Self-similar sets with optimal coverings and packings

J. Math. Anal. Appl. 334 (2007) 1088–1095 www.elsevier.com/locate/jmaa Self-similar sets with optimal coverings and packings ✩ Marta Llorente a,∗ , Manuel Morán b a Departamento de Análisis Económico: Economía Cuantitativa, Universidad Autònoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain b Departamento de Análisis Económico I: Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain Received 5 October 2006 Available online 8 January 2007 Submitted by P. Koskela Abstract We prove that if a self-similar set E in Rn with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the H s -density on the class of arbitrary subsets of Rn and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of Rn , an optimal almost covering of E by balls and an optimal packing of E.  2007 Elsevier Inc. All rights reserved. Keywords: Hausdorff measure; Packing measure; Self-similar sets; Densities; Optimal coverings 1. 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 ⊂ Rn is defined as ✩ This research has been supported by the Ministerio de Educacion y Ciencia, research project MTM2006-02372. * Corresponding author. E-mail addresses: m.llorente@uam.es (M. Llorente), m.moranca@ccee.ucm.es (M. Morán). 0022-247X/$ – see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.01.003 M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095  H s (A) = lim inf δ→0 ∞  1089  |Ui |s , i=1 where | · | stands for the (Euclidean) diameter of subsets of Rn and the infimum is taken over all δ-coverings of A,  i.e., countable collections {Ui } of subsets of Rn with diameter smaller than δ s and such that A ⊂ ∞ i=1 Ui . The spherical Hausdorff measure Hsph (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  ∞   s s P0 (A) = lim sup |Bi | : |Bi |  δ, i = 1, 2, 3, . . . , δ→0 i=1 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   ∞ ∞   s s P (A) = inf (1) Ui . P0 (Ui ): A ⊂ i=1 i=1 These definitions are sometimes awkward to work with and, in fact, determining the exact value of some metric measure for general subsets of Rn 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 Rn . 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, ∞ U and H s (E) = following the notation in that article), i.e., a covering such that E ⊂ i i=1 ∞ s i=1 |Ui | (here s is the Hausdorff dimension of E). 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 almost ∞ they sare optimal  coverings s (E) = s (E − ∞ U ) = 0, |U | , but H rather than optimal coverings, in the sense that H i i=1 i=1 i  U (see Corollary 3.4 below). The issue of the existence of optimal almost instead of E ⊂ ∞ i i=1 coverings was raised independently in [8]. 1090 M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 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. 2. 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. Definition 2.1. Let M := {1, 2, . . . , m} and Ψ = {fi }i∈M be a system of contracting similitudes of Rn . The self-similar set generated by Ψ is the unique non-empty compact set which satisfies E = SΨ (E), where SΨ is the set mapping defined by  SΨ (X) = fi (X) i∈M for X ⊂ Rn satisfying the following assumptions (A1) Ψ satisfies the strong open set condition (SOSC), i.e. there exists a bounded open set O ⊂ Rn , with O ∩ E = ∅ and such that fi (O) ⊂ O for all i ∈ M and fi (O) ∩ fj (O) = ∅ for i, j ∈ M with i = j . (A2) m  2 and Rn 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 m  ris = 1 (2) i=1 with 0 < ri < 1 being the contraction ratio of fi for i ∈ M. Moreover, E is easily seen to be an s-set, i.e., 0 < H s (E) < ∞. Given j = j1 j2 . . . jk ∈ M k , we write fj for the similitude fj := fj1 ◦ fj2 ◦ · · · ◦ fjk with contraction ratio rj = rj1 rj2 . . . rjk , k ∈ N and for A ⊂ Rn , Aj = fj (A). The self-similar set E is the image of the space of codes M =: M ∞ under the projection mapping π : M → E given by π(i) = ∞  Ei(k) , i = i1 i2 . . . ∈ M, k=1 where i(k) denotes the curtailment i1 i2 i3 . . . ik ∈ 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  Ei ∩ Ej . θ= i=j M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 1091 The natural probability measure on E or Hausdorff normalized measure µ is defined for Borel subsets of Rn by µ(A) = H s (E)−1 H s (A ∩ E). 