Generalized Moran sets Generated by Step-wise Adjustable Iterated Function Systems
GGENERALIZED MORAN SETS GENERATED BY STEP-WISEADJUSTABLE ITERATED FUNCTION SYSTEMS
TYNAN LAZARUS, QINGLAN XIA
Abstract.
In this article we provide a systematic way of creating generalizedMoran sets using an analogous iterated function system (IFS) procedure. Weuse a step-wise adjustable IFS to introduce some variance (such as non-self-similarity) in the fractal limit sets. The process retains the computationalsimplicity of a standard IFS procedure. In our construction of the general-ized Moran sets, we also weaken the fourth Moran Structure Condition thatrequires the same pattern of diameter ratios be used across a generation. More-over, we provide upper and lower bounds for the Hausdorff dimension of thefractals created from this generalized process. Specific examples (Cantor-likesets, Sierpinski-like Triangles, etc) with the calculations of their correspondingdimensions are studied. Introduction
The Moran construction is a typical way to generate self-similar fractals, and hasbeen studied extensively in the literature (e.g. [14], [5], [6], [11],[10], [17], [7], andreferences therein). In this paper, we extend ideas from iterated function systems(IFS) and Moran constructions by describing a new process that allows for thefunctions to be updated at every iteration while still maintaining the computationalsimplicity of an IFS. This process provides more variance in the limit sets (such asnon-self-similarity) using an analogous approach to an IFS procedure. We also giveestimates of the Hausdorff dimension of the limit sets created from such a process,and provide concrete examples.The classic construction of Moran sets was introduced in [14]. We reproduce thedefinition here with a more current interpretation to introduce notations.Let { n k } k ≥ be a sequence of positive integers for k ≥
1. For any k ∈ N , define(1.1) D k = { ( i , i , · · · , i k ) : 1 ≤ i j ≤ n k , ≤ j ≤ k } and D = (cid:91) k ≥ D k . We define D = ∅ . Let σ = ( σ , · · · , σ k ) ∈ D k and τ = ( τ , · · · , τ m ) ∈ D m , thendenote σ ∗ τ = ( σ , · · · , σ k , τ , · · · , τ m ). Using this notation, we may express(1.2) D k = { σ ∗ j | σ ∈ D k − , ≤ j ≤ n k } to emphasize the process of moving between generations.Suppose J := { J σ : σ ∈ D } is a collection of subsets of R N . Set(1.3) E k = (cid:91) σ ∈ D k J σ , and F = (cid:92) k ≥ E k . Mathematics Subject Classification.
Key words and phrases.
Non Self-similar Fractals, Random fractals, Adjustable IFS, Moransets, Compression, Hausdorff dimension, F -Limit sets. a r X i v : . [ m a t h . C A ] D ec T. LAZARUS, Q. XIA
We call F the limit set associated with the collection J . Definition 1.0.1 ([17]) . Suppose that J ⊂ R N is a compact set with nonemptyinterior. Let { n k } k ≥ be a sequence of positive integers, and { Φ k } k ≥ be a sequenceof positive real vectors with(1.4) Φ k = ( c k, , c k, , . . . , c k,n k ) , (cid:88) ≤ j ≤ n k c k,j ≤ , k ∈ N . Suppose that F := { J σ : σ ∈ D } is a collection of subsets of R N , where D is given in(1.1). We say that the collection F fulfills the Moran Structure provided it satisfiesthe following Moran Structure Conditions (MSC):MSC(1) J ∅ = J. MSC(2) For any σ ∈ D , J σ is geometrically similar to J . That is, there exists asimilarity S σ : R N → R N such that J σ = S σ ( J ).MSC(3) For any k ≥ σ ∈ D k , J σ(cid:63) , . . . , J σ(cid:63)n k are subsets of J σ , and int ( J σ(cid:63)i ) ∩ int ( J σ(cid:63)j ) = ∅ for i (cid:54) = j .MSC(4) For any k ≥ σ ∈ D k − , ≤ j ≤ n k , (1.5) diam ( J σ(cid:63)j ) diam ( J σ ) = c k,j . For the collection F fulfilling the MSC, the limit set F given in (1.3) is anonempty compact set. This limit set F is called the Moran set associate withthe collection F . This Moran set is self-similar, and has been studied extensivelyby many authors with various approaches (e.g. [14], [6], [9], [5], [15]).Several approaches have been used to relax MSC in order to create more generallimit sets. The dimension (e.g. Hausdorff, Box, Packing, . . . ) of these sets hasbeen a fruitful area of study. For example, in [13], MSC(2) has been expanded toaffine maps. In this setting, however, calculations of the dimension of some limitsets can become particularly difficult. One could also study the limit sets generatedby infinitely many similarities, as in [12]. In [10], the authors removed MSC(2),but required int ( J σ ) = J σ in their construction, and studied the dimension of theresulting fractals. In [7], Holland and Zhang studied a construction that replacedsimilarity maps in MSC(2) with a more general class of functions that are notnecessarily contractions. In [16], Pesin and Weiss removed the requirement forsimilarities from MSC(2), but also relaxed MSC(3) from non-intersecting basic setsto non-intersecting balls contained in the basic sets. In particular they pursuedsufficient conditions for which the Box-counting and Hausdorff dimensions coincide.For more examples of modifications to the Moran set definition, see [17] and thereferences therein.A special case of Moran sets can be constructed from an iterated function system(IFS). An iterated function system { S , S , · · · , S m } is a finite family of similaritiesfor a fixed natural number m ≥ n k = m and set S σ = S i k ◦ S i k − ◦· · ·◦ S i for σ = ( i , i , · · · , i k ) ∈ D .Then the resulting Moran set is self-similar and agrees with the attractor of the IFS { S , S , · · · , S m } . The dimension of the limit set can be quickly calculated fromthe Moran-Hutchinson formula in [6]. Using iterated function systems is a popularway to construct fractals, and has been used to great effect (e.g. [1], [9], [6], [4] ). TEP-WISE ADJUSTABLE IFS 3
