Dimensions of Fractals Generated by Bi-Lipschitz Maps

and Applied Analysis 3 prefix I = (i 1 , . . . , i n ) ∈ Σ ∗ such that I ∈ I b . However, in view of Remark 3, the number of such prefixes must be finite. Second, it is possible that S I = S I 󸀠 for distinct I, I ∈ Σ ; we identify such S I and S I 󸀠 . Last, for IFSs of contractive similitudes, I b = I b (U) and so A b = A b (U). In general, however, they need not be the same. Definition 4. Let X ⊂ R be a compact subset with X ̸ = 0 and let S i : X → X, i = 1, . . . , N, be bi-Lipschitz essential contractions.We say that {S i } N i=1 has the logarithmic distortion property (LDP) if there is a constant σ > 0 such that lim b→0 + sup I∈I b b r I|ln b| σ = 0. (10) Remark 5. In the above definition, we do not assume that the maps of the IFS are differentiable. Besides this, if {S i } N i=1 satisfies BDP, then there is a constant c > 0 such that b/r I ≤ c for all b ∈ (0, 1) and I ∈ I b . Thus LDP holds. Hence LDP is an extension of BDP. Examples of IFSs satisfying LDP but not BDP will be given in Section 4. Definition 6. Let X, {S i } N i=1 , U satisfy the hypotheses of Definition 4, U ⊆ X be a bounded invariant set that is open in the relative topology of X with L(U) > 0, and Φ be a finite subset of {S I : I ∈ Σ ∗ }. One calls a finite subcollection {φ 1 , . . . , φ k } ⊆ Φ a packing family for Φ with respect to U if the following conditions are satisfied: (i) φ 1 (U), . . . , φ k (U) are pairwise disjoint; (ii) for any φ ∈ Φ, φ(U) intersects at least one φ j (U). Denote the class of all packing families ofA b with respect to U by P U (b), and denote the class of all packing families of A b (U) byP U (b). Example 7. Let U = (0, 3), S i (x) = (1/2)(x + i), i = 0, 1, 2, 3, andΦ = {S i } 3 i=0 .Then {S 0 , S 3 }, {S 1 }, and {S 2 } are three packing families ofΦ. Definition 8. Let X, {S i } N i=1 , U satisfy the hypotheses of Definition 4 and fix λ ∈ (0, 1). Define Q λ (s) := lim n→∞ 1 n ln( inf Φ∈P U(λ n ) ∑


Introduction
In the literature on the equality of the Hausdorff and box dimensions of the attractor of an iterated function system (IFS), it is usually assumed that the generating maps are  1 and the bounded distortion property holds (see [1][2][3]).For IFSs of conformal contractions, the weak separation condition is also assumed (see [3]).These three conditions are usually imposed in order to obtain a formula for the dimensions of the attractor in terms of topological pressures (see, e.g., [4,5]).The main goal of this paper is to relax these three conditions.
There are many definitions of dimension for fractal sets.As is well known, the Hausdorff and upper box dimensions may be regarded as the smallest and the greatest values of any reasonable definition of dimension.Fox example, the packing dimension introduced by Tricot Jr. [6] always lies between these two values.Motivated by this observation, McLaughlin [7] and Falconer [1] studied conditions under which the Hausdorff and box dimensions of a fractal set are equal.As an application of the so-called implicit method, Falconer [1,Examples 2 and 3] proved the equality of the Hausdorff and box dimensions for all self-similar sets and a class of graph-directed sets (called recurrent sets), without assuming any separation condition.By assuming the  1smoothness of the maps of the IFS, the bounded distortion property (BDP), and the weak separation condition (WSC), Lau et al. [3] proved the equality of the two dimensions for self-conformal sets.Under these conditions, the authors [5] proved that the common dimension is given by the zero of some topological pressure functions.For an infinite iterated function system, by assuming the open set condition, BDP, and that the maps of the IFS are  1+ smooth, Mauldin and Urbański [4] proved that the Hausdorff dimension of the limit set is given by the zero of some topological pressure function.
The dimensions of self-affine sets have also been studied extensively, since the work of McMullen [8] and Falconer [9].Our results in this paper allow us to deal with a special class of self-affine sets.A simple example in this class is the self-affine set generated by the affine maps which arises in the study of connectedness of self-affine sets in [10] (see also [11] and the references therein).This IFS does not satisfy BDP.There are of course plenty of examples of IFSs that do not satisfy WSC or contain maps that are not  1 .
We will study such examples in Section 4. Our work is partly motivated by them.
There are two main goals in this paper.First, we would like to prove the equality of the Hausdorff and box dimensions by assuming a weaker set of conditions.We weaken the  1 -smoothness condition to the bi-Lipschitz condition and replace the bounded distortion property by a weaker logarithmic distortion property.Second, under these conditions, we would like to obtain a formula for the common dimension in terms of the zero of some topological pressure functions, without assuming any separation condition.
As some of the mappings we consider are not necessarily contractive with respect to the Euclidean metric, but contractive with respect to some other metric, for convenience we first introduce the definition of an iterated function system of essential contractions.

