[EN] In this paper, we deal with a classical problem in Fractal Geometry consisting of the calculation of the similarity dimension of IFS-attractors.
The open set condition allows to easily calculate their similarity dimension though it depends on an external open set. We contribute
a necessary condition to reach the equality among some fractal dimensions for the natural fractal structure for IFS-attractors and the
similarity dimension. That condition, weaker than the SOSC, becomes
more representative of the attractor’s self-similar structure.Both authors are partially supported by grant No. MTM2015-64373-P. The first author also
acknowledges the partial support of grant No. MTM2014-51891-P and grant No. 19219/PI/14
from Fundaci´on S´eneca of Regi´on de Murcia.Fernández-Martínez, M.; Sánchez-Granero, MÁ. (2017). Irreducible fractal structures for Moran's theorems. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 33-40. http://hdl.handle.net/10251/127022OCS334

Fernández-Martínez, Manuel

Sánchez-Granero, Miguel Ángel

[EN] In this paper, we deal with a classical problem in Fractal Geometry consisting of the calculation of the similarity dimension of IFS-attractors.
The open set condition allows to easily calculate their similarity dimension though it depends on an external open set. We contribute
a necessary condition to reach the equality among some fractal dimensions for the natural fractal structure for IFS-attractors and the
similarity dimension. That condition, weaker than the SOSC, becomes
more representative of the attractor’s self-similar structure.Both authors are partially supported by grant No. MTM2015-64373-P. The first author also acknowledges the partial support of grant No. MTM2014-51891-P and grant No. 19219/PI/14 from Fundación Séneca of Región de Murcia.Fernández-Martínez, M.; Sánchez-Granero, MÁ. (2017). Irreducible fractal structures for Moran's theorems. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 33-40. http://hdl.handle.net/10251/127022334

