Isomorphism and bi-Lipschitz equivalence between the univoque sets

In this paper, we consider a class of self-similar sets, denoted by \begin{document}$ \mathcal{A} $\end{document} , and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by \begin{document}$ U_1 $\end{document} . We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of \begin{document}$ U_1 $\end{document} , is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition \begin{document}$ U_1 $\end{document} is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.


1.
Introduction. Let {g j } N j=1 be an iterated function system (IFS) of similitudes defined on R by g j (x) = r j x + a j , where the similarity ratios satisfy 0 < r j < 1, and a j ∈ R. Hutchinson [14] proved that there exists a unique non-empty compact set K ⊂ R such that K = N j=1 g j (K).
We call K the self-similar set or attractor for the IFS {g j } N j=1 . For any x ∈ K, there at least exists a sequence (i k ) ∞ k=1 ∈ {1, . . . , N } N such that x = lim k→∞ g i1 • · · · • g i k (0).
We call such a sequence a coding of x. We can define a surjective projection map between the symbolic space {1, . . . , N } N and the self-similar set K by π((i k ) ∞ k=1 ) := lim and g j (O) ⊆ O for all 1 ≤ j ≤ N . Usually, self-similar sets with the open set condition is relatively easy to be analyzed. For instance, the Hausdorff dimension of K coincides with the similarity dimension which is the unique solution s of the equation N j=1 r s j = 1. Self-similar sets without the open set condition were analyzed by many scholars, see [11,13,18,22,23] and references therein. The main concern of these references is to calculate the Hausdorff dimension of the attractors and some associated measures. However, for the fractal structure, the self-similar sets are far beyond understood. For instance, how can we classify different fractal sets from the dynamical and fractal perspective. In dynamical systems, usually we utilize the measure-theoretic isomorphism to classify dynamical systems. In fractal geometry, the bi-Lipschitz equivalence is viewed as an appropriate definition which allows us to distinguish two fractal sets. We will make use of these two definitions to classify some fractal sets.
In this paper, we shall consider the measure-theoretic isomorphism and bi-Lipschitz equivalence between two univoque sets. Partial motivation of this consideration is from the classical β-expansions. For this case, there are many results concerning the univoque sets [7,8,12]. However, to the best of our knowledge, there are few papers considering the classification of the univoque sets. For the classical β-expansions, for different β's the Hausdorff dimension of the univoque sets may differ. Therefore, usually we cannot find a bi-Lipschitz map or an isomorphism between the univoque sets. In the setting of self-similar sets, generally, we can consider this problem. Another reason why we consider the univoque set is that it reflects, in some sense, the complicity of the structure of self-similar set. It has some intimate relation with the original self-similar set [4]. We want to find some classification results such that the original self-similar sets and their subsets have resonant phenomenon, namely, two self-similar sets are bi-Lipschitz equivalent (measure theoretically isomorphic) if and only if their associated univoque sets are bi-Lipschitz equivalent (measure theoretically isomorphic).
Many articles are devoted to the study of the bi-Lipschitz equivalence [3,9,10,16,17,19,21,25,27,29,30,31,32,33,34]. It is an important problem to construct a bi-Lipschitz map between two fractal sets. Usually, it is not easy to find such a map, in particular for the self-similar sets which are not totally disconnected [26]. Comparing with the self-similar sets, for the univoque sets it is much more difficult to construct a bi-Lipschitz map as generally the structure of the univoque sets is not clear. Moreover, the univoque sets could not be closed. This fact makes the bi-Lipschitz equivalence between the univoque sets difficult.
It is common knowledge that many fractal sets can be taken as some dynamical systems [2, Lemma 2.1]. As such we may simultaneously classify different fractal sets from the dynamical perspective as well as from the fractal point of view.