A natural question arises: Can we construct more general fractals (e.g. non-self-similar Moran type sets) using an analogous approach while preserving thecomputational simplicity of the IFS? In this paper, we present a method to do so.We first make the following observations. Note that in the above construction,(1.6) J σ ∗ i = S i ( J σ ) , for all i = 1 , ...m, and σ ∈ D. Suppose that there is a tuning parameter in the expression of the function S i (e.g. the coefficients a i , b i in a linear function S i ( x ) = a i x + b i ). One can tune thevalues of the parameter to get a comparable function. When J σ is given, applyingthe comparable function to J σ , as in equation (1.6), will not significantly change thecomputational complexity of constructing J σ ∗ i . The advantage of doing this at eachiteration is that we introduce some variance into the limit set. Another observationis about which space the functions are defined. In classical IFS constructions,the functions are usually defined on all of the ambient space R N (as in [7], thefunctions are C α diffeomorphisms on R N ). For our construction, we wish torelax the condition MSC(2) as well. Instead of restricting our attention to functionsof higher regularity defined on the whole ambient space R N , we use maps from acollection of subsets to itself.This article is organized as follows. In section 2 we find bounds for the Hausdorffdimension of the limit sets in a general metric space setting of a collection ofbounded sets, not necessarily satisfying the MSC conditions. Then in section 3we formulate the general setup for the construction of Moran-type limit sets usingthe ideas from a modified IFS procedure, as discussed in the previous paragraph.In our construction we relax MSC(2) so that the limit set is not necessarily self-similar. More importantly, we drop MSC(4) from the construction process so thatthere are no limitations on the ratios of the diameters of the sets. Specifically, theratio diam ( J σ ∗ j ) diam ( J σ ) in (1.5) is not limited to depend on just k and j , but varies with σ . This change allows us to produce a mosaic of possible fractals. An importantobservation is that the computational complexity of generating these fractals is thesame as using an analogous, standard IFS. In section 4 we give estimates of theHausdorff dimension of the limit sets created from the general construction. Finally,in section 5 we apply the results to specific examples, including modifications ofthe Cantor set, the Sierpinski triangle, and the Menger sponge.2. Hausdorff Dimension of the Limit Sets
In this section we investigate the Hausdorff dimension dim H ( F ) of the fractals F defined in (1.3), which is not necessarily satisfying MSC conditions. To start, wedetermine an upper bound for the dimension of the limit set F by considering thestep-wise relative ratios between the diameters of sets. Proposition 2.0.1.
Suppose J := { J σ : σ ∈ D } is a collection of bounded subsetsof a metric space ( X, d ) , and s > . Let E k = (cid:83) σ ∈ D k J σ , and F = (cid:84) k ≥ E k bedefined as in (1.3). If there exists a sequence of positive numbers { c k } ∞ k =1 such that lim inf k →∞ k (cid:89) i =1 c i = 0 T. LAZARUS, Q. XIA and (2.1) n k (cid:88) j =1 ( diam ( J σ ∗ j )) s ≤ c k ( diam ( J σ )) s , for all σ ∈ D k − and all k = 1 , , · · · , then dim H ( F ) ≤ s .Proof. We prove by using mathematical induction that for k = 1 , , · · · , (2.2) (cid:88) σ ∈ D k ( diam ( J σ )) s ≤ (cid:32) k (cid:89) i =1 c i (cid:33) ( diam ( J ∅ )) s . When k = 1, (2.2) follows from (2.1). Now assume (2.2) is true for some k ≥ (cid:88) σ ∈ D k +1 ( diam ( J σ )) s = (cid:88) σ ∈ D k n k +1 (cid:88) j =1 ( diam ( J σ ∗ j )) s ≤ c k +1 (cid:88) σ ∈ D k ( diam ( J σ )) s ≤ (cid:32) k +1 (cid:89) i =1 c i (cid:33) ( diam ( J ∅ )) s as desired. By the induction principle, (2.2) holds for all k = 1 , , · · · . For each k , set δ k = max { diam ( J σ ) : σ ∈ D k } > . Then, by (2.2), δ k ≤ (cid:16)(cid:81) ki =1 c i (cid:17) /s diam ( J ∅ ). Moreover, by (2.2) H sδ k ( F ) ≤ H sδ k ( E k ) ≤ (cid:88) σ ∈ D k α ( s ) (cid:18) diam ( J σ )2 (cid:19) s ≤ (cid:32) k (cid:89) i =1 c i (cid:33) α ( s ) (cid:18) diam ( J ∅ )2 (cid:19) s . Since lim inf k →∞ (cid:81) ki =1 c i = 0, there exists a sequence { k t } ∞ t =1 such thatlim t →∞ (cid:81) k t i =1 c i = 0. Thus, δ k t → t → ∞ , and H s ( F ) = lim t →∞ H sδ kt ( F ) = 0,and hence dim H ( F ) ≤ s . (cid:3) Conversely, a lower bound on the Hausdorff dimension of the limit set F can alsobe obtained as follows. Proposition 2.0.2.
