ON THE HAUSDORFF DIMENSION OF A FAMILY OF SELF-SIMILAR SETS WITH COMPLICATED OVERLAPS. 1. Introduction and Statements

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1 ON THE HAUSDORFF DIMENSION OF A FAMILY OF SELF-SIMILAR SETS WITH COMPLICATED OVERLAPS Abstract. We ivestigate the properties of the Hausdorff dimesio of the attractor of the iterated fuctio system (IFS) {γx, λx, λx + 1}. Sice two maps have the same fixed poit, there are very complicated overlaps, ad it is ot possible to directly apply kow techiques. We give a formula for the Hausdorff dimesio of the attractor for Lebesgue-almost all parameters (γ, λ), γ < λ. This result oly holds for almost all parameters: we fid a dese set of parameters (γ, λ) for which the Hausdorff dimesio of the attractor is strictly smaller. 1. Itroductio ad Statemets Our paper is motivated by a questio of Pablo Shmerki at the coferece i Greifswald i 28. The questio was the followig: Questio. What is the Hausdorff dimesio of the attractor, which is geerated by the IFS { 1 4 x, 1 3 x, 1 3 x + 2 3}? Let us deote the Hausdorff dimesio of a compact subset Λ of R by dim H Λ. For the defiitio ad basic properties of Hausdorff dimesio we refer the reader to [1] or [2]. Let us recall here the defiitio of the attractor. Let {f,..., f } be a family of cotiuous self-maps o the real lie. We will i additio assume that each f i is a cotractio, that is, f i (x) f i (y) r i x y for all x, y ad for some < r i < 1. The there exists a uique, oempty compact subset Λ of R which satisfies Λ = f k (Λ). We call this set Λ the attractor of the iterated fuctio system (IFS) {f (x), f 1 (x),..., f (x)}. k= Date: July 9, 29. Key words ad phrases. Hausdorff dimesio, Iterated Fuctio System, Trasversality Research of Báráy was supported by the EU FP6 Research Traiig Network CODY..

2 2 Let us suppose that the fuctios of the IFS are similarities i the form {f i (x) = λ i x + d i } i=, where < λ i < 1 for every i {,..., }. We say that the attractor of the IFS or the IFS itself is self-similar. It is well kow if a self-similar IFS satisfies the so called ope set coditio (OSC), i.e. there exists a ope set U such that for every i, j {,..., }, f i (U) U ad f i (U) f j (U) = if i j, the the Hausdorff dimesio of the attractor is the uique solutio of λ i s = 1, (1.1) i= see for example [4]. Eve if the OSC does ot hold, the solutio of equatio (1.1) is called similarity dimesio of the IFS. The similarity dimesio is always a upper boud for the Hausdorff dimesio of the attractor, see [1]. I the case whe the IFS has overlappig structure, i.e. the ope set coditio does ot hold, the Hausdorff dimesio of the attractor Λ of IFS {f i (x) = λ i x + d i } i= is dim H Λ = mi {1, s} for a.e. d R +1, (1.2) where s is the uique solutio of (1.1), see [11] ad [3]. I this article, we cosider the IFS {γx, λx, λx + 1}, where we assume throughout the paper that < γ < λ < 1. Let us deote the attractor of this IFS by Λ γ,λ. The problem of calculatig the Hausdorff dimesio of the attractor of this system is far from beig simple. The special property of our chose class of IFS is that the first two maps have a commo fixed poit. This fact implies that the two maps commute, so we are goig to observe a immese (icreasig expoetially uder iteratio) amout of exact ad partial overlaps i our system. Needless to say, the OSC does ot hold. Iterated fuctio systems that do ot satisfy OSC were first studied i [9], where the trasversality coditio method was first itroduced. See [7], [8] for the most geeral treatmet of this approach. Sice this time, several other methods were proposed: weak separatio coditio, fiite type coditio ad others (see, for example, [5], [6] ad [14]). However, either of those is goig to work for overlaps as severe as our system has. For this reaso, we are forced to modify the trasversality method, applyig it oly to some subsystems of the IFS (details will be preseted i the followig sectios). The mai result of this paper is as follows:

