Self-similar sets with optimal coverings and packings

Introduction

Hausdorff and packing measures, which we shall refer to as metric measures, are the natural analog, in the study of the geometric properties of fractal sets, to the notions of length, surface or n-dimensional volume in the study of smooth manifolds. For each s 0, the s-dimensional Hausdorff measure of A ⊂ R n is defined as

where | · | stands for the (Euclidean) diameter of subsets of R n and the infimum is taken over all δ-coverings of A, i.e., countable collections {U i } of subsets of R n with diameter smaller than δ and such that A ⊂ ∞ i=1 U i . The spherical Hausdorff measure H s sph (A) is obtained if in the above definition the class of covering sets is restricted to Euclidean balls, and the packing measure is defined by means of a two-step definition: first the packing premeasure is defined by

where the supremum is taken over all δ-packings of A, i.e., countable collections of disjointed Euclidean balls centered at A and with diameter smaller than δ, and the packing measure is then given by

These definitions are sometimes awkward to work with and, in fact, determining the exact value of some metric measure for general subsets of R n is a cumbersome task. In this paper we continue the study started in [5] of what can be said in this regard if we limit our attention to the particular class of self-similar sets, whose geometric structure is highly ordered and better understood than other fractal sets. In the quoted reference it was shown the relationship, in a self-similar setting, between the exact value of metric measures and the sets of optimal density, for instance, the packing measure is given by the inverse density of a ball with a maximum inverse density (see Theorem 2.2 below for the full statement). An antecedent of these results can be seen in [1], where the relationship between sets of maximal density and the exact value of the Hausdorff measure was analyzed in the real line and there also was shown that the method cannot be generalized to higher dimensions. See also [9] for the case of the Hausdorff measure in R n .

In this paper we pursue the above issue showing that if a self-similar set E satisfies the strong separation condition (SSC), i.e., its constituent parts are disjointed (see Definition 3.1), then there exist a set of minimal inverse density, a ball of minimal inverse density and a ball of maximal inverse density. Among the self-similar sets satisfying SSC are the ternary Cantor set or the attractor for the Smale horseshoe dynamics [7], which is in turn an universal model for the structure of attractors of hyperbolic dynamical systems [6]. In [1] it is shown an example of selfsimilar set in R that satisfies the strong open set condition (SOSC) (see Definition 2.1(A1)) and that does not posses any interval of minimal inverse density. Notice that SOSC is the standard separation condition needed to make amenable the analysis of self-similar geometry. Therefore, after the results in this paper, the search of additional conditions under which a set that satisfies SOSC possesses sets of optimal density is left as the main problem in this field.

In [10] it was asked which self-similar sets E admit an optimal covering (best covering, following the notation in that article), i.e., a covering such that

In [5] it is shown that the existence of a set of optimal density permits us to construct coverings or packings which give the exact value of the corresponding metric measure. But in the case of the coverings they are optimal almost coverings rather than optimal coverings, in the sense that H s ( Corollary 3.4 below). The issue of the existence of optimal almost coverings was raised independently in [8].

We show in this paper (Corollary 3.4) that if a self-similar set satisfies SSC then there exist optimal almost coverings for the Hausdorff and spherical Hausdorff measures. It is remarkable that, in spite of the apparently more awkward, two-step definition of the packing measure given above, by a little known result by L. Feng [2] the second step (1) may be omitted if the measured set is (as in our case) a compact set. Then, by Corollary 3.4 there exits an optimal packing. This confirms the observation in [5] that, in the self-similar setting, the packing measure may be more easy to handle than the Hausdorff or spherical Hausdorff measures.

Preliminaries: Self-similar sets and Hausdorff normalized measure

We now introduce the notations and basic facts on self-similar sets which are used in Section 3.

where SΨ is the set mapping defined by

for X ⊂ R n satisfying the following assumptions

A2) m 2 and R n is the smallest linear manifold that contains E (see [3]).

Assumption (A1) ensures that the Hausdorff dimension of E coincides with its similarity dimension, defined as the unique real number s such that

with 0 < r i < 1 being the contraction ratio of f i for i ∈ M. Moreover, E is easily seen to be an

The self-similar set E is the image of the space of codes M =: M ∞ under the projection mapping π : M → E given by

where i(k) denotes the curtailment i 1 i 2 i 3 . . . i k ∈ M k of i. The mapping π projects M onto E, so that for each x ∈ E there exists some i ∈M with π(i) = x. Such i is said to be an address of x.

Let θ ⊂ E be the overlapping set of E defined by

The natural probability measure on E or Hausdorff normalized measure μ is defined for Borel subsets of R n by

It is well known that under assumption (A1), H s (θ ) = 0, and then π : (M, ν) → (E, μ) is an isomorphism of measure spaces, where ν is the product measure on M, defined by ν(i) = r s i , on the cylinder sets i ∈ M k , so that μ(E i ) = r s i . Given a Radon measure μ, we define the inverse density functions f s :

andf

where C stands for the class of compact non-empty convex sets in R n and B(x, r) is the closed Euclidean ball, i.e., B(x, r) = {x ∈ R n : d(x, y) r}, with d being the Euclidean distance.

In [5] Morán proved that the packing, the Hausdorff and the spherical Hausdorff measures of a self-similar set can be computed by finding the supremum or the infimum, under suitable classes of sets, of the inverse density functions f s andf s given in (4) and (3).

In this case the invariant set E is totally disconnected in the Euclidean metric.

It is easy to see that under SSC the overlapping set is empty and each point has an unique address.