Suppose J := { J σ : σ ∈ D } is a collection of compact subsetsof Euclidean space R N , and let F be the limit set of J as given in (1.3). If forsome s > , (2.3) n k (cid:88) j =1 diam ( J σ ∗ j ) s ≥ diam ( J σ ) s for all σ ∈ D k − and all k = 1 , , · · · , then dim H ( F ) ≥ s .Proof. We first show that under condition (2.3), there exists a probability measure µ on R N concentrated on F such that for each Borel subset B of R N ,(2.4) µ ( B ) ≤ (cid:18) diam ( B ) diam ( J ∅ ) (cid:19) s . TEP-WISE ADJUSTABLE IFS 5
Let µ ( J ∅ ) = 1, and for each σ ∈ D k for k > i = 1 , · · · , n k , we inductivelyset µ ( J σ ∗ i ) = diam ( J σ ∗ i ) s (cid:80) n k j =1 diam ( J σ ∗ j ) s µ ( J σ ) . Then by Proposition 1.7 in [9], µ can be uniquely extended to a probability measureon R N , concentrated on F . For any Borel set B , the value µ ( B ) = inf (cid:40) ∞ (cid:88) i =1 µ ( J σ i ) : B ∩ F ⊂ ∞ (cid:91) i =1 J σ i and J σ i ∈ J (cid:41) . Thus, to prove (2.4) for each Borel set B , it is sufficient to prove (2.4) for J σ , ∀ σ ∈ D . We proceed by using induction on k when σ ∈ D k . It is clear for k = 0.Now assume that (2.4) holds for each σ ∈ D k for some k . Then by inductionassumption and (2.3), for each i = 1 , · · · , n k +1 , µ ( J σ ∗ i ) = diam ( J σ ∗ i ) s (cid:80) n k j =1 diam ( J σ ∗ j ) s µ ( J σ ) ≤ diam ( J σ ∗ i ) s (cid:80) n k j =1 diam ( J σ ∗ j ) s (cid:18) diam ( J σ ) diam ( J ∅ ) (cid:19) s ≤ (cid:18) diam ( J σ ∗ i ) diam ( J ∅ ) (cid:19) s . This proves inequality (2.4).Now, for any δ >
0, let { B i } be any collection of closed balls with diam ( B i ) ≤ δ and F ⊆ ∪ i B i . Then, by (2.4), (cid:88) i α ( s ) (cid:18) diam ( B i )2 (cid:19) s ≥ α ( s ) (cid:18) diam ( J ∅ )2 (cid:19) s (cid:88) i µ ( B i ) ≥ cµ (cid:32)(cid:91) i B i (cid:33) ≥ cµ ( F ) = c, where c = α ( s ) (cid:16) diam ( J ∅ )2 (cid:17) s . Thus, H s ( F ) = lim δ → H sδ ( F ) ≥ c >
0, and hencedim H ( F ) ≥ s . (cid:3) General Setup of F -Limit sets We now formalize the ideas from section 1 to give a description of the construc-tion of such fractals. We concentrate on the maps in order to take advantage of thecomputational nature of an IFS, but allow for the maps to be updated and changedat each iteration.In this section let X be a collection of nonempty compact subsets of a metricspace. Definition 3.0.1.
A mapping f : X → X is called a compression on X if f ( E ) ⊆ E for each E ∈ X .For each natural number m , let C m ( X ) = { ( f (1) , f (2) , . . . , f ( m ) ) : f i is a compression on X , i = 1 , . . . , m } . Definition 3.0.2.
Let M be a nonempty set. A mapping F : M → C m ( X )(3.1) k → f k = ( f (1) k , f (2) k , · · · , f ( m ) k ) . (3.2) T. LAZARUS, Q. XIA is called a marking of C m ( X ) by M . Each element k ∈ M is called the marker of f k .Given a marking F and an initial set E ∈ X , we will construct a generalizedMoran set from any sequence of markers in M . Note that any sequence { k (cid:96) } ∞ (cid:96) =0 in M can be represented as a mapping from the ordered set D to M . Definition 3.0.3.
Let F be a marking of C m ( X ) by M , let E be any element in X , and D be as in (1.1). Suppose (cid:126)k : D → M is a map sending σ to k σ . For each σ ∈ D and 1 ≤ j ≤ m , we recursively define J ∅ = E and(3.3) J σ ∗ j = f ( j ) k σ ( J σ ) , where f k σ is given by F as in (3.2).The limit set F = (cid:92) k ≥ (cid:91) σ ∈ D k J σ associated with J ( (cid:126)k ) = { J σ : σ ∈ D } is calledthe F -limit set generated by (cid:126)k with the initial set E .We now make two observations relating the concepts of an F -limit set with theattractor of an IFS.We first observe that the attractor of an IFS { S , S , . . . , S m } on a closed subset∆ of R N can be viewed as an F -limit set as follows.Let X = { E : E is a non-empty compact subset of ∆ , S i ( E ) ⊆ E, for all i } .Since each S i is a contraction on ∆, the set E r := ∆ ∩ B (0 , r ) is a non-emptycompact subset of ∆, and S i ( E r ) ⊆ E r for each i when r is sufficiently large. Inother words, E r ∈ X for sufficiently large r . Also, each contraction map S i actingon ∆ naturally determines a map f ( i ) : X → X given by(3.4) f ( i ) ( E ) = S i ( E ) := { S i ( x ) | x ∈ E ⊆ ∆ } for each E ∈ X . Since f ( i ) ( E ) = S i ( E ) ⊆ E , f ( i ) is a compression for each i . Set f = ( f (1) , f (2) , . . . , f ( m ) ) . For any non-empty set M , define the marking F of C m ( X ) to be the constantfunction F ( k ) = f for all k ∈ M . Thus, for each σ ∈ D k and i = 1 , . . . , m , we havethat J σ ∗ i = S i ( J σ ) from (3.3). As a result, for any map (cid:126)k : D → M , the collection J ( (cid:126)k ) = { J σ : σ ∈ D } is independent of the choice of (cid:126)k . Thus, the associated F -limitset F = (cid:92) k ≥ (cid:91) σ ∈ D k J σ agrees with the attractor of the given IFS { S , S , . . . , S m } .Conversely, let F be a marking of C m ( X ) by M where X is a collection of non-empty compact subsets of ∆. Suppose there is a mapping (cid:126)k : D → M such thatthe sequence { f k σ } σ ∈ D is constant in C m ( X ) (i.e. there exists an f ∈ C m ( X ) suchthat f k σ = f for all σ ∈ D ) and for each i = 1 , , . . . , m , there exists a contraction S i on ∆ such that equation (3.4) holds for each E ∈ X . Then the F -limit set F generated by (cid:126)k is the attractor of the IFS { S , S , . . . , S m } .Therefore, choosing (cid:126)k : D → M to be a constant map will result in a limit set F that is the attractor of an IFS. In the above sense, our approach is a generalizationof the standard IFS construction.An important observation is that replacing { k σ } σ ∈ D by another sequence { ˜ k σ } σ ∈ D in (3.3) will not change the computational complexity of the construction of J ( (cid:126)k ).Thus, generating the limit set F will have a similar computational complexity asgenerating the attractor of a comparable IFS. TEP-WISE ADJUSTABLE IFS 7
In the following section we will compute the Hausdorff dimension of the con-structed F -limit sets. In section 5 we will provide examples along with their di-mensions. 4. Hausdorff dimensions of F -Limit sets As indicated in Propositions 2.0.1 and 2.0.2 in section 2, the relative ratio be-tween the diameters of the sets plays an important role in the calculation of thedimension of the limit set. Therefore, we introduce the following definition.
