A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES. 1. Introduction

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1 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES DE-JUN FENG AND YANG WANG 1 Itroductio Let I = {φ m be a iterated fuctio system (IFS cosistig of a family of cotractive affie maps o R d Hutchiso [13] proved that there exists a uique compact set K = K(I, called the attractor of the IFS I, such that K = m φ (K Moreover, for ay give probability vector p = (p 1,, p m, ie p > 0 for all ad m p = 1, there exists a uique compactly supported probability measure ν = ν I,p such that (11 ν = p ν φ 1 This paper is devoted to the study of fudametal properties of a class of self-affie sets ad measures, such as the L q spectrum, the Hausdorff dimesio ad the etropy dimesio It is well kow that problems cocerig self-affie sets ad measures are typically difficult Questios that may be trivial i the self-similar settig are ofte itractable i the self-affie settig A tellig example is calculatig the Hausdorff ad box dimesios of the attractor of a IFS I = {φ m If all φ are similitudes ad I satisfies the so-called ope set coditio (OSC the Haudorff dimesio ad the box dimesio of the attractor K(I agree, ad they are easily computable by the formula ρ dim H(K = 1 where ρ i deotes the cotractio ratio of φ, see eg Falcoer [5] Eve without the ope set coditio the dimesio of K(I ca ofte be computed if I belogs to a more geeral class called the fiite type IFS, see eg Lalley [16] ad Ngai ad Wag [23] ad the refereces therei However this is o loger the case whe φ i are affie maps Eve uder the ope The first author is supported i part by the Special Fuds for Maor State Basic Research Proects i Chia The secod author is supported i part by the Natioal Sciece Foudatio, grats DMS ad DMS

2 2 DE-JUN FENG AND YANG WANG set coditio we kow how to compute the Hausdorff dimesio of K(I oly for very special I s, ad for which the solutios are quite otrivial McMulle [21] ad Bedford [1] idepedetly computed the Hausdorff ad box dimesios of K(I for I = {φ m i which all φ have the form (12 φ (x = [ ] k 1 x + [ ] a / b /k where all a, b are itegers, 0 a < ad 0 b < m They foud that the Hausdorff dimesio ad the box dimesio are ot the same i geeral Lalley ad Gatzouras [17], i a highly techical paper alog the same spirit of [21], computed the Hausdorff ad box dimesios for a broader class of IFS I = {φ m, i which φ map the uit square (0, 1 2 ito disoit rectagles havig certai geometric arragemet iside the uit square More precisely, i the Lalley-Gatzouras class all rectagles φ ((0, 1 2 are parallel to the axes ad have loger sides parallel to the x-axis Furthermore oce proected oto the x-axis these rectagles are either idetical or disoit Aside from a few other special cases such as the graph-directed McMulle class studied by Keyo ad Peres [14], the Lalley-Gatzouras class (which icludes the McMulle class remais the oly determiistic class of self-affie sets whose Hausdorff dimesio are kow I [10] Hu obtaied the box dimesio of a class of oliear self-affie sets i terms of the topological pressure More precisely, he cosidered expadig maps o R 2 which leaves ivariat a strog ustable foliatio ad obtaied a formula for the box-dimesio of its closed ivariats sets ivolvig topological pressures Alog aother directio, Falcoer, i a celebrated paper [4], gave a variatioal formula for the Hausdorff ad box dimesios for almost all self-affie sets uder some assumptios Later, Hueter ad Lalley [12] ad Solomyak [27] proved that Falcoer s formula remais true uder some weaker coditios We focus o the L q spectrum ad the Hausdorff ad etropy dimesios of a self-affie measure i this paper These quatities are importat basic igrediets i the study of fractal geometry, particularly i the study of multifractal pheomea As a by-product we also obtai results o dimesios of self-affie sets Let ν be a compactly supported measure i R d ad q R For each 1 let D be the set of cubes {[0, 2 d + α : α 2 Z d The L q spectrum of ν is defied as log τ (ν, q (13 τ(ν, q = lim log 2, where τ (ν, q = Q D (ν(q q,