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) = ris , on the cylinder sets i ∈ M k , so that µ(Ei ) = ris . Given a Radon measure µ, we define the inverse density functions fs : Rn × R+ → [0, ∞] and f˜s : C → [0, ∞], respectively, by fs (x, r) = (2r)s µ(B(x, r)) (3) and |A|s , (4) f˜s (A) = µ(A) where C stands for the class of compact non-empty convex sets in Rn and B(x, r) is the closed Euclidean ball, i.e., B(x, r) = {x ∈ Rn : 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 fs and f˜s given in (4) and (3). Theorem 2.2. (See [5, Corollary 7 and Theorem 10].) Let E be a self-similar set in Rn satisfying the strong open set condition for the open set O and with Hausdorff dimension s. Let µ be the natural probability measure, or Hausdorff normalized measure on E. Then (i) H s (E) = inf{|A|s /µA: A is a convex polytope}, s (E) = inf{(2r)s /µB(x, r): x ∈ Rn }, (ii) Hsph (iii) P s (E) = sup{(2r)s /µB(x, r): x ∈ E, B(x, r) ⊂ O}. 3. Optimal almost coverings and packings: The totally disjointed case Definition 3.1. Let E be the self-similar set generated by the system of contracting similitudes Ψ = {f1 , f2 , . . . , fm }. We say that the  system Ψ = {f1 , f2 , . . . , fm } satisfies the strong separation condition (SSC) if the union E = m i=1 fi (E) is disjoint. 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 ⊂ Rn , we shall write Ar := {x ∈ Rn : dist(x, A)  r} for the closed r-neighborhood or r-parallel body of A, and (A)r := {x ∈ Rn : dist(x, A) < r} for the open r-parallel body of A. Assume that a system Ψ = {f1 , f2 , . . . , fm } of contracting similitudes of Rn satisfies SSC, and let c := min dist fi (E), fj (E) . i,j ∈M (5) Then one can easily check that O := (E) 2c (6) 1092 M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 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 ⊂ Rn and i ∈ M we have fi fi−1 (A) ∩ E ⊂ A ∩ E. (7) Suppose Ψ satisfies the SSC and assume that, for some A ⊂ Rn and some i ∈ M, ∅ = A ∩ E ⊂ Ei holds. Let x ∈ A ∩ fi (E) and x = fi (y) for some y ∈ fi−1 (A) ∩ E. Then x ∈ fi (fi−1 (A) ∩ E), so A ∩ E ⊂ fi (fi−1 (A) ∩ E). This and (7) show that for any A ⊂ Rn such that ∅ = A ∩ E ⊂ Ei for some i ∈ M we have A ∩ E = fi fi−1 (A) ∩ E . (8) In particular, taking A = B(x, r) B(x, r) ∩ E = fi B fi−1 (x), r/ri ∩ E (9) whenever B(x, r) ∩ E ⊂ Ei . We now state the main results to be proved in this paper. Theorem 3.3. Let E be the invariant set of the system Ψ satisfying SSC with dimH E = s and |E| = R and let µ be the natural probability measure, or Hausdorff normalized measure on E. Then, (i) H s (E) = min{|A|s /µA: A ∈ C with c  |A|  R}, with c as in (5), s E = min{(2r)s /µB(x, r): x ∈ E and (ii) Hsph R (iii) P s (E) = max{(2r)s /µB(x, r): x ∈ E, crmin 2 c 2  r  2R},  r  R}, where rmin = min{ri : i ∈ M}. Proof. In order to prove (ii), let L := (x, r) ∈ Rn × (0, ∞): B(x, r) ∩ E ⊂ fi (E) for some i ∈ M . (10) Observe that (9) implies that for any (x, r) ∈ L there exists i ∈ M such that fs (x, r) = = H s (E)(2r)s H s (E)(2r)s (2r)s = s = µB(x, r) H (E ∩ B(x, r)) H s (fi (E ∩ B(fi−1 (x), r/ri ))) H s (E)(2r)s (2r1 )s = = fs (x1 , r1 ), ris H s (E ∩ B(x1 , r1 )) µB(x, r1 ) (11) where r1 = rri and x1 = fi−1 (x). We want to prove that inf Rn ×(0,∞) fs (x, r) = inf ER ×[ 2c ,2R] fs (x, r). (12) Should this not be the case, there would exists (x0 , r0 ) ∈ / ER × [ 2c , 2R] such that fs (x0 , r0 ) < inf ER ×[ 2c ,2R] fs (x, r). (13) M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 1093 We will show that this cannot happen. Indeed, let x ∈ / ER . Then by the definition of fs we have that fs (x, r)  (2r)s  (2R)s if r  R and fs (x, r) = ∞ otherwise. Moreover, for any xE ∈ E ⊂ ER , fs (xE , R) = (2R)s , thus (14) fs (xE , R)  fs (x, r) for all x ∈ / ER . Hence, (13) implies x0 ∈ ER . We now claim that if (13) holds and r0 < 2c then (x0 , r0 ) ∈ L. This property is a consequence of (5) if x0 ∈ E. In order to see that it also holds for x0 ∈ / E, note that E ∩ B(x0 , r0 ) = ∅ and thus Ei ∩ B(x0 , r0 ) = ∅, for some i ∈ M. The claim holds because if there exist y, z ∈ E ∩ B(x0 , r0 ) with y ∈ Ei and z ∈ Ek , with i = k, then dist(Ei , Ek )  dist(y, z)  dist(y, x0 ) + dist(x0 , z) < c 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 x0 ∈ ER and r0 < 2c . Let i ∈ M ∞ r0 −1 . It is easy to see be the address of x, i.e. π(i) = x, and k ∈ N. Let xk = fi(k) (x0 ) and ρk = ri(k) c c that ρk → ∞ and in particular for some k0 ∈ N, rk0  2 and rk0 −1 < 2 . Therefore, if we show that xk ∈ ER for all k  k0 , by our claim above, we can apply repeatedly (11) and get that fs (x0 , r0 ) = fs (xk , rk ) ∀k  k0 . (15) / ER . Suppose, on the contrary, that there exists k̃ < k0 such that xk ∈ ER for all k < k̃ and xk̃ ∈ Then (15) applied to (xk̃ , rk̃ ) together with (14) yields the desired contradiction. Thus (13) with x0 ∈ ER implies r0  2c . Assume now (13) with x0 ∈ ER and r0 > 2R. In this case, E ⊂ B(x0 , r0 ) and therefore, fs (x0 , r0 ) > (4R)s = fs (x0 , 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 fs (x, r) is a continuous function on E × (0, ∞). Therefore, the infimum given by (3) is attained on the compact set ER × [ 2c , 2R]. 