Definition 4.0.1.
For any compression g : X → X , define(4.1) U ( g ) = sup E ∈X diam ( g ( E )) diam ( E ) , and L ( g ) = inf E ∈X diam ( g ( E )) diam ( E ) . Note that, for each E ∈ X ,(4.2) L ( g ) · diam ( E ) ≤ diam ( g ( E )) ≤ U ( g ) · diam ( E ) . For any k ∈ M and f k = ( f (1) k , f (2) k , · · · , f ( m ) k ) ∈ C m ( X ), define U k = (cid:16) U ( f (1) k ) , · · · , U ( f ( m ) k ) (cid:17) ∈ R m , and L k = (cid:16) L ( f (1) k ) , · · · , L ( f ( m ) k ) (cid:17) ∈ R m . Also, for each x = ( x , · · · , x m ) ∈ R m and s >
0, denote || x || s = (cid:32) m (cid:88) i =1 | x i | s (cid:33) s . These notations, Proposition 2.0.1 and Proposition 2.0.2 motivate our main the-orem.
Theorem 4.0.2.
Let F be the F -limit set generated by a sequence { k σ } σ ∈ D withinitial set J ∅ , and s > . (a) If inf σ ∈ D {|| L k σ || s } ≥ , then dim H ( F ) ≥ s . (b) If sup σ ∈ D {|| U k σ || s } < , then dim H ( F ) ≤ s .Proof. (a) By 3.3 and 4.2, for all σ ∈ D , m (cid:88) j =1 diam ( J σ ∗ j ) s = m (cid:88) j =1 diam (cid:16) f ( j ) k σ ( J σ ) (cid:17) s ≥ m (cid:88) j =1 (cid:16) L ( f ( j ) k σ ) (cid:17) s diam ( J σ ) s ≥ diam ( J σ ) s . Thus, by Proposition 2.0.2, dim H ( F ) ≥ s .(b) Similarly, for all σ ∈ D , m (cid:88) j =1 diam ( J σ ∗ j ) s ≤ m (cid:88) j =1 (cid:16) U ( f ( j ) k σ ) (cid:17) s diam ( J σ ) s ≤ c · diam ( J σ ) s , T. LAZARUS, Q. XIA where c := sup σ { ( || U k σ || s ) s } < . By Proposition 2.0.1, dim H ( F ) ≤ s . (cid:3) Remark 4.0.3.
For practical reasons, we find that it is more convenient to rep-resent the mapping (cid:126)k : D → M by a sequence { k (cid:96) } ∞ (cid:96) =0 ⊆ M . For each σ =( i , i , . . . , i k ) ∈ D k , let(4.3) (cid:96) ( σ ) = k − (cid:88) p =0 m p i k − p be the ordering of σ in the ordered set D . Using this notation, we can rewriteDefinition 3.0.3 as follows. Definition 3.0.3’.
Let F be a marking of C m ( X ) by M , let { k (cid:96) } ∞ (cid:96) =0 be a sequencein M , and E ∈ X be a starting set. For each (cid:96) = 0 , , , · · · and j = 1 , , · · · , m ,we iteratively denote the set E m(cid:96) + j = f ( j ) k (cid:96) ( E (cid:96) ) ∈ X , where f k (cid:96) is given by F as in (3.2).Let G m (0) = 0 and for n ≥ G m ( n ) = m + m + · · · + m n = m n +1 − mm − n th generation, i.e. the cardinality of D n .The limit set(4.5) F = ∞ (cid:92) n =1 G m ( n ) (cid:91) (cid:96) = G m ( n − E (cid:96) is called the F -limit set generated by the triple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ).In the following, we will use the notation from Definition 3.0.3’ to describe theconstruction of the F -limit sets. Clearly, using this notation, Theorem 4.0.2 simplysays that if inf (cid:96) {|| L k (cid:96) || s } ≥ , then dim H ( F ) ≥ s , and if sup (cid:96) {|| U k (cid:96) || s } < , thendim H ( F ) ≤ s .When both {|| L k (cid:96) || s } ∞ (cid:96) =0 and {|| U k (cid:96) || s } ∞ (cid:96) =0 are convergent sequences, the follow-ing corollary enables us to quickly estimate the dimension of F . Corollary 4.0.4.
Let F be the limit set generated by the triple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ) .Then, (4.6) s ∗ ≤ dim H ( F ) ≤ s ∗ , where s ∗ = sup (cid:26) s : lim inf (cid:96) →∞ {|| L k (cid:96) || s } > (cid:27) , and s ∗ = inf (cid:26) s : lim sup (cid:96) →∞ {|| U k (cid:96) || s } < (cid:27) . Proof.
For any 0 < s < s ∗ , by the definition of s ∗ ,lim inf (cid:96) →∞ {|| L k (cid:96) || s } > . TEP-WISE ADJUSTABLE IFS 9
Thus, when (cid:96) ∗ ∈ N is large enough,inf (cid:96) ≥ (cid:96) ∗ {|| L k (cid:96) || s } ≥ , i.e. inf (cid:96) ≥ {|| L k (cid:96) ∗ + (cid:96) || s } ≥ . Since F ∩ E (cid:96) ∗ is the set generated by the triple ( F , { k (cid:96) ∗ + (cid:96) } ∞ (cid:96) =0 , E (cid:96) ∗ ), by Theorem4.0.2, it follows that dim H ( F ∩ E (cid:96) ∗ ) ≥ s for any (cid:96) ∗ large enough. This implies thatdim H ( F ) ≥ s for any s < s ∗ and hence dim H ( F ) ≥ s ∗ .Similarly, we also have dim H ( F ) ≤ s ∗ . (cid:3) In the following corollaries, we will see that bounds of the dimension of F canalso be obtained from corresponding bounds on L k (cid:96) and U k (cid:96) . Notation.
For any two points x = ( x , · · · , x m ) and y = ( y , · · · , y m ) in R m ,we say x ≤ y if x i ≤ y i for each i = 1 , · · · , m . Corollary 4.0.5.