3 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 3 if the limit exists Related to τ(ν, q are the L q dimesio D(ν, q ad the etropy dimesio h(ν of ν, defied respectively by (14 D(ν, q := ad (15 h(ν := lim τ(ν, q q 1 Q D ν(q log(1/ν(q log 2 if the limit exists For a similarity IFS I = {φ m with the ope set coditio ad ay probability vector p = (p 1,, p m the L q spectrum of ν = ν I,p is kow to be aalytic i q R, give by the equatio (16 p q ρ τ(ν,q = 1, where ρ deotes the cotractio ratio of φ, see Cawley ad Mauldi [2] ad Olse [24] Moreover, the Legedre trasform τ (ν, α of τ(ν, q give by { (17 τ (ν, α := if qα τ(ν, q : q R equals the Hausdorff dimesio of the set { K(α := x supp (ν : ( log ν Br (x lim r 0 + log r = α For a self-similar measure without the ope set coditio, however, the L q spectrum is geerally difficult to obtai ad is calculated for oly a few special cases, see [18, 7, 8, 19] Oe importat such special case is the class of fiite type IFS s ([23], a substatially larger class tha the class with the OSC For a fiite type IFS i R, Feg [9] expressed τ(ν, q via products of certai oegative matrices, ad proved that τ(ν, q is differetiable for q (0, As oe would expect, eve less is kow about the L q spectrum ad the Hausdorff ad etropy dimesios of a self-affie measure Kig [15] calculated τ(ν, q for ν = ν I,p where the IFS I is i the McMulle class (12 He gave a detailed multifractal aalysis for such measures Olse [25] geeralized Kig s results to dimesios d 3 Peres ad Solomyak [26] proved the existece of τ(ν, q ad h(ν for the class of self-coformal measures, ad asked whether they also exist for all self-affie measures I [6], Falcoer gave a variatioal formula for the L q -spectrum (1 < q < 2 for almost all self-affie sets uder some assumptios

4 4 DE-JUN FENG AND YANG WANG I this paper we calculate the L q spectrum ad the etropy dimesio for a class of self-affie measures i R 2 This class of self-affie measures ν I,p requires oly that the uderlyig IFS s I = {φ m satisfy the rectagular ope set coditio (ROSC It is a much larger class tha the McMulle class studied i [15] ad the Lalley-Gatzouras class Simply speakig, I = {φ m i R2 satisfies the ROSC if there is a ope rectagle T such that the maps φ i map T ito disoit rectagles parallel to the axes iside T As a applicatio we obtai the formula for the box dimesio of K(I uder the ROSC as well as the Hausdorff dimesio of ν I,p uder some additioal assumptios Our results o the box dimesio ca be viewed as a extesio of the box dimesio results by Lalley ad Gatzouras [17] ad Hu [10] 2 Statemet of Mai Results We first itroduce some defiitios ad otatios The ambiet dimesio i the rest of the paper will be set to d = 2, although most of the defiitios exted easily to higher dimesios Let I = {φ m be a affie IFS i R2 Throughout this paper we shall always assume that φ (x, y = (a x + c, b y + d with 0 < a, b < 1 for all Thus each φ maps ay rectagle (0, R 1 (0, R 2 + v to a rectagle parallel to the axes Defiitio 21 We say that I = {φ m satisfies the rectagular ope set coditio (ROSC if there exists a ope rectagle T = (0, R 1 (0, R 2 + v such that {φ (T m are disoit subsets of T For a self-affie measure ν = ν I,p associated with I ad probability vector p we shall defie the proectios ν x ad ν y of ν oto the x- ad y-axes, which we rely o heavily i this paper Let I x := {π x φ π 1 x = a x + c ad I y := {π y φ π 1 y = b y + d be the proectios of I, where π x ad π y are the caoical proectios of R 2 oto the x- ad y-axes, respectively We defie ν x = ν I x,p ad ν y = ν I y,p It is easy to check that ν x = ν π 1 x (21 Γ(d := ad ν y = ν πy 1 For ay d = (d 1, d 2,, d m we use Γ(d to deote { t = (t 1, t 2,, t m : t 0, m t = 1, m d t 0 We also use log d to deote (log d 1, log d 2,, log d m (if all d > 0 Our mai theorem cocerig the L q spectrum of ν is:

