥ C E,k ,where we set E = [0 , by convention. Remark 2.6.
For self-affine carpets possessing vacant rows, there is a simple criterion fortotally disconnectedness ([20]): Let E = K ( n, m, D ) and D possess vacant rows. Then E is totally disconnected if and only if a j < n for all 0 ≤ j ≤ m − IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 7 Structure tree of a metric space
In this section, we introduce a structure tree to describe the nested structure of a metricspace.3.1.
Tree and boundary.
Let T be a tree with a root and we denote the root by φ . Let v, v ′ be two vertices of T . The level of v is the distance from the root φ to v , and wedenote it by | v | . We say v ′ is a direct offspring of v , if there is an edge from v to v ′ , and | v ′ | = | v | + 1, and meanwhile, we say v is the parent of v ′ . We say v ′ is an offspring of v ifthere is a path from v to v ′ .In this paper, we always assume that any vertex of T has at least one direct offspring,and the number of direct offsprings of a vertex is finite. The boundary of T , denoted by ∂T , is defined to be the collection of infinite path emanating from the root; we shall denotesuch a path by v = ( v k ) ∞ k =0 , where v k is a vertex of level k in the path. In what follows,an infinite path always means that a path emanating from the root. Given 0 < ξ <
1, weequip ∂T with the metric d ξ ( u , v ) = ξ | u ∧ v | . Structure trees of self-affine carpets.
The following is an alternative descriptionof a nested structure of a metric space.
Definition 3.1.
Let X be a compact metric space and let T be a rooted tree. We call T a structure tree of X if( i ) the root is φ = X and every vertex of T is a closed subset of X ;( ii ) the vertices of the same level are disjoint;( iii ) if { v , . . . , v p } is the set of direct offsprings of v , then v = S pj =1 v j . Example 3.1.
Let T be a rooted tree. Let ∂T be the boundary of T equipped with ametric d ξ . If we identify a vertex v as the subset of ∂T consisting of the infinite pathspassing v , then T is a structure tree of ∂T . Example 3.2.
Here we give two structure trees of E = K ( n, m, D ). The first structure tree.
Let T be a tree such that the vertices of level k are W ∩ E ,where W runs over the approximate squares in E k . We set an edge from vertex u to v if v ⊂ u and | v | = | u | + 1. Clearly T is a structure tree of E . The second structure tree.
Let T be a tree such that the vertices of level k are U ∩ E , where U runs over the components of E k . Then the vertex set of T is C E ∩ E = S k ≥ ( C E,k ∩ E ). We define an edge from u to v if v ⊂ u and | v | = | u | + 1. We shall call T ∗ the structure tree of E induced by C E . For simplicity, we sometimes say that a component U is a vertex of T instead of U ∩ E .3.3. Regularity.
To study the Lipschitz classification, we wish a structure tree has niceseparation property. This motivates us to give the following definition. Let (
X, d ) be ametric space. For two set
A, B ⊂ X , we define d ( A, B ) = inf { d ( a, b ); a ∈ A, b ∈ B } . Definition 3.2.
Let T be a structure tree of the compact metric space ( X, d ). If thereexist a real number 0 < ξ < α > k ≥ u, v of level k , diam u ≤ α ξ k and d ( u, v ) ≥ α − ξ k , then we say T is ξ -regular.Similar idea has been appeared in Jiang, Wang and Xi [8], where a ξ -regular structuretree of X is called a configuration of X there. The following result is essentially containedin [8]. Theorem 3.1.