3 HD OF {γx, λx, λx + 1} 3 Theorem 1.1. Let Λ γ,λ the attractor of the IFS {γx, λx, λx + 1}. The for Lebesgue almost every < γ < λ < 1 2 where s γ,λ is the uique solutio of dim H Λ γ,λ = mi { 1, s γ,λ}, (1.3) 2λ s + γ s λ s γ s = 1. (1.4) Moreover L(Λ γ,λ ) > for Lebesgue almost every (γ, λ) such that s γ,λ > 1. More precisely, the above statemets are true for every fixed < λ < 1 2 Lebesgue almost every < γ < λ. ad Note here that the assumptio that λ < 1 2 is ot really restrictive: the attractor of our system cotais the attractor of its subsystem {λx, λx + 1}, which for λ 1 2 is a iterval ad, i particular, it has dimesio 1. Our result oly holds for almost all parameters. Ad ideed, we ca preset a family of parameter values for which the result does ot hold. Propositio { 1.2. Let q ad } p itegers ad q > p, (q, p) = 1. Let Λ λ,q,p be the attractor of λ q p x, λx, λx + 1. The { } dim H Λ λ,q,p mi 1, s λ p,q, (1.5) where s λ p,q is the uique solutio of p 1 ( ) 2λ s q p + λ k+1 s = 1. Note that this family of exceptioal parameter values is dese i k=1 {(γ, λ) : 2λ + γ < 1, γ < λ}, where the statemet of Propositio 1.2 excludes the possibility that the assertio of Theorem 1.1 holds. This implies that the fuctio (γ, λ) dim H Λ γ,λ caot be cotiuous. 2. Trasversality methods First let us itroduce the trasversality coditio for self-similar IFS with oe parameter. The defiitio correspods to the defiitio i [12],[13] which was itroduced for geeral IFS. Let U be a ope, bouded iterval of R ad Σ a fiite set of symbols. Let Ψ t = { ψ t i (x) = λ i(t)x + d i (t) } i Σ, where λ i, d i C 1 (U) ad < α λ i (t) β < 1 for every i Σ ad t U ad for some α, β (, 1). Let Λ t be the attractor of Ψ t

4 4 ad π t be the atural projectio from the symbolic space Σ N to Λ t. More precisely, let i = (i i 1... ) Σ N ad π t (i) = lim ψt i ψ t i 1 ψ t i (). (2.1) It is well-kow that the limit exists ad idepedet of the base poit. Moreover, π t is a surjective fuctio from Σ N oto Λ t. Deote σ the left-shift operator o Σ N. More precisely, let σ : (i i 1... ) (i 1 i 2... ). It is easy to see that π t (i) = ψ t i (π t (σi)). Defiitio 2.1. We say that Ψ t satisfies the trasversality coditio o a ope, bouded iterval U R, if there exists a costat C > such that for every i, j Σ N with i j it holds that L(t U : π t (i) π t (j) r) Cr for every r >, where L is the Lebesgue measure o the real lie. This defiitio is equivalet to the oes give i e.g. [12], [13]. Now let us recall the Theorem of K. Simo, B. Solomyak ad M. Urbański [12, Theorem 3.1] i the self-similar case with oe parameter. Theorem 2.2 (Simo, Solomyak, Urbański). Suppose that Ψ t satisfies the trasversality coditio o a ope, bouded iterval U. The (1) dim H Λ t = mi {s(t), 1} for Lebesgue-a.e. t U, (2) L(Λ t ) > for Lebesgue-a.e. t U such that s(t) > 1, where s(t) is the similarity dimesio of Ψ t. equatio λ i (t) s(t) = 1. i Σ More precisely, s(t) satisfies the We ca use the followig Lemma to prove trasversality which follows from [12, Lemma 7.3]. Lemma 2.3. Let U R be a ope, bouded iterval ad f i,j (t) = π t (i) π t (j). If for every i, j Σ N with i j ad for every t U f i,j (t ) = df i,j dt (t ) > (2.2) the there is trasversality o ay ope iterval V whose closure is cotaied i U.