Let t = ( t , · · · , t m ) and r = ( r , · · · , r m ) be two points in (0 , m ⊂ R m . Let s ∗ and s ∗ be the solutions to || t || s ∗ = 1 , and || r || s ∗ = 1 respec-tively, i.e. t s ∗ + t s ∗ + · · · + t s ∗ m = 1 , and r s ∗ + r s ∗ + · · · + r s ∗ m = 1 . (a) If L k (cid:96) ≥ t for all (cid:96) , then dim H ( F ) ≥ s ∗ . (b) If U k (cid:96) ≤ r for all (cid:96) , then dim H ( F ) ≤ s ∗ . (c) If L k (cid:96) = r = U k (cid:96) for all (cid:96) , then dim H ( F ) = s ∗ .Proof. (a) Let 0 < s < s ∗ . Then,inf (cid:96) {|| L k (cid:96) || s } ≥ || t || s ≥ || t || s ∗ = 1 . Thus, by Theorem 4.0.2, dim H ( F ) ≥ s for any s < s ∗ , and hence dim H ( F ) ≥ s ∗ .(b) Similarly, let 0 < s ∗ < s . Then,sup (cid:96) {|| U k (cid:96) || s } ≤ || r || s < || r || s ∗ = 1 . Thus, by Theorem 4.0.2, dim H ( F ) ≤ s for any s > s ∗ , and hence dim H ( F ) ≤ s ∗ .(c) follows from (a) and (b). (cid:3) A special case of Corollary 4.0.5 gives the following explicit formulas for thebounds on the dimension of F . Corollary 4.0.6.
Let F be the limit set generated by the triple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ) .Let t = ( t, · · · , t ) and r = ( r, · · · , r ) , for some < t, r < . (a) If L k (cid:96) ≥ t for all (cid:96) , then dim H ( F ) ≥ log m − log t . (b) If U k (cid:96) ≤ r for all (cid:96) , then dim H ( F ) ≤ log m − log r . (c) If L k (cid:96) = r = U k (cid:96) for all (cid:96) , then dim H ( F ) = log m − log r . Other types of bounds on L k (cid:96) and U k (cid:96) can also be used to provide bounds ondim H ( F ), as indicated by the following result. Corollary 4.0.7.
Let F be the limit set generated by the triple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ) . (a) If w := inf (cid:96) {|| L k (cid:96) || } ≥ , then dim H ( F ) ≥ log( m )log( m ) − log( w ) . (b) If u := sup (cid:96) {|| U k (cid:96) || } < , then dim H ( F ) ≤ log( m )log( m ) − log( u ) .Proof. (a). In this case, for s = log( m )log( m ) − log( w ) ≥
1, we have (cid:80) mj =1 (cid:16) L (cid:16) f ( j ) k (cid:96) (cid:17)(cid:17) s m ≥ (cid:80) mj =1 L (cid:16) f ( j ) k (cid:96) (cid:17) m s ≥ (cid:16) wm (cid:17) s for each (cid:96) . Thus, inf (cid:96) {|| L k (cid:96) || s } ≥ m s wm = 1 , then by Theorem 4.0.2, dim H ( F ) ≥ s .(b). In this case, for any 1 ≥ s > log( m )log( m ) − log( u ) , we have (cid:80) mj =1 (cid:16) U (cid:16) f ( j ) k (cid:96) (cid:17)(cid:17) s m ≤ (cid:80) mj =1 U (cid:16) f ( j ) k (cid:96) (cid:17) m s ≤ (cid:16) um (cid:17) s for each (cid:96) . Thus, sup (cid:96) {|| U k (cid:96) || s } ≤ m s um < . By Theorem 4.0.2, dim H ( F ) ≤ s . Hence, dim H ( F ) ≤ log( m )log( m ) − log( u ) . (cid:3) Note that this corollary generally provides better bounds on dim H ( F ) than thoseobtained from directly applying Theorem 4.0.2.5. Examples of F -Limit sets In this section we describe the construction of both classical fractals and gener-alized Moran sets in the language of Section 3, and calculate the dimension usingthe results from Section 4.5.1.
Cantor-Like Sets.
We first consider Cantor-like sets. Let X = { [ a, b ] : a, b ∈ R } be the collection of closed intervals, m = 2, and let M = [0 , ⊆ R . For each k = ( k (1) , k (2) ) ∈ M , we consider the following two maps, f (1) k : X → X [ a, b ] (cid:55)→ [ a, k (1) ( b − a ) + a ] f (2) k : X → X [ a, b ] (cid:55)→ [ k (2) ( a − b ) + b, b ] . Note that both f (1) k and f (2) k are compression maps for any k ∈ M . Thus, thisdefines a marking F : M → C ( X ) k (cid:55)→ f k = ( f (1) k , f (2) k ) . TEP-WISE ADJUSTABLE IFS 11
Here, for each k = ( k (1) , k (2) ) ∈ M , one can clearly see that diam (cid:16) f ( i ) k ([ a, b ]) (cid:17) = k ( i ) · diam ([ a, b ]) . Thus, L (cid:16) f ( i ) k (cid:17) = k ( i ) = U (cid:16) f ( i ) k (cid:17) , and hence(5.1) L k = k = U k . Let E = [0 , ∈ X be fixed. For any sequence { k (cid:96) } ∞ (cid:96) =0 ∈ M , we define thefollowing: E (0) = E E (1) = f (1) k ( E ) ∪ f (2) k ( E ) =: E ∪ E E (2) = f (1) k ( E ) ∪ f (2) k ( E ) ∪ f (1) k ( E ) ∪ f (2) k ( E ):= E ∪ E ∪ E ∪ E ... E ( n ) = n − (cid:91) i =2 n − − (cid:16) f (1) k i ( E i ) ∪ f (2) k i ( E i ) (cid:17) := n − (cid:91) i =2 n − − ( E i +1 ∪ E i +2 ) = n − (cid:91) (cid:96) =2 n − E (cid:96) . Note that when k (cid:96) = ( , ) for all (cid:96) , E ( n ) is the n th -generation of the Cantorset C and F = lim n →∞ E ( n ) = (cid:92) n E ( n ) = C .Observe that the process of constructing the sequence { E ( n ) } ∞ n =0 is independentof the values of { k (cid:96) } ∞ (cid:96) =0 . To allow for more general outcomes, we can update thelinear functions f (1) k and f (2) k simply by changing the value of k at each stage of theconstruction, which does not change the computational complexity of the process.Using this idea, we now construct some examples of Cantor-like sets by choosingsuitable sequences { k (cid:96) } ∞ (cid:96) =0 . Example 5.1.1.