Let ( X, d ) be a compact metric space and let T be a structure tree of X .If T is ξ -regular, then ( X, d ) ∼ ( ∂T, d ξ ) .Proof. For any x ∈ X , there is a unique infinite path ( v k ) k ≥ such that { x } = T k ≥ v k .Denote by f the map from ∂T to X defined by f (( v k ) k ≥ ) = x . We claim that f isbi-Lipschitz. Pick ( u k ) k ≥ , ( v k ) k ≥ ∈ ∂T , and denote x = f (( u k ) k ≥ ) , y = f (( v k ) k ≥ ). Let q be the length of common prefix of ( u k ) k ≥ and ( v k ) k ≥ , then d ξ (( u k ) k ≥ , ( v k ) k ≥ ) = ξ q and x, y ∈ u q . Since T is ξ -regular, we have α − ξ q +1 ≤ d ξ ( u q +1 , v q +1 ) ≤ d ξ ( x, y ) ≤ diam( u q ) ≤ α ξ q , which implies that f is bi-Lipschitz. (cid:3) p -Subtree and bundle map In this section, we develop several techniques on structure tree.4.1. p -subtree. Now we consider a special ‘subtree’ of a rooted tree T . Let p ≥ T ∗ be the tree whose vertices of level k consist of the vertices of T of level pk , k ≥
0. For two vertices u, v ∈ T ∗ , v is a direct offspring of u if v is a p -step offspringof u in T (that is, v is an offspring of u and | v | = | u | + p ). We call T ∗ the p -subtree of T .The following result is obvious. IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 9
Theorem 4.1.
For any ξ ∈ (0 , , it holds that ( ∂T, d ξ ) ∼ ( ∂T ∗ , d ξ p ) .Proof. Let f : ∂T → ∂T ∗ be a map defined by f (( v k ) k ≥ ) = ( v pk ) k ≥ . Clearly, f isa bijection. Pick any ( u k ) k ≥ , ( v k ) k ≥ ∈ ∂T , let t be the length of the common pre-fix of ( u pk ) k ≥ and ( v pk ) k ≥ , then d ξ p ( f (( u k ) k ≥ )) , f (( v k ) k ≥ )) = ξ pt and ξ pt + p − ≤ d ξ (( u k ) k ≥ , ( v k ) k ≥ ) ≤ ξ pt , so f is bi-Lipschitz. (cid:3) bundle map. Let T be a tree. We call B = { v , . . . , v t } a bundle of T of level k , if v , . . . , v t are vertices of T of level k and they sharing the same parent.Let S and T be two trees, and let C k and C ′ k be the sets of vertices of level k of themrespectively. A map ∆ defined on S k ≥ C k is called a bundle map from S to T , if for each k ≥
0, it holds that( i ) For u ∈ C k , ∆( u ) is a bundle of T of level k ;( ii ) { ∆( u ); u ∈ C k } is a partition of C ′ k ;( iii ) If u ′ is an offspring of u , then elements in ∆( u ′ ) are offsprings of elements in ∆( u ).For a vertex w of T , we use [ w ] to denote the set of infinite paths in ∂T passing w ; if B is a bundle of T , we denote [ B ] = S w ∈ B [ w ]. Theorem 4.2.
Let ξ ∈ (0 , , and let S and T be two trees. If there is a bundle map ∆ from S to T , then ( ∂S, d ξ ) ∼ ( ∂T, d ξ ) .Proof. The map ∆ induces a structure tree of ( ∂T, d ξ ), which we will denote by T ∗ , in thefollowing way: the vertices of T ∗ of level k are { [∆( u )]; u ∈ C k } . Clearly ∆ induces an isometry from ( ∂S, d ξ ) to ( ∂T ∗ , d ξ ). We claim that T ∗ is ξ -regular.Indeed, if u and v are vertices of S of level k , we havediam([∆( u )]) ≤ ξ k − and d ξ ([∆( u )] , [∆( v )]) ≥ ξ k . So ( ∂T ∗ , d ξ ) ∼ ( ∂T, d ξ ) and the theorem is proved. (cid:3) Homogeneous tree.
Let T be a tree with root φ . If every vertex of level k − n k number of direct offsprings where n k ≥
1, then we call T a homogeneous tree withparameter ( n k ) k ≥ .A homogeneous tree can be regarded as a symbolic space. Lemma 4.1.