Generally, for the self-similar sets without the open set condition, it is difficult to analyze. In this paper, we shall consider a class of overlapping self-similar sets as follows. Fix λ ∈ (0, 1). Let K be the attractor of {f i (x) = λx + b i } n i=1 , where n ≥ 3 and b i ∈ R for all i. Let A be the collection of all the self-similar sets satisfying the following (1) − (4) conditions, (1) 0 = b 1 < b 2 < · · · < b n = 1 − λ; (2) f i ([0, 1]) ∩ f j ([0, 1]) = ∅ for any |i − j| ≥ 2; (3) There exist i, j ∈ {1, · · · , n − 1} such that Denote by A 1 all the self-similar sets satisfying conditions (1) − (5) and by A 2 all the self-similar sets satisfying conditions (1) − (4) and (5 ). It is easy to prove that by conditions (1) is an interval and that p − i is largest, then we call ∪ p k=i f k ([0, 1]) a maximal connected component of [0,1]. This definition is the same as defined in topology. Denote by C n the number of all the maximal connected components. Define k l = k r = C n − k i , and k m = n − k i − 2k l . It is easy to find that n K = k l + k m . The subscripts l, r, m, i stand for "left", "right", "middle" and "independent", respectively. The meaning of these words will be clear, see Definition 3.10 and the example below. We use the following example to illustrate the above definitions. Example 1.1. Let K be the attractor of the IFS The basic intervals in the first level for K are as follows.
is not a maximal connected component as for this set p − i = 2 is not the largest number. Therefore C n = 3. Moreover, n K = 5, which refers to the number of the following intersections k i = 1, namely, we only have that f 5 ([0, 1]) does not intersect with other basic intervals. Moreover, k m = 3, k l = k r = 2. The first result of this paper is from the ergodic perspective. We prove the existence of the unique measures of maximal entropy with respect to the lazy map for U 1 and K, where U 1 is the closure of U 1 . For the definition of lazy map, we refer to the next section.
Then U 1 has a unique measure of maximal entropy with respect to the lazy map. Moreover, the same statement is correct for K.
The next result is giving a uniform dimensional formula of U 1 . The key idea, which we implement, is the configuration set with finite patterns [16]. ( where γ 1 is a Pisot number of the following equation: where γ 2 is a Pisot number of the following equation: where γ 3 is the appropriate root of the following equation: Morover, γ 3 is a Pisot number if and only if k m + 1 < k i .
We give some remarks upon this result. Firstly, the Pisot numbers play a crucial role in proving all the following theorems. Secondly, the significance of this result is that we give a uniform formula of the Hausdorff dimension of U 1 . Without this formula, we cannot analyze the relation between the bi-Lipschitz equivalence and measure-theoretic isomorphism for the univoque sets. A useful corollary of Theorem 1.3 is the following result.
1 ). Then the following conditions are equivalent.
where · (i) refers to the index associated with K i , i = 1, 2.
In [16], Jiang, Wang and Xi, using the idea of configuration, gave a uniform dimensional formula of K. In fact, in [16], a general dimensional formula was proved.
where γ is a Pisot number of the following polynomial The following corollary follows from Theorems 1.5, 1.3 and Corollary 1.4.
Corollary 1.6. Given i = 1 or 2. For any K 1 , K 2 ∈ A i . Then We may find some counterexamples in the class A such that dim H (U , see one example in Section 4. The next theorem is one of the main results of this paper.  are measure theoretically isomorphic with respect to the unique measures of maximal entropy, and k i . (7) K 1 and K 2 are measure theoretically isomorphic with respect to the unique measures of maximal entropy, and k i . The definition of bi(quasi)-Lipschitz equivalence is available in the next section. We emphasize that dim H (U m . Remark 1.10. When U 1 is not closed, we may still prove some resonant result. For the class A 1 , we may further classify two sub-classes, i.e. A 1 = A 1 ∪ A 1 such that we may prove the following result. Let Then the following conditions are equivalent. are Lipschitz equivalent; (4) n K1 = n K2 ; (5) K 1 and K 2 are Lipschitz equivalent; (6) K 1 and K 2 are quasi-Lipschitz equivalent; (7) dim H (K 1 ) = dim H (K 2 ). However, the proof of this result is more complicated than the method in our paper, although the main idea is similar. We shall prove this result in another paper.
This paper is arranged as follows. In section 2, we give some basic definitions which will appear in the paper. In section 3, we give the proofs of the main theorems. In section 4, we give one example. Finally, we give some remarks and open problems.

Some basic definitions and lemmas. Given
, we say f i1···in (x) is an exact overlap if there exist two different i 1 · · · i k ∈ {1, · · · , n} k and j 1 · · · j m ∈ {1, · · · , n} m such that For any k ≥ 1, let (i 1 · · · i k ) ∈ {1, · · · , n} k we call f i1···i k ([0, 1]) the k-th level basic interval. It is well known that any self-similar set can be taken as a topological dynamical system [2]. Let K be a self-similar set from the class The following lemma was proved in [2, Lemma 2.1].