5 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 5 Theorem 21 Let I = {φ m be a affie IFS i R2 satisfyig the ROSC, with φ (x, y = (a x + c, b y + d ad 0 < a, b < 1 for all Let a = (a 1, a 2,, a m ad b = (b 1, b 2,, b m Suppose that p = (p 1, p 2,, p m is ay probability vector The for q > 0 the L q spectrum of ν = ν I,p is τ(ν, q = mi(θ a, θ b, where t ( log t τ(ν y, q(log b log a + q log p θ a = if t Γ(log b log a t log a t θ b = if t Γ(log a log b ( log t τ(ν x, q(log a log b + q log p t log b We poit out that if a < b for all the the set Γ(e b is empty, ad if so we have τ(ν, q = θ a I fact we prove: Theorem 22 Uder the hypotheses of Theorem 21, assume furthermore that a b for all The τ(ν, q satisfies (22 a τ(νy,q τ(ν,q b τ(νy,q p q = 1 Theorem 22 allows us to easily calculate the L q spectrum if τ(ν y, q is kow, which is the case if I is i the McMulle or Lalley-Gatzouras class Moreoever, Theorem 21 allows us to calculate τ(ν, q, at least i theory, if the proectios of I oto the two axes are of fiite type by the result of Feg [9], makig the L q spectrum computable for a cosiderably larger class of IFS s tha the Lalley-Gatzouras class Oe of the applicatios of the above two theorems is a formula for the box dimesio of K = K(I It is easy to see that K = supp (ν ad by defiitio dim B (K is simply τ(ν, 0 Therefore we also obtai as a by-product of Theorems 21 ad 22 a formula for dim B (K: Corollary 23 Uder the hypotheses of Theorem 21, we have dim B (K = max(u a, u b, where t (log t dim B (π y (K(log b log a u a = sup t Γ(log b log a t log a ( t log t dim B (π x (K(log a log b u b = sup t Γ(log a log b t log b

6 6 DE-JUN FENG AND YANG WANG If assume furthermore that a b for all, the dim B (K satisfies (23 a dim B(K dim B (π y(k b dim B(π y(k = 1 I particular, if dim B (π y (K = 1 furthermore, the (24 a dim B(K 1 b = 1 The computable cases i the above corollary for dim B (K clearly iclude the Lalley- Gatzouras class, i which a b ad I y satisfies the OSC I [10] Hu obtaied a formula for the box-dimesio dim B (K i terms of topological pressures whe a b uiformly I a persoal commuicatio, Hu iformed us that (23 ca be also derived from his formula after some otrivial calculatios Aother applicatio of the theorems is computig the Hausdorff dimesio of a self-affie measure Let ν be a fiite Borel measure i R d It is said to be exactly dimesioal if there exists a costat c such that log ν(b r (x lim = c ν ae x R d r 0 log r Ngai [22] proved that if τ(ν, q is differetiable at q = 1 the ν is exactly dimesioal, ad dim H (ν = c = d dq τ(ν, 1 As a corollary of Theorem 22 we obtai a Ledrappier-Youg type formula (see [20] for dim H (ν: Theorem 24 Uder the hypotheses of Theorem 22, if τ(ν y, q is differetiable at q = 1 the so is τ(ν, q, ad dim H (ν = p (log p + dim H (ν y (log a log b p log a I particular if furthermore a = a ad b = b for all ad a b the dim H (ν = p log p + dim H(ν y log(a/b log a Our techique ca also be used to study the etropy dimesio, which for a Borel measure ν is defied i (15 It is kow [26] that the etropy dimesio exists for all self-similar (i fact self-coformal measures We determie the etropy dimesio for the self-affie measures with ROSC:

7 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 7 Theorem 25 Uder the hypotheses of Theorem 21, h(ν = h(ν y p (log b log a + p log p p log a h(ν x p (log a log b + p log p p log b if p (log b log a 0, otherwise 3 Some Combiatorial Results We establish two combiatorial results that will be eeded to prove our mai theorems i this paper First let us itroduce some otatios o symbolic spaces These otatios are mostly stadard We use Σ = Σ(m to deote the alphabet {1, 2,, m Wheever there is o ambiguity we shall use Σ rather tha Σ(m, as m is usually fixed i this paper The set of all words i Σ of legth is deoted by Σ, with Σ := 0 Σ ad Σ N beig the set of all oe-sided ifiite words Here we adopt the covetio that Σ 0 cotais oly the empty word Associated with Σ are two actios: The left shift actio σ ad ad the right shift actio δ, defied respectively by σ( = δ( = ad σ(i 1 i 2 i k = i 2 i k, δ(i 1 i 2 i k = i 1 i k 1 for each i 1 i 2 i k Σ with k 1 We shall use boldface letters i,, l to deote elemets i Σ or Σ N For each sequece a = (a m we may exted it to a fuctio f a : Σ R by f a ( = 1 ad f a (i = a i1 a ik for i = i 1 i k Most of the time, because there is o ambiguity, we shall use the simplified otatio a i i place of f a (i The above are geeral purpose otatios Now we itroduce some that are specific to this paper Suppose that a = (a m ad b = (b m are two sequeces with 0 < a, b < 1 for all For ay 0 < r < 1 let A r := A r (a, b = { i Σ : a δ(i r, b δ(i r, mi (a i, b i < r ad A a r := {i A r : a i b i, A b r := {i A r : a i > b i