Given r > 0, and A ⊂ R n , we shall write A r := {x ∈ R n : dist(x, A) r} for the closed r-neighborhood or r-parallel body of A, and (A) r := {x ∈ R n : dist(x, A) < r} for the open r-parallel body of A. Assume that a system Ψ = {f 1 , f 2 , . . . , f m } of contracting similitudes of R n satisfies SSC, and let

Then one can easily check that

satisfies the open set condition, and therefore the results in the previous section can be fully applied in the totally disconnected case.

Remark 3.2. Given any subset A ⊂ R n and i ∈ M we have

Suppose Ψ satisfies the SSC and assume that, for some A ⊂ R n and some i ∈ M,

. This and (7) show that for any A ⊂ R n such that

In particular, taking A = B(x, r)

We now state the main results to be proved in this paper. Proof. In order to prove (ii), let

Observe that (9) implies that for any (x, r) ∈ L there exists i ∈ M such that

where r 1 = r r i and

Should this not be the case, there would exists

We will show that this cannot happen. Indeed, let x / ∈ E R . Then by the definition of f s we have that f s (x, r) (2r) s (2R) s if r R and f s (x, r) = ∞ otherwise. Moreover, for any

for all x / ∈ E R . Hence, (13) implies x 0 ∈ E R . We now claim that if (13) holds and r 0 < c 2 then (x 0 , r 0 ) ∈ L. This property is a consequence of (5) if x 0 ∈ E. In order to see that it also holds for x 0 / ∈ E, note that E ∩ B(x 0 , r 0 ) = ∅ and thus

which contradicts the minimality property of the constant c. This completes the proof of our claim.

We now use our claim to get a contradiction from (13) with x 0 ∈ E R and r 0 < c 2 . Let i ∈ M ∞ be the address of x, i.e. π(i) = x, and k ∈ N. Let x k = f −1 i(k) (x 0 ) and ρ k = r 0 r i(k) . It is easy to see that ρ k → ∞ and in particular for some k 0 ∈ N, r k 0 c 2 and r k 0 −1 < c 2 . Therefore, if we show that x k ∈ E R for all k k 0 , by our claim above, we can apply repeatedly (11) and get that

Suppose, on the contrary, that there existsk < k 0 such that x k ∈ E R for all k <k and xk / ∈ E R . Then (15) applied to (xk, rk) together with (14) yields the desired contradiction. Thus (13) with x 0 ∈ E R implies r 0 c 2 . Assume now (13) with x 0 ∈ E R and r 0 > 2R. In this case, E ⊂ B(x 0 , r 0 ) and therefore, f s (x 0 , r 0 ) > (4R) s = f s (x 0 , 2R), in contradiction with (13). This completes the proof of (12).

In [4] it is proved that, under assumptions (A1) and (A2), the intersection of E with any C 1 manifold is an H s -null set, and therefore a μ-null set. This shows that f s (x, r) is a continuous function on E × (0, ∞). Therefore, the infimum given by (3) is attained on the compact set

]. This completes the proof of part (ii). The proof of (i) follows by a similar argument. Recall that the class C of all non-empty compact convex subsets of R n is closed with respect to the Hausdorff distance d H ,

We are going to use the characterization

that follows directly from [5,Corollary 7], where the infimum is taken over the smaller class of convex polytopes. As before, we observe that letting

where D = f −1 i (A). Therefore, the same argument used in (ii) proves that we can restrict (16) to the class of convex sets A ∈ C with |A| c 2 . The upper bound of |A| follows directly if we note that for any convex set A with |A| > |E| = R we havẽ

where conv(·) stands for the convex hull of subsets of R n . We now check thatf s is a lower semicontinuous function of C with the Hausdorff distance. Let C n be a sequence of compact convex sets that converges to C ∈ C in the Hausdorff metric. Then, given δ > 0, there exists a natural number n 0 such that C n ⊂ C δ for n > n 0 , and |C n | → |C| as n → ∞. Therefore

which shows thatf s is l.s.c. and the infimum in Theorem 2.2, part (i) is attained on the compact subset {A ∈ C with c |A| R} ⊂ C. This completes the proof of part (i). We now prove part (iii

Proof. By Theorem 5, part (i) in [5], associated with any subset U withf s (U ) = α there exists an almost covering {U i } such that H s (E − ∞ i=1 U i ) = 0 and ∞ i=1 |U i | s = α. By Corollary 7, part (i) in the quoted reference H s (E) coincides with the infimum off s on the class of convex polytopes, which in turn clearly coincides with the infimum off s on the class of closed convex sets. By Theorem 3.3 above, such infimum is attained in some closed convex set. This completes the proof of part (i). The same argument is valid to prove part (ii) if we replace closed convex sets with Euclidean balls. Lastly, part (iii) follows in the same way from Theorem 10 in the quoted reference and part (iii) in Theorem 3.3 above. 2

For completeness we describe the construction of optimal coverings and packings from sets of optimal density. This is an application of the self-similar tiling principle (see [5,Lemma 4]). We show, for instance, how an optimal packing can be obtained. Let B ⊂ O be a closed ball centered at E and with maximal inverse density. Then we can find a disjoint collection of cylinders {E i } i∈J with J ⊂ ∞ k=1 M k and such that E − B = i∈J E i . Let I = ∞ k=1 J k , where J k is the set of all words that can be obtained by concatenation of k words in J . Then the collection of balls {B i } i∈I together with B itself is a best packing for E. The optimal coverings from sets of minimal inverse density are obtained in the same way.