RiuNet

WATS'17Workshop on Applied Topological StructuresValencia, 11-12 July 2017pp. 33 – 40Irreducible fractal structures for Moran’s theoremsM. Ferna´ndez-Mart´ınez a,1 and M.A. Sa´nchez-Granero ba Departamento de Ciencias, University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720Santiago de la Ribera (Murcia, SPAIN). (manuel.fernandez-martinez@cud.upct.es)b Departamento de Matema´ticas, Universidad de Almer´ıa, 04120 La Can˜ada de S.U. (Almer´ıa, SPAIN)(misanche@ual.es)AbstractIn this paper, we deal with a classical problem in Fractal Geometry con-sisting of the calculation of the similarity dimension of IFS-attractors.The open set condition allows to easily calculate their similarity di-mension though it depends on an external open set. We contributea necessary condition to reach the equality among some fractal di-mensions for the natural fractal structure for IFS-attractors and thesimilarity dimension. That condition, weaker than the SOSC, becomesmore representative of the attractor’s self-similar structure.Keywords: fractal structure; Hausdorff dimension; similarity dimen-sion; open set condition; IFS-attractor.MSC: 28A80.1Both authors are partially supported by grant No. MTM2015-64373-P. The first author alsoacknowledges the partial support of grant No. MTM2014-51891-P and grant No. 19219/PI/14from Fundacio´n Se´neca of Regio´n de Murcia.This work is licensed under a Creative Commons License CC BY-NC-ND 4.0Editorial Universitat Polite`cnica de Vale`ncia33M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-Granero1. IntroductionA classical problem in Fractal Geometry deals with determining under what condi-tions on the pieces of a strict self-similar set K, the equality between the similarityand the Hausdorff dimensions of K stands. In this way, a classical result con-tributed by P. A. P. Moran in the forties (c.f. [12, Theorem III]) states that underthe open set condition (OSC in the sequel), a property required to the pieces ofK to guarantee that their overlaps are thin enough, the desired equality holds.Afterwards, Lalley introduced the strong open set condition (SOSC) by furtherrequiring that the (feasible) open set provided by the OSC intersects the attractorK. The next chain of implications and equivalences stands in the case of Euclideanself-similar sets and is best possible (c.f. [14]):(1) SOSC⇔ OSC⇔ HαH(K) > 0⇒ dimH(K) = α,where HαH is the α−dimensional Hausdorff measure, dimH denotes the Hausdorffdimension, and α is the similarity dimension of K. A counterexample due toMattila allows to guarantee that the last implication in Eq. (1) cannot be inverted,in general. Accordingly, the OSC becomes only sufficient to reach the equalitybetween those dimensions. A further extension of the problem above takes placein the more general context of attractors on complete metric spaces. Schief alsoexplored such a problem and justified the following chain of implications (c.f. [15]):(2) HαH(K) > 0⇒ SOSC⇒ dimH(K) = α,i.e., the SOSC is necessary for HαH(K) > 0 and only sufficient for dimH(K) = α.Once again, the above-mentioned result due to Mattila implies that Eq. (2) isbest possible. From both Eqs. (1) and (2), it holds that the SOSC is a sufficientcondition on the pre-fractals of K leading to dimH(K) = α.In this paper, we shall make use of the concept of a fractal structure (first sketchedin [3]) to explore and characterize a novel separation property in both contexts:Euclidean attractors and self-similar sets in complete metric spaces. Such a sep-aration property, weaker than the OSC, becomes necessary to reach the equalitybetween the similarity dimension of the attractor and its Hausdorff dimension.34Irreducible fractal structures for Moran’s theorems2. Preliminaries2.1. The open set condition. We say that F = {f1, . . . , fk} (or its attractor K,as well) is under the Moran’s OSC (c.f. [12]) if there exists a nonempty open subsetV ⊆ Rd such that the images fi(V) are pairwise disjoint with all of them containedin V , called a feasible open set. The strong open set condition (SOSC) stands, ifand only if, it holds, in addition to the OSC assumptions, that V∩K 6= ∅ (c.f. [10]).Schief proved that both the OSC and the SOSC are equivalent on Euclidean spaces(c.f. [14, Theorem 2.2]).2.2. Fractal structures. Fractal structures were first sketched by Bandt andRetta in [3] and formally introduced afterwards by Arenas and