Let k (cid:96) = (cid:16) (cid:96) +14 (cid:96) +6 , (cid:96) +58 (cid:96) +16 (cid:17) for (cid:96) ≥
0, and let F be the F -limit setgenerated by the triple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ). In Figure 1 we plot the usual Cantor set C (in blue) below the set F (in red) to illustrate the comparison. We can see thatthe set F has the same basic shape as the Cantor set C , but is no longer strictlyself-similar.In order to compute the Hausdorff dimension of the new Cantor-like set F , weapply Corollary 4.0.4. Note that by 5.1,lim (cid:96) →∞ || L k (cid:96) || s = lim (cid:96) →∞ || k (cid:96) || s = 2 s . So, s ∗ = sup s { lim inf (cid:96) →∞ || L k (cid:96) || s > } = sup s (cid:40) s > (cid:41) = 12 . Similarly, we also have s ∗ = . By (4.6), dim H ( F ) = .In the next example, we will construct a random Cantor-like set as follows. Figure 1.
Comparison of classical Cantor set (blue) and newCantor-like set (red)
Example 5.1.2.
For each (cid:96) ≥
0, we take k (cid:96) = (cid:0) q (cid:96) , − q (cid:96) (cid:1) where q (cid:96) is a randomnumber between and . Let F be the corresponding F -limit set generated by thetriple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ). We plot F in Figure 2. In this example, the total length ofthe n th generation E ( n ) is chosen to be ( ) n , while the scaling factors of the leftsubintervals at each stage are randomly chosen. Figure 2.
A randomly generated Cantor-like set
TEP-WISE ADJUSTABLE IFS 13
We now estimate the dimension of F . By (5.1), (cid:18) , (cid:19) ≤ L k (cid:96) = k (cid:96) = U k (cid:96) ≤ (cid:18) , (cid:19) . By Corollary 4.0.6, log(2) − log(1 / ≤ dim H ( F ) ≤ log(2) − log(3 / . That is, 13 ≤ dim H ( F ) ≤ log(2)log(8 / ≈ . . Example 5.1.3.
In this example, we create a sequence { k (cid:96) } ∞ (cid:96) =0 that results in alimit set with a given measure, e.g. 1/3. Of course, the classic example of sucha limiting set is the fat Cantor set. For a different approach, let (cid:80) ∞ n =0 a n be anyconvergent series of positive terms with limit L . We consider a sequence { k (cid:96) } ∞ (cid:96) =0 defined in the following way.Let n ≥ (cid:96) with 2 n − − ≤ (cid:96) ≤ n −
2, define k (cid:96) = ( b n , b n ) where b := L − a (cid:0) L (cid:1) and b n := L − (cid:80) n − i =0 a i (cid:16) L − (cid:80) n − i =0 a i (cid:17) for n ≥ . With this sequence { k (cid:96) } ∞ (cid:96) =0 , one can find that the length of each interval in the n th generation is b b · · · b n = L − (cid:80) n − i =0 a i n · L .
Thus, the total length of the n th generation is L − (cid:80) n − i =0 a i L = 1 − L n − (cid:88) i =0 a i which converges to 1/3 as desired.As an example, we take the convergent series ∞ (cid:88) n =0 n ! = e and use it to create the F -limit set F with measure 1/3. The first few generations are shown in Figure 3.5.2. Sierpinski Triangle.
The Sierpinski triangle is another well known fractal.Following the general setup in Section 3, we take(5.2) X = { ( A, B, C ) | A, B, C ∈ R } representing the collection of all triangles ∆ ABC in R , m = 3, and M = [0 , ⊆ R . For each k = (cid:0) k (1) , k (2) , k (3) , k (4) , k (5) , k (6) (cid:1) ∈ M and i = 1 , , f ( i ) k : X → X as f (1) k ( A, B, C ) = (
A, A + k (1) ( B − A ) , A + k (2) ( C − A )) f (2) k ( A, B, C ) = ( B + k (4) ( A − B ) , B, B + k (3) ( C − B )) f (3) k ( A, B, C ) = ( C + k (5) ( A − C ) , C + k (6) ( B − C ) , C )for every ( A, B, C ) ∈ X . Figure 3.
Fractal of measure created by using (cid:80) ∞ n =0 1 n ! = e Note that each f ( i ) k is a compression map for i = 1 , , k ∈ M . Thus,this defines a marking F : M → C ( X ) k (cid:55)→ f k = ( f (1) k , f (2) k , f (3) k ) . Of course, to prevent overlaps we can require that k (1) + k (4) ≤ , k (2) + k (5) ≤ , k (3) + k (6) ≤
1. When each of the inequalities are strict, the images of f ( i ) k arethree disconnected triangles, as illustrated in Figure (4a). When all equalities hold,the images are connected, as illustrated in Figure (4b). (a) (b) Figure 4.