Let T be a homogeneous tree with parameters ( n k ) k ≥ . Then ( ∂T, d ξ ) ∼ (Ω , d ξ ) , ∗ where Ω = Q ∞ k =1 { , , . . . , n k − } .Proof. Clearly we can label a vertex u ∈ T of level k as i . . . i k ∈ Q kj =1 { , , . . . , n j − } ,and the direct offsprings of i . . . i k are i . . . i k i k +1 where i k +1 ∈ { , , . . . , n k +1 − } . Hencethis labeling gives us an isometry between ∂T and Ω. (cid:3) Proof of Theorem 1.2
In this section, we prove Theorem 1.2. Let E = K ( n, m, D ) ∈ M t,v,r . Recall that N = D and s = E . Then we have a j = 0 or a j = N/s . For k ≥
1, we define(5.1) θ k = 1 / ( N k s ℓ ( k ) − k ) , n k = N s ℓ ( k ) − ℓ ( k − − , where we set ℓ (0) = 0 by convention. (The n k defined above coincide with that in (1.4).)Notice that n k ≥ N . Let us denote µ = µ E to be the uniform Bernoulli measure of E .Let W be an approximate square in E k . By Remark 2.3, W has n k +1 direct offsprings.Hence, by Lemma 2.4,(5.2) µ ( W ) = ( n · · · n k ) − = θ k . The coin lemma.
The following lemma is motivated by Xi and Xiong [22] whichdeals with the fractal cubes. Recall that C E,k is the collection of components of E k . Lemma 5.1. (Coin Lemma)
Let k be an integer such that ℓ ( k ) > k . If U ∈ C E,k , thenthere exist V , . . . , V q ∈ C E,k +1 which are direct offsprings of U , such that P qj =1 µ ( V j ) = θ k .Proof. We pick an h ∈ { , , . . . , m − } such that a h = 0; such h exists since E possessesvacant rows. Let S , S , S and S be the four approximate squares in U which locate atthe most top-left corner, the most top-right corner, the most bottom-left corner, and themost bottom-right corner, respectively. (We remark that S i and S j may coincide.)Write S = Q ( x , y ) where y = y . . . y ℓ ( k ) . Recall that D y k +1 = { x ; ( x, y k +1 ) ∈ D} , andits cardinality is less than n by Remark 2.6. Let b ∈ { , , . . . , n − } \ D y k +1 , and denoteby z the left-bottom point of S . Then by Remark 2.3(i), the horizontal lines passing z + (0 , h +0 . m ℓ ( k )+1 ) and the vertical line passing z + ( b +0 . n k +1 ,
0) make a cross, and this crossdivides the direct offsprings of S into four disjoint parts. Especially, the offsprings in theleft-top part are isolated and thus build one or several components of E k +1 ; let us denotethe collection of these components by U .Denote S = Q ( x ′ , y ′ ) and let z ′ be the left-bottom of S . We have y = y ′ since bothof them are located in the top row of U , and it follows that W is a direct offspring of S if and only if W + ( z ′ − z ) is a direct offspring of S . Hence, shifting the above cross by IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 11 z ′ − z , the new cross divides the direct offsprings of S into four disjoint parts, which arecongruent to the four disjoint parts of S respectively. Especially, the offsprings in theright-top part build one or several components of E k +1 , and we denote the collection ofthese components by U . It is seen that X V ∈U ∪U µ ( V ) = { j ; a j > j > h } s · θ k . Similar as above, there exist U and U which are two collections of components of E k +1 contained in S and S respectively, such that X V ∈U ∪U µ ( V ) = { j ; a j > j < h } s · θ k . (We remark that some of U j , j = 1 , . . . ,
4, may be the empty set.) The lemma is provedin this case. (cid:3)
Homogeneous tree.
Let T be a homogeneous tree with parameter ( n k ) k ≥ . Let ν be the uniform measure on ∂T . Then for a vertex u of T of level k , we have(5.3) ν ([ u ]) = k Y i =1 n i ! − = θ k . The following theorem asserts that the tree T is a symbolic representation of E . Theorem 5.1.