5 HD OF {γx, λx, λx + 1} 5 3. Proofs Let us fix a λ (, 1 2 ). Without loss of geerality we ca assume that γ = cλ, where < c < 1. Let ψ c(x) = cλx, ψc 1 (x) = λx ad ψc 2 (x) = λx + 1. We ote that ψ1 c, ψc 2 do ot deped o c. Let us deote Σ = =1 Σ ad for every 1 let ψi c = ψc i ψi c 1 ψi c where i Σ. We ote that Ψ c = {ψ c, ψc 1, ψc 2 } does ot satisfy the trasversality coditio, sice for every i, j {, 1} N ad < c < 1 we have π c (i) π c (j). I order to prove Theorem 1.1, we are goig to itroduce well-chose systems Ψ c which do satisfy trasversality. Moreover, we are goig to show that the attractor of Ψ c is cotaied i, ad has dimesio arbitrarily close to, the attractor Λ cλ,λ of IFS Ψ c, so that we are able to coclude iformatio o the dimesio of Λ cλ,λ by studyig these subsystems with trasversality. First of all, we have to cosider some properties of the atural projectio. Let i Σ N ad deote by i() the first elemets of i, moreover deote by i i() the umber of is i i(). Similarly, let i i(, l) be the umber of is betwee the th ad lth elemets of i. We ote that i() is the empty word ad i i( 1, l) := i i(l). The π c (i) = δ ik c i(k) λ k, (3.1) k= where δ i = 1 if i = 2, else δ i =. We ca write the atural projectio i aother form. Let α i 2 be the umber of 2 s i i, more precisely, let αi 2 = lim i() if i is a ifiite legth word of symbols ad α i 2 = 2i( i ) if i is fiite. If i is ifiite the α2 i ca be equal to ifiite. We will use the otatio i l for the place of the lth 2 i i (here i ca be fiite or ifiite) for every 1 l α2 i. We defie i := 1 ad if α2 i is fiite, the let i := i. We ote that if i does ot cotai ay 2 s the α i 2 +1 π c (i) =. Observe that with this otatio π c ca be writte as α i 2 π c (i) = c i( i k ) λ i k, k=1 we ote that we defie the empty sum as. If α2 i, the let P i = { i( i l ) : l 1} else let P i =. Moreover let } } m i k {j = sup : i( i j) = k, rk {j i = if : i( i j) = k. (3.2)

6 6 for k P i. We ote that P i ca be fiite or ifiite, ad if P i is fiite the m i max P i ca be ifiite. Therefore π c (i) = mi c k d i k (λ), where k di k (λ) = k P i l=r i k λ i l. (3.3) We call i the degree ad i m i k rk i lowerdeg respectively. the lower degree of d i k (λ) deoted by deg ad Lemma 3.1. For every i Σ N ad λ (, 1 2 ) we have (1) lowerdegd i k (λ) k for every k P i, (2) deg d i k (λ) + l k + 1 lowerdegdi l (λ) for every k, l P i, k < l, (3) λ l k d i k (λ) di l (λ) for every k, l P i, k < l. Proof. Sice i(k) k, (1) is obvious. Let k, l P i, k < l. Sice deg d i k (λ) = i ad lowerdegd i m i l (λ) = i betwee k rl, i the i ad i elemets of i there have to be l k zeros. Therefore m i k rl i l k = i( i, i ) i i 1. m i k rl i rl i m i k This completes the proof of (2). The property (3) is a easy cosequece of (2) by usig the fact λ < 1 2. Now we are goig to defie the families Ψ c of iterated fuctio systems for which trasversality holds. First of all, we defie sets of fiite legth words of symbols Σ i by iductio. Let Σ 1 = {1, 2} ad for every 1 let Σ +1 = {i} {1i, 2i}. (3.4),i 1 For example Σ 2 = {1, 2} 2 {2} ad Σ 3 = {1, 2} 3 {12, 22, 21, 22, 2} etc. Obviously, for every 1, Σ Σ. Lemma 3.2. For every 1 ad for every i, j Σ, i = j if ad oly if α i 2 = αj 2 ad l αi 2 = αj 2, i l = j l ad i( i l, i l+1 ) = j( j l, j l+1 ) (3.5) Proof. The implicatio ( i = j ) = (3.5) is obvious for every 1. We are goig to prove the other directio by iductio. For = 1, (3.5) = ( i = j ) is trivial. Let us suppose that Σ satisfies the statemet ad let i, j Σ +1 such that i, j satisfy (3.5). Therefore i = 2i ad