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 Rn is closed with respect to the Hausdorff distance dH , dH (E, F ) = inf{δ: E ⊂ Fδ and F ⊂ Eδ }. We are going to use the characterization H s (E) = inf |A|s /µA: A ∈ C (16) 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 L̃ := A ∈ C: A ∩ E ⊂ fi (E) for some i ∈ M , (8) implies that for any A ∈ L̃ there exists i ∈ M such that (|A|/ri )s |A|s = = f˜s fi−1 (A) = f˜s (D), f˜s (A) = µA µ(fi−1 (A)) (17) where D = fi−1 (A). Therefore, the same argument used in (ii) proves that we can restrict (16) to the class of convex sets A ∈ C with |A|  2c . The upper bound of |A| follows directly if we note 1094 M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 that for any convex set A with |A| > |E| = R we have f˜s (A)  |A|s > R s = f˜s (conv E), where conv(·) stands for the convex hull of subsets of Rn . We now check that f˜s is a lower semicontinuous function of C with the Hausdorff distance. Let Cn 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 n0 such that Cn ⊂ Cδ for n > n0 , and |Cn | → |C| as n → ∞. Therefore |C|s |Cn |s = → f˜s (C) as δ → 0, lim inf f˜s (Cn )  lim inf n→∞ n→∞ µ(Cδ ) µ(Cδ ) which shows that f˜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). By the continuity of fs (x, r) the assertion in part (iii) is proved if we show that the supremum in Theorem 2.2, part (iii) is attained by fs in E  × [ cr2min , R]. Let B(x, r) ⊂ O with x ∈ E. Then obviously r  R because otherwise B(x, r) ⊂ m i=1 (Ei )c/2 and (Ei )c/2 are open and disjoined, thus if r  R, B(x, r) would intersect each Ei , in contradiction with its connectivity. On the other hand, if r < cr2min , and i = i1 i2 i3 . . . is the address of x, then (x), rri−1 ). If B(x, r) only can intersect Ei1 and, proceeding as in (11) we get fs (x, r) = fs (fi−1 1 1 crmin crmin −1 crmin crmin −1 −1 −1 c rri1  2 we are done since r < 2 implies rri1 < 2 ri1  2 < R. If rri1  2 we can repeat this procedure until the radius exceeds cr2min . This completes the proof of part (iii). ✷ We now state the results concerning the existence of optimal coverings and packings. Corollary 3.4. If the hypotheses of Theorem 3.3 hold then s (i) There exists covering of E, i.e. a collection of sets {Ui } such that  an optimal H -almost ∞ s (E) = s H s (E − ∞ U ) = 0 and H |U i| . i=1 i i=1 s -almost covering of E, i.e. a collection of Euclidean balls {B } (ii) There exists an optimal Hsph i s (E − ∞ B ) = 0 and H s (E) = ∞ |B |s . such that Hsph i i i=1 i=1 sph (iii) There exists an optimal packing of E, i.e. a collection {Bi } of disjointed balls centered at  s E and such that P s (E) = ∞ i=1 |Bi | . Proof. By Theorem 5, part (i) in [5], associated with f˜s (U ) = α there exists  with any subset U ∞ s an almost covering {Ui } such that H s (E − ∞ U ) = 0 and i=1 i i=1 |Ui | = α. By Corollary 7, s ˜ part (i) in the quoted reference H (E) coincides with the infimum of fs on the class of convex polytopes, which in turn clearly coincides with the infimum of f˜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. ✷ 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 M. Llorente, M. Morán / J. Math. Anal. Appl. 334 (2007) 1088–1095 1095 ∞   k k k {Ei }i∈J with J ⊂ ∞ k=1 J , where J is k=1 M and such that E − B = i∈J Ei . Let I = the set of all words that can be obtained by concatenation of k words in J . Then the collection of balls {Bi }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. References [1] E. Ayer, R. Strichartz, Exact Hausdorff measure and intervals of maximal density for Cantor sets, Trans. Amer. Math. Soc. 351 (9) (1999) 3725–3741. [2] D. Feng, J.G. Hua, Some relations between packing premeasure and packing measure, Bull. London Math. Soc. 31 (1998) 665–670. [3] J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (5) (1981) 713–747. [4] P. Mattila, On the structure of self-similar fractals, Ann. 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