8 8 DE-JUN FENG AND YANG WANG Suppose that c = (c m is aother sequece of positive real umbers The obective of this sectio is to evaluate several limits Set ( Θ a = Θ a log i A (c := lim a c i r, ad r 0 + log r ( Θ b = Θ b log i A (c := lim b c i r r 0 + log r Similarly, for ay probability vector p = (p 1, p 2,, p m set Ω a = Ω a i A (c, p := lim a p i log c r i, ad r 0 + log r Ω b = Ω a i A (c, p := lim b p i log c r i r 0 + log r We prove the followig results: Propositio 31 Give sequeces a = (a m ad b = (b m with all a, b i (0, 1 let e a = log b log a ad e b = log a log b (i If a b for some 1 m the Θ a (c exists, ad Θ a t ( log t + log c (c = if, t Γ(log b log a t log a where for ay vector d Γ(d is defied i (21 (ii If a > b for some 1 m the Θ b (c exists, ad Θ b t ( log t + log c (c = if t Γ(log a log b t log b (iii Θ a (c (resp Θ b (c is cotiuous with respect to c if it exists Propositio 32 Uder the assumptios of Propositio 31, ad let p be a probability vector (i If p (log b log a > 0 the (ii If p (log b log a < 0 the (iii If p (log b log a = 0 the lim r 0 + Ω a (c, p = p log c p log a, ad Ωb (c, p = 0 Ω b (c, p = p log c p log b, ad Ωa (c, p = 0 i A a p i log(b r i /a i = lim log r r 0 + i A b p i log(a r i /b i = 0, log r

9 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 9 ad lim r 0 + i A r p i log c i log r = p log c p log a, We eed to first prove some lemmas For ay i = i 1 i 2 i Σ let [i] Σ N deote the i-cylider [i] := { Σ N : k = i k for 1 k Lemma 33 For ay 0 < r < 1, {[i] : i A r is a partitio of Σ N Proof It is clear that {[i] : i A r are distict subsets i Σ N Furthermore, for ay = Σ N there exists a smallest such that mi(a 1 a, b 1 b < r Therefore [ 1 ] ad [ 1 ] A r This proves the lemma Lemma 34 Let = m with each N The 1 ( log! ( log = t log t + O, 1! 2! m! where t = Proof We apply Stirlig s formula log(q! = q log q q log q + O(1 Thus log(! = log + O(log Sice m is fixed i our settig, m log(! = ( log + O(log = Now log = log(t = log t + log It follows that 1 ( log! = log 1! 2! m! = log + O(log log ( log + O ( log t log t + O For each i = i 1 i 2 i Σ we use i = to deote the legth of i ad i = #{k : i k = to deote the umber of occureces of the letter i i Lemma 35 There exists a costat C > 1 such that C 1 log r 1 i C log r 1 for ay 0 < r < 1 2 ad i A r

10 10 DE-JUN FENG AND YANG WANG Proof Let s + = max {a, b : 1 m ad s = mi {a, b : 1 m The we have s i a i, b i s i + for ay i Σ The lemma follows by settig C = max ( log s, log s log 2 1 Note that the coditio 0 < r < 1 2 ca be replaced with 0 < r < r 0 for ay fixed r 0 < 1 Proof of Propositio 31 We shall prove part (i of the propositio oly, as part (ii follows from a idetical argumet ad part (iii is rather obvious To prove (i we estimate the sum i A a r c i For ay i = i 1 i 2 i A a r we observe that i = i is also i A a, where is ay permutatio of δ(i = i 1 i 1, which gives c i = c i The umber of distict such i is precisely ( 1!/ m! where := δ(i Let T (i := ( 1! m! m c = 1 c i i =i c i where rus through all permutatios of δ(i We prove that for sufficietly small r we have (31 mi {c sup T (i c i O(log m r 1 sup T (i i A a r i A a i A a r r The left iequality is clear To see the right iequality we have from Lemma 35 that i C log r 1 for ay i A a r Whe i rus through A a r the umber of distict vectors ( δ(i 1, δ(i 2,, δ(i m is bouded by (C log r 1 m = O(log m r 1 Also there are at most m choices for the last letter of i The right iequality i (31 the follows Now for ay i = i 1 i i +1 A a r set t = Lemma 34, log T (i = O the other had we have a i = m log r ( t log t + t log c + O ( log a = a i +1 < r m where = δ(i ad = δ(i By a ( log t log a + O Hece Combiig the two estimates yields m log T (i ( t log t + t log c ( log (32 = log r m t + O log a