Let E = K ( n, m, D ) ∈ M t,v,r and let T be a homogeneous tree withparameters ( n k ) k ≥ defined as (5.1) . Then E ∼ ( ∂T, d /n ) .Proof. Let S be the structure tree of E induced by the components in E k , k ≥
0. (Thisis the second structure tree in Section 3.2.) Let us denote the root of S and T by φ S and φ T respectively. Clearly, S is 1 /n -regular by Lemma 2.5, and hence E ∼ ( ∂S, d /n ) byTheorem 3.1.Let L be the constant in Lemma 2.5, and let p be an integer such that(5.4) N p − ≥ L and ℓ ( p − > p − . Let S ∗ and T ∗ be the p -subtree of S and T , respectively. We shall construct a bundle map∆ from S ∗ to T ∗ such that ∆ is measure preserving.First, we define ∆( φ S ) = φ T . Suppose that ∆ has been defined on S kj =0 C E,pj already.We are going to extend ∆ to C E,p ( k +1) .Pick U ∈ C E,pk . Let V = { V , . . . , V r } be the set of offsprings of U in C E,p ( k +1) , that is, V j ’s are direct offsprings of U in S ∗ . Let g be the number of members of U ; by Lemma2.5 we have g ≤ L . We claim that ∗ Claim. The collection V has a partition (5.5) V = V ′ ∪ · · · ∪ V ′ g , such that µ ( V ′ j ) = θ pk for every j = 1 , . . . , g . Let U = { U , . . . , U q } be the set of offsprings of U in C E,p ( k +1) − . Clearly θ pk ≤ µ ( U ) ≤ qL θ p ( k +1) − and it follows that q ≥ θ pk θ p ( k +1) − L ≥ N p − L ≥ L . Denote δ = θ p ( k +1) − . Since the rank of U i is no less than p − ℓ ( p − > p − U i can be divided into two collections V si and V bi ,such that the total measure of V si is δ (the small collection), and the total measure of V bi is ( N i − δ ≤ ( L − δ (the residual collection), where N i is the number of members of U i . Therefore, we have a partition of V given by q [ i =1 {V si , V bi } . Now we regard δ as one ‘dollar’ and regard each V si as a one-dollar coin. We regard V bi as a big coin that its value varies from 0 to L −
1. The total wealth of these coins is gM ‘dollars’, where M = Q p − j =1 n pk + j . Then the claim holds due to the following facts: first,the value of every coin is no larger than L −
1; secondly, we have plenty of one-dollarcoins (actually, the number is no less than q , and q ≥ L ≥ gL since g ≤ L ). Our claimis proved.By the induction hypothesis on ∆, ∆( U ) is a bundle of T ∗ of level k , which we write as∆( U ) = { w , . . . , w t } . Clearly t = g , since µ ( U ) = gθ pk , ν ([∆( U )]) = tθ pk , and ∆ is measure preserving.Let us regard θ p ( k +1) as a ’cent’. Then for any V ∈ V , µ ( V ) is a multiple of ‘cent’, andthe ν -measure of a vertex of T of level p ( k + 1) is one ‘cent’.For j ∈ { , . . . , g } , V ′ j is a collection of p -step offsprings of U which we write as V ′ j = { v j, , . . . , v j,h j } ;accordingly we take a partition W j, ∪ · · · ∪ W j,h j of direct offsprings of w j in T ∗ satisfying W j,i = µ ( v j,i ) /θ p ( k +1) . We define(5.6) ∆( v j,i ) = W j,i , j ∈ { , . . . , g } , i ∈ { , . . . , h j } . IPSCHITZ CLASSIFICATION OF BEDFORD-MCMULLEN CARPETS 13
Repeating the above process for all U , we extend ∆ to vertices of S ∗ of level k + 1. It isnot hard to check that the three requirements in the definition of bundle map still holdand ∆ is still measure preserving.Hence, ∆ is a bundle map from S ∗ to T ∗ , and ( ∂S ∗ , d /n p ) ∼ ( ∂T ∗ , d /n p ) by Theorem4.2. On the other hand, by Theorem 4.1, we have E ∼ ( ∂S, d /n ) ∼ ( ∂S ∗ , d /n p ) and( ∂T, d /n ) ∼ ( ∂T ∗ , d /n p ), so(5.7) E ∼ ( ∂S ∗ , d /n p ) ∼ ( ∂T ∗ , d /n p ) ∼ ( ∂T, d /n ) . The theorem is proved. (cid:3)
Proof of Theorem 1.1.