Motivated by this lemma, we define the orbits.
an orbit set of x.
Clearly, for a generic point x ∈ K, it may have multiple codings. In other words, it has multiple orbits. The following definition is to define a special orbit, namely, the lazy orbit.
We call such map the lazy map, and the orbit {T i1···i k (x) : k ≥ 0} the lazy orbit. We denote the lazy orbit of x by Evidently, we have a dynamical description of U 1 .
Now we introduce some definitions related with the bi-Lipschitz equivalence. Two metric spaces (X 1 , d 1 ) and (X 2 , d 2 ) are said to be bi-Lipschitz equivalent if there exists a bijection φ : X 1 → X 2 and a constant c > 0 such that We simply say X 1 and X 2 are Lipschitz equivalent if they are bi-Lipschitz equivalent. For simplicity, we denoted by A 1 A 2 when they are Lipschitz equivalent. We say that X 1 and X 2 are quasi-Hölder equivalent [29] if there exists a bijection φ : In particular, when t = 1, we say that X 1 and X 2 are quasi-Lipschitz equivalent [29]. Clearly, if X 1 and X 2 are bi-Lipschitz equivalent, then they are quasi-Lipschitz equivalent. Moreover, dim H (X 1 ) = dim H (X 2 ), provided that X 1 and X 2 are quasi-Lipschitz equivalent. An algebraic number is called a Pisot number if all its conjugates have modules strictly smaller than 1. The Pisot number plays a pivotal role in this paper, it has many useful properties, see [1] and references therein. There are many definitions regarding the ergodic theory, for instance, the unique measure of maximal entropy, isomorphism. We do not introduce them in details. The reader can find these definitions in the book [28].

Proofs of main theorems.
3.1. Proof of Theorem 1.2. In this section, we shall give the proofs of the main results introduced in Section 1. We first prove Theorem 1.2. In [15], Jiang and Dajani proved that for the self-similar sets with the exact overlaps they gave an algorithm which can calculate the dimension of U 1 , see [15,Theorem 2.18] or [4].
The key idea of [15] is partitioning the self-similar sets and constructing a graphdirected self-similar set with the open set condition. Then the univoque set U 1 can be identified with a graph-directed self-similar set. This idea enables us to find the individual example rather than a class of examples. Nevertheless, the partition idea allows us to find the unique measure of maximal entropy for U 1 . For convenience, we introduce how to give a partition for any K ∈ A. Let . For two consecutive members s i and s i+1 of P, we call {s i , s i+1 } an admissible pair if there exists a j such that Define For an admissible pair {s i , s i+1 }, there exist at most two j's satisfying (2) and we denote by α(i) the smaller j. Dynamically, we choose the lazy algorithm. It is not For this case, we delete the isolated point from In what follows, we always obey this rule. To illustrate this modification, we give the following example.
Example 3.1. Let K be the attractor of the IFS The point 2/9 is an isolated point in [s 1 , s 2 ] K . However, 2/9 is an accumulation point in [s 2 , Proof. The first statement follows from the definitions of partition P and [s i , s i+1 ] K . For the second statement, we only need to consider the image of [s i , s i+1 ] K under the expanding map f −1 α(i) , i.e. it suffices to prove that However, this is a directed consequence of the definitions of For the open set condition [22], we can check it directly from the equation in the second statement.
Lemma 3.2 is essentially giving a Markov partition [20] for K. Note that by the reformulation (1), for the univoque point, its lazy orbit never hits the set V . Roughly speaking, if we delete these [s i , s i+1 ] K , i ∈ Q which are associated with V in the above graph-directed self-similar set, then we obtain a subset that is equal to the univoque set (up to a countable set). Therefore, we have the following construction. Let Q * be the subset of Q defined in (3) by deleting those j such that for some . For this Q * , we are allowed to define a new Markov chain as follows.
In terms of this rule, we define a new directed graph with vertex set The corresponding similitude between these two vertices is f α(i) . Therefore, we have constructed a graph-directed self-similar set K * satisfying the open set condition [22].