11 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 11 The coditio a i b i is equivalet to (33 t log b + b i m +1 t log a + a i +1 The propositio follows from (32 ad (33, by lettig teds to We ow prove Propositio 32 We will eed to ivoke the followig Large Deviatio Priciple: Lemma 36 (Large Deviatio Priciple Let p = (p 1,, p m be a probability vector For ay ε > 0 there exists a ω = ω(ε > 0 such that i B (ε p i < e ω for all sufficietly large, where (34 B (ε := with Σ = Σ(m { i Σ : i p > ε Proof Stadard The reader may see [3, Theorem 2110] for a proof Proof of Propositio 32 As with Propositio 31, we prove (i oly The others are proved usig idetical argumets Assume that p (log b log a = m p (log b log a = δ 0 > 0 For ay η > 0 let ε = ε(η = η/m where M = 2m ( log c + log a + log b The it is easily verified that for ay i Σ \ B (ε we have 1 log c i p log c < η, as well as Therefore (35 1 log a i p log a < η, 1 log b i log c i log a i = p log b < η m p log c m p log a + O(η Note that for this ε > 0 there is a ω = ω(ε > 0 such that i B (ε p i < e ω for all 0

12 12 DE-JUN FENG AND YANG WANG By Lemma 34, for ay i A r we have C 1 log r 1 i C log r 1 Let r > 0 be sufficietly small so that C 1 log r 1 0 We ow decompose A r ito A r,1 ad A r,2 with A r,1 = A r \ A r,2, ad A r,2 = A r ( B (ε By observig that log c i log r C 0 := C max 1 m { log c we obtai p i log c i C 0 p i C 0 e kω log r i A r,2 i A r,2 C 1 log r 1 k C log r 1 Hece p i log c i i A r,2 log r teds to 0 as r 0+ O the other had, because a i < r a δ(i we have log r = log a i + O(1 If i A r,1 the by (35 p i log c i log r i A r,1 = i A r,1 p i = p logc p log a 1 ( p logc p log a + O(η i A r,1 p i + O(η Sice i A r p i = 1 because {[i] : i A r is a partitio of Σ N ad p i e kn(ε 0 i A r,2 C 1 log r 1 k C log r 1 as r 0, we must have lim r 0 i A r,1 p i = 1 Now because m p (log b log a = δ 0 > 0, A r,1 A a r wheever η (ad hece ε is sufficietly small It follows that p i log c i (36 log r p i log c i log r p i log c i log r i A r,1 i A r i A a r Takig limit r 0 yields Ω a = p logc p log a To see that Ωb = 0 we oly eed to observe that by (36, Ω a + Ω b p i log c i = lim = Ω a r 0 log r i A r 4 Proof of Theorem 21 We adopt the followig defiitio from [26]: Defiitio 41 Let K be a compact set i R d Fix M, ε > 0 ad N N A coverig {G i i=1 of K by Borel sets is said to be (M, ε, N good if diam (G i Mε for all i, ad ay ε-cube i R d itersects at most N elemets i the coverig

13 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 13 Lemma 41 Let M, q > 0 ad N, d N There exists a costat C 1 = C 1 (M, N, d, q such that for ay compactly supported probability measure ν o R d ad ay (M, 2, N-good Borel coverig {G i of supp (ν we have C 1 1 τ (ν, q i (ν(g i q C 1 τ (ν, q where τ (ν, q is defied i (13 Proof See [26], Lemma 22 Let {φ m be a IFS i Rd For ay i = i 1 i 2 i Σ, Σ = {1, 2,, m we let φ i deote φ i1 φ i2 φ i Lemma 42 Let I = {φ m be a IFS i Rd ad p = (p 1, p 2,, p m be a probability vector The for ay compact set F we have ν I,p (F = lim p i i B where B = {i Σ : φ i (K F ad K = K(I Proof Stadard Lemma 43 Uder the assumptios of Theorem 21, for ay i Σ we have ν(φ i (K = p i, where ν := ν I,p ad K = K(I Proof Let T be the ope rectagle for the ROSC, so {φ (T m are disoit ope rectagles i T First we cosider the case i which φ (T T for some 1 m Without loss of geerality we assume that φ 1 (T T Let l Σ By Lemma 42 we have { B k = i Σ k : φ i (K φ l (K {l Σ k Hece ν(φ l (K = lim k i B k p i p l We prove the coverse I fact we prove ν(φ l (T p l Let ad set C k = {i Σ k : φ i (K φ l (T C 1 k = {i C k : φ i (K φ l (T, C 2 k = C k \ C 1 k