First generation of disconnected and connected trianglesIn the case of the connected sets, the values of k = (cid:0) k (1) , k (2) , k (3) , k (4) , k (5) , k (6) (cid:1) are determined by k (1) , k (2) , k (3) since k (4) = 1 − k (1) , k (5) = 1 − k (2) , k (6) = 1 − k (3) .In this sense, we may also view k = (cid:0) k (1) , k (2) , k (3) (cid:1) as a vector in [0 , ⊆ R .To create the normal Sierpinski triangle, we choose(5.3) E = (cid:20) − / / √ / (cid:21) , TEP-WISE ADJUSTABLE IFS 15 the equilateral triangle of unit side length, and k (cid:96) ∈ M to be the constant sequence k (cid:96) = k = (1 / , / , / , / , / , /
2) so that each iteration maps a triangle to threetriangles of half the side length with the desired translation. In this case the F -limitset generated by ( F , { k (cid:96) } ∞ (cid:96) =0 , E ) corresponds to the standard Sierpinski Triangle.To generate Sierpinski-like fractals, we now adjust the values of the markingparameters { k (cid:96) } ∞ (cid:96) =0 . For each k = ( k (1) , k (2) , · · · , k (6) ) ∈ M and 1 ≤ i ≤ U (cid:16) f ( i ) k (cid:17) = sup ( A,B,C ) ∈X diam (cid:16) f ( i ) k ( A, B, C ) (cid:17) diam (( A, B, C )) = max (cid:110) k (2 i − , k (2 i ) (cid:111) , and L (cid:16) f ( i ) k (cid:17) = inf ( A,B,C ) ∈X diam (cid:16) f ( i ) k ( A, B, C ) (cid:17) diam (( A, B, C )) = min (cid:110) k (2 i − , k (2 i ) (cid:111) . When k is bounded, i.e. if λ ≤ k ( j ) ≤ Λ < j = 1 , · · · ,
6, then U k ≤ r := ( r, · · · , r ) and L k ≥ s := ( s, · · · , s ) , where r = max { − λ, Λ } and s = min { − λ, Λ } .Following our general process, we construct some random Sierpinski-like sets byintroducing randomness into the choice of the sequence { k (cid:96) } ∞ (cid:96) =0 . Example 5.2.1.
Let { k (cid:96) } ∞ (cid:96) =0 = (cid:110)(cid:16) k (1) (cid:96) , k (2) (cid:96) , k (3) (cid:96) (cid:17)(cid:111) ∞ (cid:96) =0 be a sequence in [0 , witheach k ( i ) (cid:96) a random number between given numbers λ and Λ for each i = 1 , , F be the F -limit set generated by ( F , { k (cid:96) } ∞ (cid:96) =0 , E ). Then the 6 th generationof the construction results in images like Figure 5. Here, in Figure 5a, λ = andΛ = ; while in Figure 5b, λ = 0 .
45 and Λ = 0 .
55. Note that the sets are no longerself-similar. (a)
Each k ( i ) (cid:96) is random in [ , ]. (b) Each k ( i ) (cid:96) is random in [0 . , . Figure 5.
Generation 6 of Random Sierpinski triangleIn Figure 5b, we pick λ = 0 .
45 and Λ = 0 .
55. By Corollary 4.0.6,log( m ) − log( s ) ≤ dim H ( F ) ≤ log( m ) − log( r ) , where m = 3, r = 0 .
55 and s = 0 .
45. That is,1 . ≤ dim H ( F ) ≤ . . Example 5.2.2.
As in Example 5.2.1, but replacing E with ˜ E = (cid:20) (cid:21) , the7 th generation of the construction results in an image like Figure 6, when λ = and Λ = . Figure 6.
Generation 7 of a Random Sierpinski triangle
Example 5.2.3.
For each (cid:96) = 0 , , · · · , let k (cid:96) = (cid:16) k (1) (cid:96) , k (2) (cid:96) , · · · , k (6) (cid:96) (cid:17) where k (1) (cid:96) = 12 + a (cid:96) √ (cid:96) + 1 , k (2) (cid:96) = 1 − k (1) (cid:96) ,k (3) (cid:96) = 12 + b (cid:96) √ (cid:96) + 1 , k (4) (cid:96) = 1 − k (3) (cid:96) ,k (5) (cid:96) = 12 + c (cid:96) (cid:96) + 1 , k (6) (cid:96) = 1 − k (5) (cid:96) . for random numbers a (cid:96) , b (cid:96) , c (cid:96) ∈ [ − , ]. Let F be the F -limit set F generated by( F , { k (cid:96) } ∞ (cid:96) =0 , E ). Then the seventh generation of the construction of F results inan image like Figure 7.In this case, we can calculate the exact value of the Hausdorff dimension of F .Indeed, by Corollary 4.0.4,lim (cid:96) →∞ ( || U k (cid:96) || s ) s = 32 s = lim (cid:96) →∞ ( || L k (cid:96) || s ) s . Thus, dim H ( F ) = log(3)log(2) .5.3. Menger Sponge.
Let(5.4) X = (cid:8) ( O, A, B, C ) | O, A, B, C ∈ R (cid:9) representing the collection of all rectangular prisms ( OABC ) in R , m = 20 , and M = (cid:110)(cid:16) k (1) , k (2) , k (3) , k (4) , k (5) , k (6) (cid:17) ∈ [0 , : k (1) ≤ k (2) , k (3) ≤ k (4) , k (5) ≤ k (6) (cid:111) . TEP-WISE ADJUSTABLE IFS 17
Figure 7.
Generation 6 of a Sierpinski-type triangle with con-trolled dimensionFor each k ∈ M and i = 1 , , . . . ,
20, we can define affine transformations f ( i ) k : X → X as follows.For any k = ( k (1) , k (2) , k (3) , k (4) , k (5) , k (6) ) ∈ M , define T = (cid:2) k (1) k (2) (cid:3) , R = (cid:2) k (3) k (4) (cid:3) , S = (cid:2) k (5) k (6) (cid:3) . Let I = { ( a, b, c ) | ≤ a, b, c ≤ a, b, c ∈ Z , and no two of a, b, c equal to 2 } . For each ( a, b, c ) ∈ I and k ∈ M , define M k ( a, b, c ) = − ( T ( a ) + R ( b ) + S ( c )) T ( a ) R ( b ) S ( c )1 − ( T ( a + 1) + R ( b ) + S ( c )) T ( a + 1) R ( b ) S ( c )1 − ( T ( a ) + R ( b + 1) + S ( c )) T ( a ) R ( b + 1) S ( c )1 − ( T ( a ) + R ( b ) + S ( c + 1)) T ( a ) R ( b ) S ( c + 1) . Note that the set I contains 20 elements, so we can express it as I = { ( a i , b i , c i ) | ≤ i ≤ } . For each k ∈ M and 1 ≤ i ≤
20, we consider the affine transformation f ( i ) k : X → X given by(5.5) f ( i ) k ( O, A, B, C ) = M k ( a i , b i , c i ) OABC for every (
O, A, B, C ) ∈ X .Note that for i = 1 , . . . ,
20 and k ∈ M , f ( i ) k is a compression. Thus, we candefine a marking F by F : M → C ( X ) k (cid:55)→ f k = ( f (1) k , . . . , f (20) k ) . Using this, for any starting rectangular prism E = ( O, A, B, C ) ∈ X , we cangenerate a sequence of sets that follows a similar construction to the Menger Sponge. Example 5.3.1.