Proof. First, by the reformulation (1), we have U 1 ⊂ K * . Conversely, take x ∈ K * . Then we can find a lazy coding (i n ) ∈ {1, 2, · · · , n} N (we choose the lazy algorithm) such that If the orbit of x never hits the endpoints of every [s j , s j+1 ] K , j ∈ Q \ Q * , denoted the set of these endpoints by W (clearly, we have W ⊂ P), then x ∈ U 1 . Here the set W can be defined as follows: If there exists some smallest n 0 such that Proof. By Lemma 3.3, it suffices to prove that C ⊂ U 1 . We suppose without loss of generality that For any x ∈ C \ W ⊂ K * , there exists a smallest k ≥ 1 and i 1 · · · i k ∈ {1, · · · , n} k such that T i1···i k (x) hits W for the first time (here we emphasize that we choose the lazy orbit of x), i.e. T i1···ij (x) / ∈ W for any 1 ≤ j ≤ k − 1 and T i1···i k (x) ∈ W , or x / ∈ W and T i1 (x) ∈ W. In other words, f i1···i k (x) is not an exact overlap. We may assume that Clearly, Here, we use a fact f 1 (1) = π(2 ∞ ). Hence, it suffices to prove that x l ∈ U 1 for any l ≥ 1. Evidently, By the assumption, f n ([0, 1]) ∩ f n−1 ([0, 1]) = ∅, it follows that i + 1 ≤ n − 1 and that t 0 ≤ n − 1. Therefore, f i1···i k (i+1)2 l t0 (x) is not an exact overlap (if f t1···tp (x) is an exact overlap and f t1···tp−1 (x) is not an exact overlap, then t p should be 1 or n). Moreover, by the maximum of t 0 . Thus, x l = f i1···i k (i+1)2 l t0 (1) is a univoque point for any l ≥ 1. With a slight change, we can prove W ⊂ U 1 . Now U 1 = K * follows from The graph-directed construction in Lemma 3.2 has an associated matrix which is defined as follows: S = (s i,j ) l×l , l = Q, where The corresponding subshift of finite type generated by S is defined as follows: , · · · , l} N : s i k ,i k+1 = 1 for any k}. Similarly, we may define the adjacent matrix with the directed graph of K * , denoted by S * . The subshift of finite type generated by this matrix is denoted by Σ * .
The following lemma can be checked directly.
Lemma 3.5. Let K ∈ A. Then the associated graph-directed construction of K and K * have irreducible matrices S and S * , respectively.
Proof. Without loss of generality, we prove only one case, i.e.
First, we prove that S is irreducible. It suffices to show that the directed graph of K is strongly connected. For any vertex [s i , s i+1 ] K , if i ∈ Q * , then there exists some 1 ≤ j ≤ n − 1 such that Therefore, the vertex [s i , s i+1 ] K can reach any other vertices. If Hence, the vertex [s i , s i+1 ] K can reach any other vertices. If i ∈ Q\Q * , then we may repeat the above discussion (the same as the case f j (K) ⊃ f jn (K) = f (j+1)1 (K)). Now we have proved that the directed graph of S is strongly connected. With analogous discussion, we are able to prove that the graph of S * is also strongly connected.
By Lemma 3.5, it follows that the associated subshift of finite type generated by the matrix S, i.e. Σ, has a unique measure of maximal entropy (Parry measure [24]), denoted by µ. On the other hand, we may construct a natural map φ between Σ and K by where for each [s in , s in+1 ] K , there is a unique f jn (we choose the lazy algorithm) such that [s in , s in+1 ] K ⊂ f jn (K). Proof. We first prove that φ is a bijection. Note that we chose the lazy orbit when we give a Markov partition of K, therefore, φ is well-defined and it is one-to-one. Moreover, it is also onto. Therefore, φ is a bijection. It is easy to check that where L is the lazy map. Hence, we finish the proof.
Similarly, for the graph-directed self-similar set K * = U 1 , we can obtain the following lemma.

3.2.
Proof of Theorem 1.3. Now we define a set which offers another method that can calculate the Hausdorff dimension of U 1 . Before we introduce this approach, we recall some definitions.
is an exact overlap, then any univoque point of U 1 cannot be in f i1···i k (K). Therefore, if we need to find all the univoque points, we should delete all possible exact overlaps in every k-th level. This simple observation leads to the following construction. Let k ∈ N ≥2 . Define E k = {f j1···j k (x) : j 1 · · · j k ∈ {1, · · · , n} k , f j1···j k (x) is an exact overlap}.