14 14 DE-JUN FENG AND YANG WANG Note that Ck 1 = {l Σk Hece lim k i Ck 1 p i = p l O the other had, Ck {i 2 = i 1 i 2 i k : i 1 i 2 i l, i 1 for > Hece i Ck 2 p i < (1 p 1 k 0 as k Thus lim k i C k p i = p l It follows that (41 ν(φ l (K ν(φ l (T = p l By cosiderig the iteratios of the IFS I it is clear that the above proof exteds to the case i which there exists a i Σ such that φ i (T T It remais to prove the lemma whe φ i (T T for all oempty i Σ I this case it is clear that K is cotaied i a lie parallel to oe of the axes, say, the horizotal axis The ν is idetical to its proectio ν x oto the x-axis up to a traslatio Furthermore the proectio IFS I x must satisfy the OSC (ad it is self-similar Therefore the lemma still holds Propositio 44 Let ν be a self-similar probability measure i R with supp (ν [c, d] For ay q, δ > 0 there exist costats C 1, C 2 > 0 depedig o ν, q, δ such that for ay > 0 we have C 1 τ(ν,q δ where I i = [c + (i 1(d c, c + i(d c ] (ν(i i q C 2 τ(ν,q+δ i=1 Proof It is kow that for a self-similar measure the L q spectrum exists, see Peres ad Solomyak [26] Set = d c, which is the legth of each iterval I i By defiitio ad Lemma 41 we have lim log i=0 (ν(i i q log = τ(ν, q Thus for ay δ > 0 there exists a 0 such that for all > 0 we have τ(ν,q δ (ν(i i q τ(ν,q+δ i=0 Now for 1 0 we simply choose C 1 ad C 2 to satisfy the iequalities of the propositio Proof of Theorem 21 Let r > 0 be sufficietly small We costruct a coverig {G i of supp (ν as follows: Let T be the ope rectagle associated with the ROSC for the IFS I Without loss of geerality we may assume that T is a uit square For a = (a m ad

15 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 15 b = (b m defie the subsets of A r, A a r ad A b r of Σ with Σ = {1, 2,, m as i Sectio 3 For ay i A a r by defiitio b i a i, ad we set w a (i := [b i /a i ] Note that φ i (T is a closed rectagle of width a i ad height b i ; so w a (i is the aspect ratio of the rectagle rouded off to a iteger We ow cut φ i (T horizotally ito w a (i equal rectagles of width a i ad height b i /w a (i Call these smaller rectagles {Ri,k a wa(i k=1 Similarly for ay i A b r by defiitio a i > b i, ad we set w b (i := [a i /b i ] φ i (T is a closed rectagle of width a i ad height b i We ow cut φ i (T vertically ito w b (i equal rectagles of width a i /w b (i ad height b i Call these smaller rectagles {R b i,k w b(i k=1 Observe that if s = mi{a, b ad s + = max{a, b the each Ri,k a ad Rb i,k has width ad height betwee s r ad r/s + Furthermore { C r = Ri,k a : i Aa r, 1 k w a (i { Ri,k b : i Ab r, 1 k w b (i is a coverig of supp (ν It follows that C r is a (M, r, N-good coverig of supp (ν with M = s /2 ad N = 4 The key is to estimate w a(i k=1 (ν(ra i,k q for i A a r ad w b (i k=1 (ν(rb i,k q for i A b r We make the followig claim: Claim: For ay δ > 0 there exist costats C 1 ad C 2 idepedet of i ad r such that (42 C 1 p q i (w a(i τ(νy,q δ w a(i (ν(ri,k a q C 2 p q i (w a(i τ(νy,q+δ k=1 Proof of Claim: The combiatio of Lemma 42 ad Lemma 43 implies that ν(ri,k {p a = lim : Σ, φ (K φ i (T, φ (K R a i,k = lim {p : {i Σ i, φ (K R a i,k = p i lim {p : Σ, φ i φ (K R a i,k But observe that the set { Σ : φ i φ (K Ri,k a is precisely the set { (43 Σ : φ y (K y I i,k, where φ y := π y φ π 1 y, K y = π y (K ad I i,k := [ c+(k 1(d c w a(i (So by assumptio actually d c = 1 Propositio 44 ow asserts that C 1 p q i (w a(i τ(νy,q δ w a(i, c+k(d c w a(i ] with [c, d] = π y (T (ν(ri,k a q C 2 p q i (w a(i τ(νy,q+δ k=1

