Research Article | Open Access
Dimensions of Fractals Generated by Bi-Lipschitz Maps
On the class of iterated function systems of bi-Lipschitz mappings that are contractions with respect to some metrics, we introduce a logarithmic distortion property, which is weaker than the well-known bounded distortion property. By assuming this property, we prove the equality of the Hausdorff and box dimensions of the attractor. We also obtain a formula for the dimension of the attractor in terms of certain modified topological pressure functions, without imposing any separation condition. As an application, we prove the equality of Hausdorff and box dimensions for certain iterated function systems consisting of affine maps and nonsmooth maps.
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 and the bounded distortion property holds (see [1–3]). For IFSs of conformal contractions, the weak separation condition is also assumed (see ). 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.  always lies between these two values. Motivated by this observation, McLaughlin  and Falconer  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 -smoothness of the maps of the IFS, the bounded distortion property (BDP), and the weak separation condition (WSC), Lau et al.  proved the equality of the two dimensions for self-conformal sets. Under these conditions, the authors  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 smooth, Mauldin and Urbański  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  and Falconer . 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  (see also  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 . 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 -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 , equipped with the Euclidean metric, and let , , 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 are essential contractions with respect to (or simply essential contractions). In this case one calls 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 ). 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 , equipped with the Euclidean metric, and let , , 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 . For , we denote by the length of and write ( is defined to be the identity). We also denote simply by and let be the word obtained from by deleting its last alphabet.
Let denote the Euclidean norm. Define
For any , by writing we obtain the following sets of inequalities: These inequalities will be used repeatedly.
Assumption 2. Throughout this paper we assume that ; equivalently, , , are bi-Lipschitz.
Remark 3. It is possible that . Since all , , are essential contractions, converges uniformly to as tends to infinity. As a consequence, we also have .
For any , we let , , , , , , and denote, respectively, the Hausdorff dimension, packing dimension, box dimension, -dimensional Hausdorff measure, -dimensional Lebesgue measure, Euclidean diameter, and interior of . Given an IFS on , a nonempty set (not necessarily open) is said to be invariant if . We say that is open if it is open in the relative Euclidean topology of .
Fix an invariant set and let . Define
We make a few remarks concerning these sets of indices or mappings. First, since can be greater than , for , it is possible that there are more than one prefix such that . However, in view of Remark 3, the number of such prefixes must be finite. Second, it is possible that for distinct ; we identify such and . Last, for IFSs of contractive similitudes, and so . In general, however, they need not be the same.
Definition 4. Let be a compact subset with and let , , be bi-Lipschitz essential contractions. We say that has the logarithmic distortion property (LDP) if there is a constant such that
Remark 5. In the above definition, we do not assume that the maps of the IFS are differentiable. Besides this, if satisfies BDP, then there is a constant such that for all and . 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 satisfy the hypotheses of Definition 4, be a bounded invariant set that is open in the relative topology of with , and be a finite subset of . One calls a finite subcollection a packing family for with respect to if the following conditions are satisfied: (i) are pairwise disjoint;(ii)for any , intersects at least one .
Denote the class of all packing families of with respect to by , and denote the class of all packing families of by .
Example 7. Let , , , and . Then , , and are three packing families of .
Definition 8. Let , , satisfy the hypotheses of Definition 4 and fix . Define We call (resp., ) the lower (resp., upper) topological pressure function (with scale ). If , we denote the common function by and call it a topological pressure function (with scale ). Note that is fixed and is the variable of the functions and .
Remark 9. The above and are similar to those in , 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.
Theorem 10. Let , , satisfy the hypotheses of Definition 6. Fix any and any sequence of packing families , where . Then (a) for all ;(b).
For some applications, it is easier to treat than and . Similar to Definition 8, we define We have the following theorem.
Theorem 11. Let , , satisfy the hypotheses of Definition 6. Fix any and any sequence of packing families , where . Then
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 , the method we use appears to be simpler (see Section 4).
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.
2. Properties of Topological Pressures
In this section we prove some basic properties of the topological pressure functions. Let be an IFS of bi-Lipschitz essential contractions on a compact subset . The following inequalities will be used repeatedly, for any , and any :
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 , , and satisfy the hypotheses of Theorem 10 and let . Then both and are real-valued, strictly decreasing, and continuous functions on that tend to and as tends to and , respectively. Moreover, and is convex on .
Proof. Since uniformly as , there is an integer such that for all such that . Let , , and . Write with . Then we have for some constant . Hence (7) implies
It follows that
Using (16)–(18) and a similar derivation as that in [5, Proposition 2.3] gives
Hence and are real-valued, . Moreover, since , we have and .
Next, for any , by using (16)–(18), we get Exactly the same inequalities hold for . Therefore, and are strictly decreasing and continuous on . The convexity of follows from Hölder’s inequality.
By using the inequalities in (15), we can prove the following proposition in the same way.
Proposition 15. Under the same assumptions of Proposition 14, both and are real-valued, strictly decreasing, and continuous functions on that tend to and as tends to and , respectively. Moreover, and is convex on .
We now state some simple consequences of LDP.
Lemma 16. Assume the same hypotheses on , , and as in Theorem 10. Let , and let be defined as in (4), and let be defined as in Definition 4. The following hold.(a)There is a constant such that (b)There exists a constant such that
Proof. (a) By Definition 4, we have
Hence , and the conclusion follows.