Roughly speaking, E k is a family of similitudes which are exact overlaps. Define f i1i2 ((0, 1)).
Note that for any k ≥ 1, E k is a union of some basic intervals with length λ k and some basic intervals deleting some appropriate sets. These appropriate sets are associated with the exact overlaps (see the following example). We denote the cardinality of all the closed intervals in E k by (E k ), k ≥ 1, i.e. if I is a basic interval in E k with length k, then we count this interval for one time. If I is a closed interval generated by a basic interval deleting some appropriate intervals (these intervals are associated with the exact overlaps, moreover, we delete at most two appropriate intervals), then we also count this closed interval for one time. We shall use the following example to illustrate the above definitions.
Example 3.8. Let K be the attractor of the IFS Note that f 13 (x) = f 21 (x) is an exact overlap.
Therefore, (E 2 ) = 7. Generally, we can count the number of closed intervals of E k , k ≥ 3 in this way.

Lemma 3.3 can be expressed in the following way.
Lemma 3.9. Let K ∈ A. Then U 1 \ U 1 is at most a countable set.
Lemma 3.3 gives an algorithm which may calculate the Hausdorff dimension of U 1 . However, we cannot directly obtain Theorem 1.3 as for different IFS's, the matrices may be distinct. Hence, we need to find a uniform approach which can calculate the Hausdorff dimension of U 1 . Our idea of solving this problem is motivated by the Ngai and Wang's finite type condition [23]. First, we give the following definition of types.
In Section 1, we define some numbers, namely k i , k l , k r , k m . Now we use the above definition to give an explanation to these numbers. Example 3.11. Let K be the attractor of the IFS The basic intervals in the first level for K is as follows.
are the right types.
Now, it is easy to see that k i , k l , k r , k m refer to the numbers of all possible independent, left, right and middle types in E 1 , respectively. In terms of Definition 3.10, given K ∈ A with the IFS {f i } n i=1 , suppose there are k i independent types, k l left types, k r right types, k m middle types in E 1 . Clearly, To prove Theorem 1.3, we need the idea of configuration which was the main tool of [16]. Definition 3.13. Let (X, d, {D k } k , {δ k } k ) be a configuration set. We say that X is a configuration set of finite pattern if the following conditions are satisfied: (1) δ k = λ k for some λ ∈ (0, 1); (2) there is a surjective label mapping l : ∪ ∞ k=0 D k → {1, 2, · · · , m} and a transition matrix M = (a ij ) m×m such that for any 1 ≤ i, j ≤ m, any k ≥ 0 and any A ∈ D k with l(A) = i, The following result was proved in [16].  Proof. First, we prove that for any interval A ⊂ E k , c 1 λ k ≤ |A| ≤ c 2 λ k . The right inequality is clear if we take c 2 = 1. For the left inequality, |A| ≥ λ k − 2λ k+1 . By the condition of A, it follows that 1 > 2λ, therefore, there is some c 0 > 0 such that 1 > (2 + c 0 )λ. Thus, |A| ≥ λ k − 2λ k+1 ≥ c 0 λ k+1 . Now we take c 1 = c 0 λ and prove the left inequality. Next, we show that if . Hence, we have proved that U 1 is a configuration set. Now, we prove that U 1 is a configuration set with finite patterns. By the definition of A, there are only 4 types for the univoque sets. Given one type from E k , k ≥ 1, without loss of generality, we may assume that it is an independent type. In E k+1 the independent type generates (under the IFS {f i } n i=1 ) k i independent types, k l left types, k r right types, and k m middle types. For other three types, the discussion is similar, see Lemma 3.19, 3.20, 3.21, The following lemma was proved by Akiyama [1], which is crucial for the proof of Theorems 1.7 and 1.3.  f (x) = x n + a n x n−1 + · · · + a 0 ∈ Z[x], a 0 = 0.