16 16 DE-JUN FENG AND YANG WANG for some costats C 1 ad C 2, provig the claim By a idetical argumet we also have costats C 1 ad C 2 such that (44 C 1p q i (w b(i τ(νx,q δ for ay i A b r w b (i (ν(ri,k b q C 2p q i (w b(i τ(νx,q+δ k=1 To complete the proof of our theorem, D C r (ν(d q = It follows from (42 ad (44 that i A a r i A a r w a(i k=1 w b (i (ν(ri,k a q + (ν(ri,k b q i A a r w a(i i A b r k=1 C 1 p q i (w a(i τ(νy,q δ (ν(ri,k a q C 2 p q i (w a(i τ(νy,q+δ, ad similarly C 1 Note that i A b r k=1 w b (i i A b r k=1 i A a r p q i (w b(i τ(νx,q δ (ν(ri,k b q C 2p q i (w b(i τ(νx,q+δ b i 2a i i A b r w a (i b i a i Applyig Propositio 31 twice with c = {c m c = p q ( b a τ(νy,q δ ad c = p q ( b a τ(νy,q+δ respectively, ad with δ 0, yields wa(i log( i A lim a r k=1 (ν(ra i,k q = if r 0 log r t Γ(e a where e a := log b log a Similarly log i A lim b r r 0 wb (i k=1 (ν(rb i,k q log r t ( log t τ(ν y, qe a + q log p t log a ( t log t τ(ν x, qe b + q log p = if t Γ(e b t log b set to be where e b := log a log b The proof is fially complete by observig that for ay A B > 0 we have log A < log(a + B log A + log 2,, Proof of Theorem 22 It is clear from the proof of Theorem 21 that if all a b the τ(ν, q = θ a where θ a is give i Theorem 21 I this case Γ 0 := Γ(log b log b = {t =

17 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 17 (t 1,, t m : t 0 ad m t = 1 Hece m t ( log t τ(ν y, q log b + q log p (45 τ(ν, q = τ(ν y, q + if t Γ 0 m t log a We first simplify the otatio Set A = 1/a ad B = p q b τ(νy,q (46 τ(ν y, q τ(ν, q = sup t Γ 0 Let θ be the uique real root of the equatio m m t log B t m t log A A θ from the fact that A > 1 ad B > 0 for all The m t log B t m t log A θ = m t log A θ B t m t log A The (45 becomes B = 1 (the existece of θ follows Note that f(x = log x is covex By Jese s Iequality, t log A θ B ( m A θ B ( m log t log A θ B = 0 t t The = i the first iequality is achieved whe t = A θ B for 1 m It follows τ(ν y, q τ(ν, q = θ This proves the theorem 5 Proof of Theorem 25 Let ν be a compactly supported probability measure i R d For ay ε > 0 ad N N a family of Borel sets {G i is called a (ε, N-good partitio coverig of supp (ν if the followig coditios are met: (i i G i supp (ν ad ν(g i G = 0 for all i (ii Ay cube of side ε itersects at most N elemets of {G i ad diam (G i < ε Set f(x = x log(1/x = x log x ad defie h (ν = Q D f(ν(q, where D is the stadard partitio of R d by cubes of sides 2 defied i Sectio 1 We have Lemma 51 Let {G i be a (2, N-good partitio coverig of supp (ν where ν is ay compactly supported probability measure i R d The f(ν(g i h (ν C i where C = max (log N, log 2 d

18 18 DE-JUN FENG AND YANG WANG Proof It is easy to check that for all x 1,, x k 0 we have (51 f( k i=1 x i k i=1 f(x i f( k i=1 x i + ( k i=1 x i log k Write D = {Q ad cosider the refiemet G = {G i Q of the (2, N-good partitio coverig {G i Note that diam (G i < 2 implies that G i itersects at most 2 d cubes i D It follows from (51 that (52 f(ν(g i f(ν(g i Q f(ν(g i + log(2 d ν(g i Coversely, also by (51 we have (53 f(ν(q i f(ν(g i Q f(ν(q + log(n ν(q Summig up (52 over i yields (54 f(ν(g i i i, f(ν(g i Q i f(ν(g i + log(2 d, ad summig up (53 over yields (55 h (ν i, f(ν(g i Q h (ν + log N The lemma ow follows by combiig (54 ad (55 Propositio 52 Let ν be a self-similar probability measure i R with supp (ν [c, d] For ay δ > 0 there exist costats C 1, C 2 > 0 depedig o ν ad δ such that for ay > 0 we have (h(ν δ log + C 1 f(i i (h(ν + δ log + C 2, i=1 where I i = [c + (i 1(d c, c + i(d c Proof The propositio ca be proved by usig Lemma 51 ad a argumet idetical to that of Propositio 44 Proof of Theorem 25 For ν = ν I,p we may assume without loss of geerality that ν(l = 0 for ay lie L i R 2, for otherwise ν is i essece a self-similar measure i the oe dimesio, leavig us with othig to prove