Definition 1.
Let  be a nonempty compact subset of R  , equipped with the Euclidean metric, and let   :  → ,  = 1, . . ., , be a finite family of mappings.If there exists a metric  on  such that all the   are contractions with respect to , then one says that {  }  =1 are essential contractions with respect to  (or simply essential contractions).In this case one calls {  }  =1 an iterated function system (IFS) of essential contractions.Some IFSs of affine mappings are not necessarily contractions with respect to the Euclidean metric but are essential contractions (see [12]).Some of the IFSs we consider in this paper are defined by matrices that are powers of a single matrix (see Example 23).They are also essential contractions.
In order to state our conditions and results, we first introduce some basic definitions and notations.Let  be a nonempty compact subset of R  , equipped with the Euclidean metric, and let   :  → ,  = 1, . . ., , be essential contractions with respect to some metric .It is well known that there exists a unique nonempty compact subset  ⊆ , called the attractor, such that (see [13,14]).The set  is independent of the metric .For such an IFS, we define with Σ 0 := {0}.For  = ( 1 , . . .,   ) ∈ Σ  , we denote by || =  the length of  and write   :=   1 ∘ ⋅ ⋅ ⋅ ∘    ( 0 is defined to be the identity).We also denote  = ( we obtain the following sets of inequalities: These inequalities will be used repeatedly. We make a few remarks concerning these sets of indices or mappings.First, since  can be greater than 1, for ( 1 ,  2 , . ..) ∈ Σ ∞ , it is possible that there are more than one prefix  = ( 1 , . . .,   ) ∈ Σ * such that  ∈ I  .However, in view of Remark 3, the number of such prefixes must be finite.Second, it is possible that   =    for distinct ,   ∈ Σ * ; we identify such   and    .Last, for IFSs of contractive similitudes, I  = I *  () and so A  = A *  ().In general, however, they need not be the same.Definition 4. Let  ⊂ R  be a compact subset with  ∘ ̸ = 0 and let   :  → ,  = 1, . . ., , be bi-Lipschitz essential contractions.We say that {  }  =1 has the logarithmic distortion property (LDP) if there is a constant  > 0 such that (ii) for any  ∈ Φ, () intersects at least one   ().
Denote the class of all packing families of A  with respect to  by P  (), and denote the class of all packing families of A *  () by P *  ().
Remark 9.The above   () and   () are similar to those in [5], but they are different, since packing families are used here.
The functions   ,   , and   depend on .However, they have a common zero (independent of ), as is shown in the following main theorem.
In the following example, Theorem 11 is used in computing the dimension of the attractor.Although the dimension of the self-affine set can also be computed by the method by Bárány [11], the method we use appears to be simpler (see Section 4).
Remark 13.Theorem 11 makes dimension computation easier.The computation would be very complicated if we use Theorem 10 or the definitions of the Hausdorff or box dimensions.
The rest of this paper is organized as follows.In Section 2 we establish some basic properties of the topological pressure functions.Section 3 is devoted to the proof of the main theorems.In Section 4 we illustrate our main results by some examples.