Proof. Since f and g are two irreducible polynomials over Z[x], we have (f, g) = 1 over Z[x]. We may implement the Euclidean algorithm for f and g. There exist some . . .
where γ 1 is a Pisot number of the following equation: Proof. By Lemma 3.15, U 1 is a configuration set with finite patterns. Therefore, we only need to find the transition matrix M . We suppose that For the other case, the proof is similar. We consider the relation between 4 different types. First, we consider the offspring of independent type. Clearly, an independent type can generate k i independent types, k l left types, k r right types, and k m middle types. A left type can generate k i − 1 independent types, k l left types, k r right types, and k m middle types. A right type can generate k i independent types, k l − 1 left types, k r right types, and k m middle types. Finally, a middle type can generate k i − 1 independent types, k l − 1 left types, k r right types, and k m middle types. We may use the following matrix to describe this process.
Then we can calculate the Hausdorff dimension of U 1 in terms of the spectral radius of the above matrix.
where γ 1 is largest real root of the following equation: Note that n K = k m + k l , n = k m + k l + k i + k r . Therefore, the above equation is exactly In the remaining of the proof, we prove that γ 1 is a Pisot number. Let Note that f (n) = (2n − 1)n K > 0 and f (n − 2) = −2(n − 2) 2 − n K + 2n K (n − 2).
However, this is also impossible as It is easy to check by the fact n − 2 ≥ n K that Therefore, by Lemma 3.16 and 3.17, we have that f (x) is an irreducible polynomial. Now, we prove that f (x) is the minimal polynomial of γ 1 . If the degree of γ 1 is strictly smaller than 3, then by the conclusion obtained above, we have that γ 1 is not an integer, therefore the degree of the minimal polynomial of γ 1 must be 2. We let this polynomial be g(x) = ax 2 + bx 2 + c. Namely, g(γ 1 ) = 0. Using the Euclidean algorithm again, we may find some such that where ∂(r(x)) = 1, ∂(r 2 (x)) = 0, i.e. r 2 (x) is a constant. Since f is irreducible over Z[x], we have (f, g) = 1. Thus, (f, g) = 1 over C [x]. However, f (x) and g(x) have a common root γ 1 . Therefore, we conclude that f (x) is the minimal polynomial of γ 1 . Now, we have proved that γ 1 is a Pisot number.
where γ 2 is a Pisot number of the following equation: Proof. We consider the relation between 4 different types. The proof is similar as Then we can calculate the Hausdorff dimension of U 1 in terms of the spectral radius of the above matrix. Note that 0 and 1 are also the roots of the characteristic polynomial of M . The Pisot property is trivial as n > k m + 2.
where γ 3 is the appropriate root of the following equation:

Moreover, γ 3 is a Pisot number if and only if
Then we can calculate the Hausdorff dimension of U 1 in terms of the spectral radius of the above matrix. The remaining proof is trivial.

3.3.
Proof of Theorem 1.7. The following results were proved in [16].
where γ is a Pisot number of the following polynomial Lemma 3.23. If K 1 , K 2 ∈ A, then the following four conditions are equivalent: (2) K 1 and K 2 are quasi-Lipschitz equivalent; (3) K 1 and K 2 are Lipschitz equivalent; Now, we prove some lemmas.
Therefore, b | a 2 . By the assumption (a, b) = 1, we have b = 1. With a similar discussion, we may show that √ p ∈ N.
Proof. Firstly, we suppose that K 1 , K 2 ∈ A 1 . By Lemmas 3.19, 3.20 and 3.21, it follows that 1 ), then γ i = γ j , where γ i = γ j is a Pisot number. By Lemmas 3.19, 3.20 and 3.21, γ i is the root of i + k (2) m = 0. Hence, in terms of Lemmas 3.17 and 3.18 and the relation n K = k m + k l , it follows that n = m, n K1 = n K2 . Therefore, using Lemma 3.23, we conclude dim H (K 1 ) = dim H (K 2 ).
Secondly, suppose that K 1 , K 2 ∈ A 2 . Then dim H (K 1 ) = dim H (K 2 ) implies that γ i = γ j is a Pisot number or an algebraic number. If γ i = γ j is a Pisot number, then the proof is similar as the previous cases. If γ i = γ j is an algebraic number, then which implies by Lemmas 3.24 and 3.25 that n = m, n K1 = n K2 , and dim H (K 1 ) = dim H (K 2 ).
1 ). Proof. By Theorems 1.5, 1.3 and Corollary 1.4, it follows that the Hausdorff dimension of K and U 1 are uniquely determined by n and n K = k m + k l .
We finish this subsection by giving a proof of Theorem 1.7.