19 A CLASS OF SELF-AFFINE SETS AND SELF-AFFINE MEASURES 19 We adopt the same otatios from the proof of Theorem 21 For ay r > 0 sufficietly small let C r be the coverig of supp (ν give by { { C r = Ri,k a : i Aa r, 1 k w a (i Ri,k b : i Ab r, 1 k w b (i as i the proof of Theorem 21 It follows that C r is a (r, N-good partitio coverig of K = supp (ν for some suitable N idepedet of r f(x = x log(1/x = x log x We first estimate w a(i k=1 f(ν(ra i,k for ay i Aa r By (43 we have ν(ri,k a = p { i lim Σ : φ y (K y I i,k = p i ν y (I i,k where K y ad I i,k are as i the proof of Theorem 21 Thus w a(i k=1 f(ν(r a i,k = p i w a(i k=1 We estimate Q C r f(ν(q for f(ν y (I i,k p i log p i It ow follows from Propositio 52 that for ay δ > 0 there are C 1 ad C 2 idepedet of r such that (56 p i (h(ν y δ log w a (i p i log p i +C 1 p i w a(i k=1 f(ν(r a i,k p i(h(ν y +δ log w a (i p i log p i +C 2 p i Similarly for ay i A b r there exist C 1 ad C 2 idepedet of r such that (57 p i (h(ν x δ log w b (i p i log p i +C 1p i Now, observe that b i 2a i w b (i k=1 f(ν(r b i,k p i(h(ν x +δ log w b (i p i log p i +C 2p i w a (i b i a i So log(b i /a i log 2 log w a (i log(b i /a i This meas we may replace w a (i i (56 with b i /a i, with oly the costats C 1 ad C 2 modified Now for (56 we apply Propositio 32 twice, with c = p 1 (b /a h(νy δ ad c = p 1 (b /a h(νy +δ respectively, ad set δ 0 Set e a := log b log a ad e a := log a log b It follows that (58 lim r 0 wa(i i A a r k=1 f(ν(ra i,k log r = h(νy p e a p log p, p log a

20 20 DE-JUN FENG AND YANG WANG if m p (log b log a > 0, ad 0 if m (59 lim r 0 i A b r wb (i k=1 f(ν(rb i,k log r p (log a log b > 0 Similarly, = h(νx p e b p log p, p log b if m p (log a log b > 0, ad 0 if m p (log b log a > 0 Combiig (58 ad (59 yields h(ν G C (510 lim r f(ν(g y p e a p log p, if p e a > 0, p log a = r 0 log r h(ν x p e b p log p, if p e b > 0 p log b To estimate the left-had side of the above equatio whe m p (log a log b = 0, by summig (56 over i A a r ad (57 over i A b r we have W r,1 f(ν(g W r,2 G C r with ad W r,1 := C 3 + p i (h(ν y δ log(b i /a i + p i (h(ν x δ log(a i /b i i A a r W 2,r := C 4 + p i (h(ν y + δ log(b i /a i + p i (h(ν x + δ log(a i /b i i A a r i A b r i A b r i A r p i log p i i A r p i log p i, where C 3 ad C 4 are costats idepedet of r Applyig Propositio 32 (iii we have G C (511 lim r f(ν(g p log p = r 0 log r p log a wheever p e a = 0 Note that C r is a (r, N-good partitio coverig of supp (ν for some costat N idepedet of r Takig r = 2 ad applyig Lemma 51 we have h (ν h(ν = lim log 2 = lim G C f(ν(g 2 G C = lim r f(ν(g log 2 r 0 log r The theorem ow follows by combiig (510 ad (511 Ackowledgemet The authors thak Dr Hui Rao for some helpful discussios about the box-dimesio of self-affie sets Refereces [1] T Bedford, Crikly curves, Markov partitios ad box dimesio i self-similar sets, PhD Thesis, Uiversity of Warwick, 1984 [2] R Cawley ad R D Mauldi, Multifractal decompositios of Mora fractals Adv Math 92 (1992,

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