Sa´nchez-Graneroto characterize non-Archimedean quasi-metrization (c.f. [1]).By a covering of a nonempty set X , we shall understand a family Γ of subsetssuch that X = ∪{A : A ∈ Γ}. Let Γ1 and Γ2 be two coverings of X . The notationΓ2 ≺ Γ1 means that Γ2 is a refinement of Γ1, i.e., for all A ∈ Γ2, there existsB ∈ Γ1 such that A ⊆ B. Moreover, Γ2 ≺≺ Γ1 denotes that Γ2 ≺ Γ1, andadditionally, for all B ∈ Γ1, it holds that B = ∪{A ∈ Γ2 : A ⊆ B}. Thus, afractal structure on X is a countable family of coverings Γ = {Γn}n∈N such thatΓn+1 ≺≺ Γn, for all natural number n. The covering Γn is called level n of Γ. Afractal structure is said to be finite if all its levels are finite coverings.Definition 1 (c.f. [2], Definition 4.4). Let F be an IFS whose attractor is K. Thenatural fractal structure on K as a self-similar set is given by the countable familyof coverings Γ = {Γn}n∈N, where Γ1 = {fi(K) : i ∈ Σ}, and Γn+1 = {fi(A) : A ∈Γn, i ∈ Σ}.3. Fractal dimensions for fractal structuresLet Γ be a fractal structure on a metric space (X, ρ). We shall define An(F ) asthe collection consisting of all the elements in level n of Γ that intersect a subset Fof X . Mathematically, An(F ) = {A ∈ Γn : A ∩ F 6= ∅}. Further, let diam (Γn) =sup{diam(A) : A ∈ Γn}, and diam (F,Γn) = sup{diam(A) : A ∈ An(F )}, as well.35M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-GraneroDefinition 2 (c.f. [5, Definition 4.2] and [7, Definition 3.2]). Assume thatdiam (F,Γn)→ 0 and consider the following expression for k = 3, 4:Hsn,k(F ) = inf{∑diam (Ai)s : {Ai}i∈I ∈ An,k(F )},where(i) An,3(F ) = {Al(F ) : l ≥ n}.(ii) An,4(F ) = {{Ai}i∈I : Ai ∈ ∪l≥nΓl, F ⊆ ∪i∈IAi,Card (I) < ∞}. Here,Card (I) denotes the cardinal of I.In addition, let Hsk(F ) = limn→∞Hsn,k(F ). By the fractal dimension III (resp.,IV) of F , we shall understand the (unique) critical point satisfying the identitydim kΓ(F ) = sup{s ≥ 0 : Hsk(F ) =∞} = inf{s ≥ 0 : Hsk(F ) = 0}.4. Moran’s type theorems under the OSCOne of the main goals in this paper is to explore some separation conditions forIFS−attractors in the context of fractal structures. It is worth pointing out thatthe main ideas contributed hereafter first appeared in [13].IFS conditions. Let (X,F) be an IFS, where X is a complete metric space, F ={f1, . . . , fk} is a finite collection of similitudes on X , and K is the IFS−attractorof F . Moreover, let Γ be the natural fractal structure on K as a self-similar set(c.f. Definition 1), and ci be the similarity ratio of fi ∈ F .All the results contributed along this paper stand under the IFS conditions above.Next, we recall the concept of similarity dimension for IFS−attractors.Definition 3. Let F be an IFS and K its attractor. By the similarity dimensionof K, we shall understand the unique solution α > 0 of the equation∑ki=1 csi = 1.In other words, the similarity dimension of K is the unique value α > 0 such thatp(α) = 0, where p(s) =∑ki=1 csi − 1.Along the sequel, α will denote the similarity dimension of an IFS−attractor. Itis worth noting that (without any additional assumption) HαH(K) < ∞ for anyIFS-attractor K (c.f. [8, Proposition 4 (i)]).36Irreducible fractal structures for Moran’s theoremsTheorem 4 (IFS). (c.f. [5, Theorem 4.20]) dim 3Γ(K) = α, and Hα3 (K) ∈ (0,∞).Moran’s Theorem (1946) (EIFS). OSC⇒ dimH(K) = α, andHαH(K) ∈ (0,∞).By a Moran’s type theorem, we shall understand a result that yields the equal-ity between a fractal dimension dim of an IFS−attractor K and its similaritydimension, namely, dim (K) = α.Corollary 5 (EIFS). (c.f. [5, Corollary 4.22]) OSC⇒ dim H(K) = dim3Γ(K) = α.Lemma 6. (c.f. [7, Proposition 3.5 (3)]) Let Γ be a finite fractal structure on ametric space (X, ρ), F be a subset of X, and assume that diam (F,Γn)→ 0. Thendim H(F ) ≤ dim4Γ(F ) ≤ dim3Γ(F ).Corollary 7 (IFS). dimH(K) ≤ dim4Γ(K) ≤ dim3Γ(K) = α.Theorem 8 (EIFS). OSC⇒ dimH(K) = dim4Γ(K) = dim3Γ(K) = α.To conclude this section, we recall two key results explored by Schief (c.f. [14, 15]).Theorem 9.(EIFS) SOSC⇔ OSC⇔ HαH(K) > 0⇒ dimH(K) = α.