Proof of Theorem 1.7. By Lemmas 3.26, 3.23, Corollaries 1.4 and 1.6, the first 5 conditions are equivalent. First, we prove that if K 1 and K 2 are measure theoretically isomorphic with respect to the unique measures of maximal entropy, then dim H (K 1 ) = dim H (K 2 ). Since K 1 , K 2 ∈ A i , i = 1 or 2 , we may give the Markov partitions for K 1 and K 2 , respectively. The associated matrices of the Markov partitions are irreducible, and therefore we may find the unique measures of maximal entropy for the subshifts of finite type generated by the irreducible matrices, respectively. By Lemma 3.6, K 1 , K 2 have unique measures of maximal entropy. By the assumption, i.e. K 1 and K 2 are measure theoretically isomorphic with respect to the unique measures of maximal entropy, it follows that the corresponding subshifts of finite type of the matrices are measure theoretically isomorphic. Therefore, the spectral radii of these two matrices are equal. By Theorem 1.5, it follows that dim H (K 1 ) = dim H (K 2 ). Similar statement is still correct for U 1 ). Therefore, in terms of Corollary 1.4 and the discussion above, we prove one direction.
Conversely, If dim H (U l , by Corollary 1.4, it follows that k m . Thus, the graph-directed self-similar sets for U are measure theoretically isomorphic with respect to the unique measures of maximal entropy. Analogous proof is still correct for K.

3.4.
Proof of Theorem 1.8. It is natural to give a checkable condition under which U 1 is closed. By virtue of some discussion from the symbolic space, we have the following result. We first prove that the following lemma. Proof. Since f 1 (I) ∩ f 2 (I) = ∅, by the fourth condition of A, f 1n = f 21 . We prove this lemma via some cases.
The following two cases can be proved in a similar way.   Denote It is easy to give the following decomposition We call L u1 ∪ M u1 ∪ R up the I pattern and each M ui , 2 ≤ i ≤ p the II pattern. With this decomposition, we have the following lemma. The proof is similar as Lemma 3.15. Then U 1 is a configuration set with two patterns, i.e. I pattern and II pattern. Moreover, the matrix M is n − n K n K n − n K − 2 n K .
We prove Lemma 3.33 in terms of the configuration set with finite patterns. First, we introduce a proposition which is useful to find the bi-Lipschitz map.
be two configuration sets. If there exists a bijection η : ∪ ∞ k=0 D k → ∪ ∞ k=0 G k such that for all k, A ∈ D k if and only if η(A) ∈ G k , and that B ⊂ A with A ∈ D k and B ∈ D k+1 implies that η(B) ⊂ η(A), then X Y.
Proof of Proposition 3.36. For any x ∈ X, there exists a unique sequence A 1 , A 2 , A 3 , · · · , with A k ∈ D k such that {x} = ∩ ∞ i=1 A i . Then a mappingη : X → Y is defined as . Now we shall show that the mapping is a bi-Lipschitz mapping.
For any x = x there exists some k such that x, x ∈ A with A ∈ D k , but x ∈ B, x ∈ B for distinct B, B ∈ D k+1 . It follows from the definition of configuration set that In the same way, we obtain that as required.
The following corollary is a consequence of Proposition 3.36.
Corollary 3.37. Let X 1 and X 2 be two configuration sets of finite patterns. If their transition matrices coincide and l 1 (X 1 ) = l 2 (X 2 ) for the corresponding label mappings l 1 and l 2 , then X 1 X 2 .
Proof of Theorem 1.8. It suffices to prove that dim H (U 1 ) implies U 4. One example. In this section, we give one example which illustrates that generally dim H (U 1 ) cannot imply that dim H (K 1 ) = dim H (K 2 ). Moreover, dim H (U  The basic intervals in the first level for K 1 and K 2 are as follows.  where η is the appropriate root of the following equation: x 2k+1 − nx 2k + 2(k m + k l )x k − (k m + k l ) = 0.
However, for any n ≥ 3 and k ≥ 2, we cannot prove that the algebraic number η is also a Pisot number (Partial results can be ontained if the coefficients of the above polynomial satisfying certain conditions, for instance, we may use the Rouché theorem if n > 3(k m +k l )+1). As we mentioned in introduction, the Pisot property (Corollary 3.18) is essential for the classification results (Theorem 1.7, Corollary 1.9).