Properties of Topological Pressures
In this section we prove some basic properties of the topological pressure functions.Let {  }  =1 be an IFS of bi-Lipschitz essential contractions on a compact subset  ⊂ R  .The following inequalities will be used repeatedly, for any  ⊆ , and any  ∈ Σ * : We first state some basic properties of the topological pressures, without assuming LDP.The proof of the following proposition is similar to that of [5, Proposition 2.3]; we will only give an outline.Proposition 14.Let , {  }  =1 , and  satisfy the hypotheses of Theorem 10 and let  ∈ (0, 1).Then both   () and   () are real-valued, strictly decreasing, and continuous functions on R that tend to −∞ and ∞ as  tends to ∞ and −∞, respectively.Moreover,   (0) ≥   (0) ≥ 0 and   () is convex on R.
By using the inequalities in (15), we can prove the following proposition in the same way.We now state some simple consequences of LDP.
Lemma 16.Assume the same hypotheses on , {  }  =1 , and  as in Theorem 10.Let 0 <  0 < 1,   and let   be defined as in (4), and let  > 0 be defined as in Definition 4. The following hold.
(a) There is a constant  1 > 0 such that Proof.(a) By Definition 4, we have Hence /( 1 | ln |  ) ≤   , and the conclusion follows.
The following proposition follows easily from Lemma 17 and its proof.
Thus, the definitions of   ,   , and   are independent of the choices of the invariant open set  ⊂  and the packing families.Furthermore, in Definition 8,   can be replaced by   .
In the following, the open set  will not be mentioned unless it is necessary.
In order to obtain a lower estimate for the Hausdorff dimension in Theorem 10, we need the mass distribution principle (Lemma 19) and Proposition 20 below.

Proof of the Main Theorems
This section is devoted to the proofs of the main theorems.
Proof of Theorem 10.In order to apply Proposition 20, we first use Proposition 18 to require, in addition, that  ⊇ .Let  and  be the zeroes of   () and   (), respectively.By Proposition 18,  and  are independent of the choice of the packing family.Proposition 14 implies that both   (0) and   (0) are real numbers.We first prove Substituting  =   and  =  , into (21) gives Hence by using Proposition 18 and the fact that   (0) and   (0) are real numbers, we have Equation ( 44) now follows from the equalities   (0) = − ln  and   (0) = − ln .
Next, we prove Suppose, on the contrary,  > dim  .Then there exists  such that dim   <  < .We will derive a contradiction.By (44),   () = ( − ) ln  > 0. Choose a sequence of packing families {  , }   =1 of A   with respect to , where  > 0. Then by using (38), there exists an integer  > 0 such that To this end we first prove  ≥ dim  .Let  > .
Finally, we prove dim   = dim  .Let  be as above.For any  > 0 and  ∈ (0, 1), choose a packing family {  , }   =1 of A   with respect to .Let and let   := #B  , the cardinality of B  .According to (52), we define Then (52) implies Therefore, Remark 22.For IFSs consisting of  1 conformal contractions and satisfying BDP and WSC (see [3]), Theorem 1.1 of [5] gives a method for computing dim   by solving the equation () = 0. We remark that, in computing the function (), the sum in the definition of () is over distinct maps, and thus in numerical computations the following two types of mistakes may occur: (a)   ̸ =   , but numerical approximations show   =   ; (b)   =   , but numerical approximations show   ̸ =   .
In view of Corollary 21 and the definition of packing families, the formula dim   = lim  → ∞ (ln   / ln ) is numerically much more stable.
Proof of Theorem 11.In view of (15), we have Thus, by using (37) and (38) we need only prove For any  ∈ (0, ] and any two packing families {  1 , . . ., where the first and fourth inequalities follow from ( 22) and (21), respectively, the second, fifth, and last ones follow from (15), and the third one follows from the definition of I *  ().We assume, without loss of generality, that  2 ≥ Therefore, there is a constant  3 > 0 such that By interchanging the roles of the two packing families, it can be proved in the same way that there exist constants  4 > 0 and  5 > 0 such that It now follows from these inequalities and Theorem 10 that (61) holds.The proof is complete.

Examples
In this section we illustrate the applications of our results by some examples.
we see that {  }  =1 is an IFS of essential contractions.For the matrix , by the Jordan decomposition theorem, there is an invertible complex matrix  such that where each   is a Jordan block with all diagonal entries being the same and equal to 1 in modulus.Since all eigenvalues of  are 1 in modulus, using (71), it is not difficult (see, e.g., [5]) to prove that

Proposition 18 .
Let  and {  }  =1 satisfy the hypotheses of Theorem 10.Then for any nonempty invariant open set  ⊂  with L  () > 0, and any sequence of packing families {  , }   =1 of A   , one has
Denote the new IFS {  , }   =1 and let   be its attractor.Then this IFS satisfies OSC with  being an OSC set.Since  ⊇ , by applying Proposition 20 to the new IFS {  }   =1 and noticing that   ⊆ , we get dim   ≥ dim  (  ) ≥ , a contradiction.Thus dim   ≥ .