7 HD OF {γx, λx, λx + 1} 7 j = 2j or i = i or 1i ad j = j or 1j, where i, j Σ. I the first case by usig the iductioal assumptio we have i = j. I the secod case we will show that i = i if ad oly if j = j. Let us suppose that i = i the j( j 1 ) = i( i 1 ) = i 1 = j 1. I the middle equatio we used (3.4). Therefore j = j. The reversed directio is similar. By usig the iductioal assumptio we have i = j. Lemma { 3.3. } For every arbitrary small ε > ad for every 2 the system Ψ c = ψi c satisfies the trasversality coditio o c (ε, 1 ε). Proof. We ote that λ (, 1 2 ) is fixed. Let ε > be arbitrary small but fixed. We are goig to prove trasversality of Ψ c by usig Lemma 2.3. More precisely, we are goig to show that (2.2) holds o U = ( ε 2, 1). Let us suppose that c ( ε 2, 1). Let ĩ, j Σ N, such that ĩ = (i i 1 i 2 {... ), } j = (j j 1 j 2... ) ad i j. We ca defie the atural projectio of Ψ c = ψi c i similar way as i (2.1). Deote the atural projectio of Ψ c by π c. Let us assume that π c (ĩ) = π c ( j) for a c ( ε 2, 1). Let i = ĩ, j = j as elemets of Σ N. Therefore π c (ĩ) = π c(i) ad π c ( j) = π c (j). If mi P i = mi P j the there is the same umber of zeros before the i 1th elemet of i ad before j 1 th elemet of j. If i 1 > j 1 the some simple algebra, usig that λ < 1 2, shows that π c (i) < π c (j), which is a cotradictio. Likewise, we caot have i 1 < j 1, so ecessarily i 1 = j 1. However, if i 1 = j 1, the we have that ( ) π c (i) π c (j) = c mi P i λ i 1 π c (σ j 1 +1 i) π c (σ j 1 +1 j), where σ is the left-shift operator of Σ N = {, 1, 2} N. Sice we supposed that i j Σ, by Lemma 3.2 ad c > ε 2 we ca assume without loss of geerality that mi P i > mi P j. Let us deote l j = mi P j. Therefore f j,i (c) = π c ( j) π c (ĩ) = π c(j) π c (i) = c l j (λ)+ c l j d j l j (λ) 1 + c k d j k (λ) k P i c k d i k (λ) = c k l dj j k (λ) (λ) c k l di j k (λ) k P i d j. l j (λ)

8 8 Let us defie g j,i (c) = 1 + c k l j dj k (λ) (λ) k P i c k l j di k (λ) (λ). Note that, sice c > ε 2 > oe has f j,i(c ) = if ad oly if g j,i (c ) =. O the other had, it is easy to see that if g j,i (c ) =, the Therefore it is eough to prove df j,i dc (c ) = dg j,i dc (c ) =. dg j,i dc (c) = g j,i(c) >. Let us suppose that dg j,i dc (c ) = for some c ( ε 2, 1). The dg j,i = c dc (c ) = c (k l j ) c k l j 1 d j k (λ) (λ) (k l j ) c k l j 1 d i k (λ) k P i d j l j (λ) (k l j ) c k l j (k l j 1) c k l j By usig (3) of Lemma 3.1 we have d j k (λ) (λ) + d j k (λ) (λ) c k l j d i k (λ) k P i (λ) = (k l j 1) c k l j λ k l j + c k l j c k l j d j k (λ) (λ) c k l j d i k (λ) k P i (λ). d j k (λ) (λ) c k l j d i k (λ) k P i (λ) We ca give a upper boud for the first term, (k l j 1) c k l j λ k l j (k 1)c k λ k = (c λ) 2 (1 c λ) 2 < 1. k=1 I the last iequality we used that λ < 1 2. Therefore by the defiitio of g j,i(c) we have that < g j,i (c ). Usig Lemma 2.3 for U = ( ε 2, 1) we have that Ψc satisfies trasversality o V = (ε, 1 ε). Let us deote the attractor of Ψ c = { } ψi c by Λ cλ,λ Σ.