Let T be a homogeneous tree with parameters ( n k ) k ≥ . ByTheorem 5.1 and Lemma 4.1, we have E ∼ ( ∂T, d /n ) ∼ ∞ Y k =1 { , , . . . , n k − } , d /n ! , which proves the first assertion.If σ = p/q ∈ Q , then ℓ ( pk ) = qk for k ≥
0. It follows that p Y j =1 n pk + j = (cid:18) Ns (cid:19) p · s ℓ ( pk + p ) − ℓ ( pk ) = N p s q − p . Let T ∗ be the p -subtree of T and denote N ∗ = N p s q − p , then E ∼ ( ∂T ∗ , d /n p ) ∼ ( { , , . . . , N ∗ − } ∞ , d /n p ) . The second assertion is proved. (cid:3)
Recall that the Hausdorff dimension of a self-affine carpet is log m (cid:16)P m − j =0 a σj (cid:17) ([2, 16]). Proof of Theorem 1.2.
Let N ′ be the cardinality of the digit set of F , and let s ′ be thenumber of non-vacant rows of F .(i) Let σ = p/q ∈ Q . If E and F have the same Hausdorff dimension, then N p s q − p =( N ′ ) p ( s ′ ) q − p := N ∗ . By Theorem 1.1, we have that both E and F are equivalent to thesymbolic space ( { , , . . . , N ∗ − } ∞ , d /n p ) , so E ∼ F . That E ∼ F implies dim H E =dim H F is folklore. Assertion (i) is proved.(ii) Let σ ∈ Q c . If E and F share the same distribution sequence up to a permutation,then N = N ′ and s = s ′ . By Theorem 1.1, E and F are Lipschitz equivalent to the samesymbolic space (Ω , d /n ) determined by ( n k ) k ≥ where n k = N s ℓ ( k ) − ℓ ( k − − , so E ∼ F .The necessity part is guaranteed by Proposition 1.2. The second assertion is proved. (cid:3) ∗ Symbolic space
Recall that λ is a metric on D ∞ defined by λ (( x , y ) , ( x ′ , y ′ )) = max { d /n ( x , x ′ ) , d /m ( y , y ′ ) } . The proof of Theorem 1.3.
First we define a structure tree of D ∞ . For two words x and i , we denote x ✁ i if x is a prefix of i . For k ≥
1, we call[ x , y ] = { ( i , j ) ∈ D ∞ ; x ✁ i , y ✁ j } an approximate square of D ∞ of rank k , if x = x . . . x k , y = y . . . y ℓ ( k ) are two wordssuch that ( x j , y j ) ∈ D for 1 ≤ j ≤ k and y j ∈ E for j > k . It is seen that the approximatesquares of the same rank are disjoint.We denote by C k the collection of approximate squares of rank k of D ∞ , especially weset C = D ∞ by convention. Let T be the structure tree of D ∞ induced by ( C k ) k ≥ .We claim that T is 1 /n -regular. Indeed, if [ x , y ] and [ x ′ , y ′ ] are two approximate squaresof rank k , an easy calculation shows that λ ([ x , y ] , [ x ′ , y ′ ]) ≥ /n k anddiam([ x , y ]) ≤ max { /n k , /m ℓ ( k ) } ≤ m/n k . It follows that T is 1 /n -regular, so by Theorem 3.1, we have ( ∂T, d /n ) ∼ ( D ∞ , λ ).On the other hand, every vertex [ x , y ] of T of level k has n k +1 := N s ℓ ( k +1) − ℓ ( k ) − direct offsprings (see Remark 2.3), so T is a homogeneous tree with parameters ( n k ) k ≥ .Therefore, ( D ∞ , λ ) ∼ ( ∂T, d /n ) ∼ E where the last equivalence is due to Theorem 5.1. (cid:3) References [1] J. Barral and M. Mensi,
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Institute of applied mathematics, College of Science, Huazhong Agriculture University,Wuhan,430070, China.
E-mail address : [email protected] Department of Mathematics and Statistics, Central China Normal University, Wuhan,430079, China
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