(IFS) HαH(K) > 0⇒ SOSC⇒ dimH(K) = α.Theorem 9 is best possible due to Mattila’s counterexample.5. Towards a necessary condition for Moran’s type theoremsIn this section, we introduce a novel separation condition for each level of thenatural fractal structure Γ that any IFS−attractor can be endowed with (c.f. Def-inition 1). Such a separation property is equivalent to Γ being irreducible.Definition 10. We shall understand that F satisfies the level separation property(LSP) if the two following conditions hold for each level of Γ:LSP1: A◦ ∩B◦ = ∅, for all A,B ∈ Γn : A 6= B.LSP2: A◦ 6= ∅, for each A ∈ Γn,37M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-Granerowhere the interiors have been considered in K.It is worth pointing out that the LSP does not depend on an external open set,unlike the OSC. Let Γ be a covering of X . Recall that Γ is a tiling provided thatall the elements of Γ have disjoint interiors and are regularly closed, i.e., A◦ = Afor each A ∈ Γ. A fractal structure Γ is called a tiling if each level Γn of Γ is atiling itself.Theorem 11 (IFS). The following are equivalent:(i) Γ irreducible.(ii) dim 4Γ(K) = dim3Γ(K) = α.(iii) LSP.(iv) LSP2 and Ai ⊆ Aj implies j ⊑ i.(v) Γ tiling.(vi) Hα4 (K) > 0.Definition 12. We shall understand that F is under the weak separation condi-tion (WSC) if any of the equivalent statements provided in Theorem 11 stands.Corollary 13 (IFS). SOSC⇒WSC, and the reciprocal is not true, in general.The following Moran’s type theorem holds for both fractal dimensions III and IVprovided that F is under the WSC.Theorem 14 (IFS). WSC⇔ dim 4Γ(K) = dim3Γ(K) = α.6. ConclusionIn this section, we summarize all the results contributed along this paper.Theorem 15. Consider the following statements:(i) HαH(K) > 0.(ii) SOSC.(iii) OSC.(iv) dimH(K) = dim4Γ(K) = dim3Γ(K) = α.(v) dim 4Γ(K) = dim3Γ(K) = α.38Irreducible fractal structures for Moran’s theorems(vi) Γ irreducible.(vii) Γ tiling.(viii) Hα4 (K) > 0.The next chains of implications and equivalences stand:(EIFS) (i)⇔ (ii)⇔ (iii)⇒ (iv)⇒ (v)⇔ (vi)⇔ (vii)⇔ (viii).(IFS) (i)⇒ (ii)⇒ (iv)⇒ (v)⇔ (vi)⇔ (vii)⇔ (viii).To conclude this paper, we provide two comparative theorems (one for each con-text, EIFS or IFS) involving our results vs. those obtained by Schief.Theorem 16 (EIFS, comparative theorem).HαH(K) > 0⇔ OSC⇔ SOSC⇒ dimH(K) = α.WSC⇔ Hα4 (K) > 0⇔ dim4Γ(K) = α.Theorem 17 (IFS, comparative theorem).HαH(K) > 0⇒ SOSC⇒ dimH(K) = α.WSC⇔ Hα4 (K) > 0⇔ dim4Γ(K) = α.Both statements in Theorem 17 (Schief’s and our’s) can be combined into thefollowing summary result standing in the general case:Corollary 18 (IFS). HαH(K) > 0⇒ SOSC⇒ dim H(K) = α⇒WSC, whereWSC⇔ Hα4 (K) > 0⇔ dim4Γ(K) = α.Interestingly, Corollary 18 highlights that the WSC becomes necessary to reach theequality between the Hausdorff and the similarity dimensions of IFS-attractors.In other words, if the natural fractal structure which any IFS-attractor can beendowed with is not irreducible, then a Moran’s type theorem cannot hold.39M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-GraneroReferences[1] F. G. Arenas and M. A. Sa´nchez-Granero, A characterization of non-Archimedeanlyquasimetrizable spaces, Rend. Istit. Mat. Univ. Trieste 30 (1999), no. suppl., 21–30.[2] F. G. Arenas and M. A. Sa´nchez-Granero, A characterization of self-similar symbolic spaces,Mediterranean Journal of Mathematics 9 (2012), no. 4, 709–728.[3] C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, FundamentaMathematicae 141 (1992), no. 3, 257–268.[4] K. Falconer, Fractal geometry. Mathematical Foundations and Applications, 1st ed., JohnWiley & Sons, Ltd., Chichester, 1990.[5] M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-Granero, Fractal dimension for fractal structures:A Hausdorff approach, Topology and its Applications 159 (2012), no. 7, 1825–1837.[6] M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-Granero, Fractal dimension for fractal structures,Topology and its Applications 163 (2014), 93–111.[7] M. Ferna´ndez-Mart´ınez and M. A. Sa´nchez-Granero, Fractal dimension for fractal structures:A Hausdorff approach revisited, Journal of Mathematical Analysis and Applications 409(2014), no. 1, 321–330.[8] J. E. Hutchinson, Fractals and self-similarity, Indiana University Mathematics Journal 30(1981), no. 5, 713–747.[9] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), no. 2, 153–155.[10] S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Uni-versity Mathematics Journal 37 (1988), no. 3, 699–710.[11] J. M. 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Irreducible fractal structures for Moran's theorems

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Irreducible fractal structures for Moran's theorems

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