9 HD OF {γx, λx, λx + 1} 9 Propositio 3.4. For every < λ < 1 2 ad Lebesgue almost every < c < 1 lim dim H Λ cλ,λ Σ = dim H Λ cλ,λ. Proof. First we ote that by usig the Lemma 3.3 ad Theorem 2.2 we have for every ε >, 1 ad almost every ε < c < 1 ε { } dim H Λ cλ,λ Σ = mi 1, s c,λ where s c,λ is the uique solutio of ( c i λ ) s = 1. (3.6) Sice for every ε > the dimesio formula (3.6) holds for a.e. c (ε, 1 ε), it holds for a.e. c (, 1). We ote that s c,λ is a bouded, mootoe icreasig sequece, therefore it is coverget. Let s c,λ be the limit of this sequece. The lower boud is trivial sice for every, Λ cλ,λ Σ Λ cλ,λ. Therefore mi { s c,λ, 1} dim H Λ cλ,λ. for every < λ < 1 2 ad Lebesgue almost every < c < 1. Now we prove the upper boud. It is easy to see that the iterval the covex hull of Λ cλ,λ. By usig the fact [ ] 1, 1 λ is ψ c ψ c 1(x) ψ c 1 ψ c (x) ad < c < 1 we have that for every i Σ there exists j Σ such that ([ ]) ([ ]) ψi c 1, ψ c 1 j,. 1 λ 1 λ We ote how to fid such j Σ. The positios of the 2 s i j are the same as i i. Before the first appearace of 2 ad betwee ay two cosecutive appearaces of 2, we keep the same umber of s as i i, but chage the order so that all 1 s come before all s. After the last occurrece of a 2 (or everywhere if there are o 2 s i i), we replace all s by 1 s. { ([ ])} For each N, we cosider the coverig of Λ cλ,λ give by ψi c 1, 1 λ, ad ote that the diameters of sets i the cover are at most λ. Therefore by usig the defiitio of Hausdorff measure (see [1]) we have Hλ s (Λcλ,λ ) ( c i λ ) s, 1 λ

10 1 where H s δ (Λ) = if { i U i s : Λ i U i, U i < δ}. Let ε > be arbitrary small. The H s c,λ +ε λ (Λ cλ,λ ) ( c i λ 1 λ ) s c,λ +ε ( ) 1 s λ ε c,λ +ε 1 λ which teds to as. Therefore by the defiitio of Hausdorff dimesio (see [1]) dim H Λ cλ,λ s c,λ + ε where ε > was arbitrary. Thus the propositio is proved. Now we are able to prove Theorem 1.1. Proof of Theorem 1.1. Let Σ the followig set of symbols {}}{ Σ = {1, 2, 2, 2,...,... 2}. (3.7) { } Let Λ cλ,λ Σ the attractor of ψi c. Notice that for every i Σ ca be decomposed i Σ as a juxtapositio i = j 1 j k, where each j r is i Σ. Therefore for every < λ < 1 2 ad Lebesgue almost every < c < 1, Λcλ,λ Σ Λ cλ,λ, ad therefore by Σ usig Propositio 3.4, 1 dim H Λ cλ,λ = lim dim H Λ cλ,λ Σ lim dim H Λ cλ,λ. Σ The lower boud is trivial, therefore dim H Λ cλ,λ = lim dim H Λ cλ,λ. (3.8) Σ { } We use the fact that for every 2, ψi c satisfies the trasversality coditio i Σ o (ε, 1 ε) for all ε >. As the proof of this claim is very similar to the proof of Lemma 3.3, we omit it. By Theorem 2.2 ad by similar argumets as i the begiig of the proof of Propositio 3.4, we have for every 1 ad almost every < c < 1 where s c,λ dim H Λ cλ,λ Σ is the uique solutio of k=1 { = mi 1, s c,λ } 1 ( 2λ s + c k λ k+1) s = 1.