Let(5.6) E = be the cube of unit side length and choose k (cid:96) ∈ M to be the constant sequence k (cid:96) = k = (1 / , / , / , / , / , / F -limit set F generated by thetriple ( F , { k (cid:96) } ∞ (cid:96) =0 , E ) is the classical Menger sponge.Now we consider variations of Menger Sponge. For each k = ( k (1) , k (2) , · · · , k (6) ) ∈M and 1 ≤ i ≤ U (cid:16) f ( i ) k (cid:17) = sup ( O,A,B,C ) ∈X diam (cid:16) f ( i ) k ( O, A, B, C ) (cid:17) diam (( O, A, B, C ))= sup ( O,A,B,C ) ∈X diam ( M k ( a i , b i , c i )[ O, A, B, C ] (cid:48) ) diam (( O, A, B, C ))= max { T ( a i +1 ) − T ( a i ) , R ( b i +1 ) − R ( b i ) , S ( c i +1 ) − S ( c i ) } . Similarly, L (cid:16) f ( i ) k (cid:17) = min { T ( a i +1 ) − T ( a i ) , R ( b i +1 ) − R ( b i ) , S ( c i +1 ) − S ( c i ) } . When k (2 j ) = 1 − k (2 j − for each j = 1 , ,
3, it is easy to check that (cid:88) i =1 U ( f ( i ) k ) s = (cid:88) i =1 max { T ( a i +1 ) − T ( a i ) , R ( b i +1 ) − R ( b i ) , S ( c i +1 ) − S ( c i ) } s = 8 max { k (1) , k (3) , k (5) } s + 4 max { − k (1) , k (3) , k (5) } s +4 max { k (1) , − k (3) , k (5) } s + 4 max { k (1) , k (3) , − k (5) } s . Example 5.3.2.
Let ˜ E = . Let (cid:0) k (1) , k (2) , k (3) , k (4) , k (5) , k (6) (cid:1) ∈ M where each k ( i ) is a random number in[0 , k (1) ≤ k (2) , k (3) ≤ k (4) , k (5) ≤ k (6) . Then thefirst generation E (1) of the construction results in a set like Figure 8. Example 5.3.3.
Let k (cid:96) = (cid:16) k (1) (cid:96) , k (2) (cid:96) , k (3) (cid:96) , k (4) (cid:96) , k (5) (cid:96) , k (6) (cid:96) (cid:17) ∈ M with each k (2 j − (cid:96) arandom number between given parameters λ and Λ and k (2 j ) (cid:96) = 1 − k (2 j − (cid:96) for each j = 1 , ,
3. Let F be the F -limit set generated by ( F , { k (cid:96) } ∞ (cid:96) =0 , E ). Then the thirditeration of the construction of F results in images like Figure 9. Here, in Figure9a the parameters λ = 0 and Λ = , while in Figure 9b the parameters λ = 0 . . TEP-WISE ADJUSTABLE IFS 19
Figure 8.
First generation of a randomly generated Menger sponge (a) λ = 0 , Λ = (b) λ = 0 . , Λ = 0 . Figure 9.
Generation 3 of random Menger spongeWe now calculate the dimension of the limit fractal F illustrated by Figure 9bin Example 5.3.3. Note that in general, when λ ≤ k (2 j − ≤ Λ for each j = 1 , , || U k || s ) s = (cid:88) i =1 U (cid:16) f ( i ) k (cid:17) s ≤ s + 12 max { − λ, Λ } s . Similarly, ( || L k || s ) s ≥ λ s + 12 min { − , λ } s . In particular, when λ = 0 .
32 and Λ = 0 .
35, for any s > . || U k || s ) s ≤ s + 12 max { − λ, Λ } s ≤ ∗ . s + 12 ∗ . s < ∗ . . + 12 ∗ . . ≈ . . By Theorem 4.0.2, dim H ( F ) ≤ . s ≤ . || L k || s ) s ≥ λ s + 12 min { − , λ } s ≥ ∗ . s + 12 ∗ . s ≥ ∗ . . + 12 ∗ . . ≈ . . By Theorem 4.0.2 again, dim H ( F ) ≥ . . ≤ dim H ( F ) ≤ . . Example 5.3.4.
For each (cid:96) ≥
0, let k (cid:96) = (cid:16) k (1) (cid:96) , k (2) (cid:96) , · · · , k (6) (cid:96) (cid:17) where k (1) (cid:96) = 13 + ( − (cid:96) (cid:96) + 1) , k (2) (cid:96) = 1 − k (1) (cid:96) ,k (3) (cid:96) = 13 − ( − (cid:96) (cid:96) + 1) , k (4) (cid:96) = 1 − k (3) (cid:96) ,k (5) (cid:96) = 13 + ( − (cid:96) (cid:96) + 1) , k (6) (cid:96) = 1 − k (5) (cid:96) . Let F be the F -limit set generated by ( F , { k (cid:96) } ∞ (cid:96) =0 , E ). Then the third generationof the construction of F leads to an image like Figure 10. Figure 10.
Generation 3 of random Menger sponge with con-trolled dimensionIn this case, we can still calculate the exact Hasudorff dimension of F . By directcomputation, lim (cid:96) →∞ ( || U k (cid:96) || s ) s = 203 s = lim (cid:96) →∞ ( || L k (cid:96) || s ) s . Thus, by Corollary 4.0.4, dim H ( F ) = log(20)log(3) ≈ . References [1] Barnsley, M.F.,
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Tynan Lazarus, Department of Mathematics, Universityof California at Davis, 1 Shields Ave, Davis, USA ([email protected])([email protected])