(b) As mentioned in Remark 3, the IFS is not necessarily contractive in the Euclidean metric. Nevertheless, since as , there exists some such that when . For any with , let with ; that is, can be decomposed into parts, with of them having length and one of them having length . Hence . Taking logarithm, we have Hence the set , is bounded. Let be a constant such that
Let . The definition of shows that there is a decomposition with so that (since it is possible that for some ). Substituting and into (21) yields We need only prove (22) for small , since for any given the set is finite. Without loss of generality, we can assume for any . Using (8) and the facts that and , we have . As , we have and thus . As , we get From (25) and (28), we see that there exists some constant such that . Combining the above estimates, we get That is, . Similarly, we can show that . Thus, (22) holds and this completes the proof.
For IFSs satisfying LDP, the definitions of the topological pressures are independent of the choice of the invariant open set and the packing families. To see this we need the following lemma.
Lemma 17. Assume that and satisfy the hypotheses of Theorem 10, are fixed, and , are nonempty invariant open subsets of with and . Then there is a constant , depending only on , and , such that for any and any two packing families and ,
Proof. Let , . By using the definition of , the disjointness of , and the equality , we get
Let be a ball with radius and center . Then contains a ball with radius and center . For each , , and both and have diameters bounded above by . Let . Then for . Therefore, (31) implies
By using the inequality from Lemma 16, we get and so with . Interchanging the roles of the two packing families and using the same argument, we get . Hence Also, by Lemma 16, we have and . Hence for all , By symmetry, . Therefore, The conclusion for the case follows by letting . The proof for the case is similar; we omit the details.
The following proposition follows easily from Lemma 17 and its proof.
Proposition 18. Let and satisfy the hypotheses of Theorem 10. Then for any nonempty invariant open set with , and any sequence of packing families of , one has 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.
Lemma 19 (see, e.g., [13, Theorem 4.9]). Let , a positive Borel measure on with , and . If there is a constant such that for any , then .
Recall that an IFS on satisfies the open set condition (OSC) if there exists a nonempty bounded invariant open (in the relative topology of ) set , called an OSC set, such that and for all .
The following result is similar to that of [15, Theorem 10.3] where the strong separation condition is used; we include a proof for convenience.
Proposition 20. Let be the attractor of an IFS satisfying the hypotheses of Theorem 10. If OSC holds with an OSC set satisfying and , then .
Proof. Choose such that . Let and let be the invariant probability measure associated with the weights (see ); that is, .
For any and sufficiently small , let Then OSC implies that are disjoint and the fact that implies that . The definition of implies that (see (15) also), and Lemma 16 implies . Hence Thus there is a constant such that Combining OSC and the fact that gives Using , (41), together with the fact that , we get Since , . Hence inequality (43) implies for all . The conclusion follows by using Lemma 19.
3. 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 and are real numbers.
We first prove
Substituting and into (21) gives Hence Since by using Proposition 18 and the fact that and are real numbers, we have Equation (44) now follows from the equalities and .
Next, we prove Suppose, on the contrary, . Then there exists such that . We will derive a contradiction.
By (44), . Choose a sequence of packing families of with respect to , where . Then by using (38), there exists an integer such that
Denote the new IFS by and let be its attractor. Then this IFS satisfies OSC with being an OSC set. Since , by applying Proposition 20 to the new IFS and noticing that , we get , a contradiction. Thus .
Now, we prove To this end we first prove . Let . Then and by the fact that (Proposition 14). For every integer , choose a sequence of packing families of with respect to .
For any , there is at least one such that . Choose . Since (14) implies , , is contained in the ball with radius . Since , we have Hence is a -cover of . By (48), Hence, for infinitely many integers . Therefore, and thus . Since is arbitrary, we conclude that .
Since , by combining this with (49), we get . Equation (51) now follows by substituting this into (44).
Finally, we prove . Let be as above. For any and , choose a packing family of with respect to . Let and let , the cardinality of . According to (52), we define Then (52) implies . If , then is contained in . Since are disjoint, we have Therefore, Hence It follows immediately that . Since , the proof is complete.
A similar argument shows the following corollary.
Corollary 21. Assume the same hypotheses of Lemma 17. Also, for any given , and for , assume that the following conditions hold.(1)For any , there is at least one such that intersects .(2)For any , there are at most maps such that .
Then Theorem 10 holds by replacing the packing families with , .
Remark 22. For IFSs consisting of conformal contractions and satisfying BDP and WSC (see ), Theorem 1.1 of  gives a method for computing by solving the equation . 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 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 and any two packing families and . Similar to the proof of Lemma 17, let and .
Using LDP, we have 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 . We assume, without loss of generality, that . It follows that
By using (22) we see that is contained in a ball with center in and radius . Hence it follows from (22) again that Therefore, there is a constant such that
By interchanging the roles of the two packing families, it can be proved in the same way that there exist constants and such that
Now, for any two sequences and , by combining (63)–(66), we have Similarly, It now follows from these inequalities and Theorem 10 that (61) holds. The proof is complete.
In this section we illustrate the applications of our results by some examples.
Example 23. Let be a real matrix, , and let , , be an IFS with , and . Assume that all eigenvalues of have moduli . Then (a)LDP is satisfied and thus the conclusions of Theorems 10 and 11 hold;(b)BDP holds if and only if there is a real invertible matrix and a real orthogonal matrix such that . In this case, the attractor is similar to a self-similar set generated by the IFS with replaced by ;(c)if OSC holds, then is the unique solution of the equation
Proof. Letting and using the following norm in :
we see that 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 in modulus.
(a) We need only show that the IFS satisfies LDP. Let Since all eigenvalues of are in modulus, using (71), it is not difficult (see, e.g., ) to prove that for some constant .
For any , let for some with . Then the and defined in (4) become