11 It is easy to see by (3.8) that s cλ,λ = lim s c,λ HD OF {γx, λx, λx + 1} 11 2λ s + k=1 is the uique solutio of ( c k λ k+1) s = 1. (3.9) We ote that the fuctio f 1 (s) = 2λ s + λs γ s 1 γ is strictly mootoe icreasig for s every γ, λ (, 1), moreover lim s + f 1 (s) = + ad lim s f 1 (s) =. Therefore the equatio (3.9) has a uique solutio, which also satisfies the followig equatio: 2λ s + (cλ) s (cλ 2 ) s = 1. (3.1) By similar argumets oe ca prove that (3.1) has uique solutio as well, which was the first statemet of Theorem 1.1. Now we prove the measure claim of Theorem 1.1. If s cλ,λ > 1, the s c,λ large eough, so that, by trasversality, almost every c. Sice Λ cλ,λ Σ > 1 for has positive Lebesgue measure for Λ cλ,λ Σ Λ cλ,λ, this completes the proof of Theorem 1.1. Fially, we are provig Propositio 1.2. Proof { of Propositio 1.2. Let q, p itegers } such that (q, p) = 1 ad q > p ad let ψ (x) = λ q p x, ψ 1 (x) = λx, ψ 2 (x) = λx + 1. It is easy to see that ψ 1 (x) ψ 1 (x), ψ } {{ (x) ψ } 1 1 (x). }{{} p q Therefore if we have i Σ N = {, 1, 2} N the we ca choose j Σ N p ( Σ p is defied as i (3.7)) such that π(i) = π(j). Sice, wheever there are at least p cosecutive zeros i i, we ca replace each block of p cosecutive zeros by blocks of q cosecutive oes, ad the we ca rearrage the zeros ad oes betwee two cosecutive twos, by movig the oes to the frot. Therefore dim H Λ λ,q,p = dim H Λ λ,q,p Σ p where Λ λ,q,p Σ p is the attractor of IFS { ψ i }i Σ p. Sice the Hausdorff dimesio of a self-similar set is always at most the miimum of the similarity dimesio (see (1.1)) ad the dimesio of ambiet space, we have dim H Λ λ,q,p Σ p mi { } 1, s λ q,p

12 12 where s λ q,p is the uique solutio of which was to be proved. p 1 ( ) 2λ s q p + λ k+1 s = 1, k=1 Ackowledgmet. I am grateful to Pablo Shmerki ad Micha l Rams for useful discussios. Refereces [1] K. J. Falcoer, Fractal Geometry: Mathematical Foudatios ad Applicatios, Wiley, 199. [2] K. J. Falcoer, Techiques i Fractal Geometry, Wiley, [3] K. J. Falcoer, The Hausdorff Dimesio of Some Fractals ad Attractors of Overlappig Costructio, J. Stat. Phys., 42(1/2), (1987), [4] J. E. Hutchiso, Fractals ad Self-Similarity, Idiaa Uiv. Math. Joural, Vol. 3, No. 5, (1981), [5] K.-S. Lau, S.-M. Ngai ad H. Rao, Iterated fuctio systems with overlaps ad self-similar measures, J. Lodo Math. Soc. (2) 63 (21), o. 1, [6] Sze-Ma Ngai ad Yag Wag, Hausdorff dimesio of self-similar sets with overlaps, J. Lodo Math. Soc., 2, 63, (21), [7] Y. Peres ad B. Solomyak, Absolute cotiuity of Beroulli covolutios, a simple proof, Math. Research Letters, 3, o. 2, (1996), [8] Y. Peres ad B. Solomyak, Self-similar measures ad itersectios of Cator sets, Tras. Amer. Math. Soc. 35, o. 1 (1998), [9] M. Pollicott ad K. Simo, The Hausdorff dimesio of λ-expasios with deleted digits, Tras. Am. Math. Soc., 347, o. 3, (1995), [1] Hui Rao ad Zhi-Yig We, A Class of Self-Similar Fractals with Overlap Structure, Advaces i applied Mathematics, 2, (1998), [11] K. Simo ad B. Solomyak, O the Dimesio of Self-Similar Sets, Fractals, 1, No. 1, (22), [12] K. Simo, B. Solomyak ad M. Urbański, Hausdorff dimesio of limit sets for parabolic IFS with overlaps, Pacific J. Math., 21, o. 2, (21), [13] K. Simo, B. Solomyak, ad M. Urbański, Ivariat measures for parabolic IFS with overlaps ad radom cotiued fractios, Tras. Amer. Math. Soc. 353 (21), [14] Marti W. Zerer, Weak separatio properties for self-similar sets, Proc. of the Am. Math. Soc., 124, o. 11, (1996), Balázs Báráy, Śiadeckich 8, P.O. Box 21, -956 Warszawa 1, Polad address: balubsheep@gmail.com