Iterated Function Systems with Overlaps and Self-Similar Measures
ITERATED FUNCTION SYSTEMS WITH OVERLAPS AND SELF-SIMILAR MEASURES
KA-SING LAU, SZE-MAN NGAI HUI RAO
A
The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.
1. Introduction N d We will call a family of contractive maps oSjqj=" on an iterated function system (IFS). An iterated function system will generate an invariant compact subset K l N j=" Sj K, and if, further, the system is associated with a set of probability weights N owjqj=", then it will generate an invariant measure N −" µ l wj µ @ Sj . (1.1) i=" In order to obtain sharp results on the invariant set K or the invariant measure µ,it is often assumed that the maps are similitudes (or the extension to self-conformal maps). The corresponding K and µ are called the self-similar set and the self-similar measure respectively. For the iteration, it is often assumed that the iterated function system satisfies the open set condition (OSC) [10]. One of the advantages of the open set condition is that the ‘generic’ points of the set K can be uniquely represented in a symbolic space, and the dynamics of the iterated function system can be identified with the shift operation in the symbolic space. Without the open set condition, the iteration has overlaps; then such a representation will break down, and it is more difficult to handle the situation. The simplest case of an iterated function system with overlaps is provided by the maps " S" x l ρx, S# x l ρxj1, x ? with # ! ρ ! 1. (1.2) " The invariant measure µ associated with the weights # has been studied for a long time in the context of Bernoulli convolutions [27]. Recently Solomyak [23, 28] proved
Received 18 September 1999; revised 18 February 2000. 2000 Mathematics Subject Classification 28A80 (primary), 42B10 (secondary). The first author’s research was partially supported by an HK RGC grant. The second author’s research was partially supported by a postdoctoral fellowship from CUHK and by the NSF grant DMS-96-32032. The third author’s research was partially supported by a postdoctoral fellowship from CUHK. J. London Math. Soc. (2) 63 (2001) 99–116. ' London Mathematical Society 2001. 100 - , - that for almost all such ρ, the measure µ is absolutely continuous. This solves a conjecture of Erdo$ s. The ‘transversality’ argument used in [23] has also been used to consider a variety of iterated function systems with overlaps (by Peres and Solomyak [24], and by Peres and Schlag [22]). Other considerations on such systems with overlaps related to digit expansion can be found in [11, 12, 25, 26]. In [17] Lau and Ngai introduced a weak separation condition (WSC) on an iterated function system of similitudes that has overlaps. This condition is weaker than the open set condition, and includes many of the important overlapping cases (see the examples in Section 2). In particular, it includes the system in equation (1.2) where " ρ− is a PV number. The multifractal structure of µ related to these numbers has been studied in detail in [8, 9, 16–18]. In this paper we continue the study of the weak separation condition (see N d Definition 2.1). We will restrict our attention to similitudes oSjqj=" on with the same contraction ratio, that is,
Sj x l Aj xjbj l ρRj xjbj (1.3) where 0 ! ρ ! 1 and Rj is an orthogonal transformation. It is known that the self- similar measure µ is either absolutely continuous or continuously singular, but it is rather difficult to determine the dichotomy. Our main purpose in this paper is to study this problem. We prove the following theorem.
N T 1.1. Let oSjqj=" be an IFS as in equation (1.3) and assume that it satisfies d the WSC. Suppose that wj " ρ for at least one j. Then the self-similar measure is singular.
For the proof of the theorem we observe that the weak separation condition n implies that any ball Bρ (x) contains a (uniformly) bounded number of Sσ(x!), QσQ l n. This property yields the key Proposition 2.3 which allows us to retain some control over counting the overlaps. We can then use it to handle the product measure on the symbolic space and the self-similar µ as its projection. By using the Lebesgue density theorem and Theorem 1.1 we have the following interesting result.
N T 1.2. Let oSjqj=" be as aboe. If the self-similar measure µ is absolutely " d d continuous, then the density function f l Dµ ? L ( ) is actually in L_( ).
The second theorem allows us to determine the absolute continuity by checking # the existence of the L -density, which is simpler for self-similar measures. By assuming a slightly stronger condition on the iterated function system, which we call WSC* (see §4), we can make use of the self-similar identity (1.1) and the correlation of µ to construct a nonnegative, irreducible transition matrix TI (where I stands for the identity map on d) and prove the following theorem.
N T 1.3. Suppose that oSjqj=" is an IFS as in equation (1.3) and satisfies the WSC*. Then µ is absolutely continuous if and only if d λmax l ρ (1.4) where λmax is the maximal eigenalue of TI. 101 d d We will see that λ & ρ always holds (Proposition 4.4). When λ " ρ , we can max # max use λmax to determine the L -dimension of µ:dim#(µ) l Qlog λmax\log ρQ (Theorem 4.1). The construction of the TI is quite direct, and can be implemented for the concrete cases. We organize the paper as follows. We give the definition of the weak separation condition in Section 2, together with some examples and properties. Theorems 1.1 and 1.2 are proved in Section 3. In Section 4 we will define the transition matrix TI and prove Theorem 1.3. We also discuss some examples and the construction of the matrix TI. To conclude the paper, we consider in Section 5 an extension of the classical Bernoulli convolution associated with the PV numbers, and prove the singularity in such a case. This extends a result of Erdo$ s[27]. " We remark that for the case Sj(x) l #(xjj), where j l 0,…, N on , if the µ in Theorem 1.3 is absolutely continuous and if we let f l Dµ be the Radon Nikodym derivative of µ, then the eigenvalue of TI with the second-largest magnitude can be used to determine the regularity of f. In other words, if we let
λg l maxoQλQ:λ is an eigenvalue of TI, λ λmaxq, # " then the L -Lipschitz exponent of f is given by #(Q(log λg )\(log 2)Qk1). The result is a special case in [20]. The proof is more complicated than that of Theorem 1.3 because the corresponding eigenvalue and eigenvector may not be positive. We conjecture that the result is still true in the present setup of the weak separation condition.
2. The weak separation condition d d Throughout the paper we assume that Sj: ,- , where 1 % j % N, are contractive similitudes defined by
Sj x l Aj(xjdj) l Aj xjbj, d where bj l Aj dj ? , Aj l ρRj with 0 ! ρ ! 1 and Rj orthonormal matrices. We will use Σn to denote the set of multi-indices σ l ( j",…, jn) and use QσQ l n to denote the length of σ. Let Sσ l Sj" @…@ Sjn. Then
Sσ(x) l Aj"…jn xjAj"…jn−" bjnjAj"…jn−# bjn−"j…jbj" n l Aj"…jn xj Aj"…ji−" bji. i=" n n i−" In particular, if Aj l A for all j then Sσ(x) l A xji=" A bj . N i Let owjqj=" be a set of probability weights associated with the iterated function N N −" system oSjqj=", and let µ be the self-similar measure defined by µ l j=" wj µ @ Sj as in equation (1.1). For a fixed x!, we define the discrete measure µn by d µnoxqlowσ :Sσ(x!) l x, QσQ l nq, x ? where wσ l wjn. It is well known that oµnq converges to µ weakly. Moreover, if we let c l maxjQbjQ\(1kρ), then supp µ is contained in a ball of radius c. For x l _ i=" Aj"…ji−" bji, it is easy to show that n n n µ(Bρ (x)) % µn(Bcρ (x)) % µ(B#cρ (x)) where Br(x) denotes the open ball of radius r centered at x [17]. If the iterated function n system satisfies the open set condition, then the term µn(Brρ (x)) is quite easy to handle. In the following paragraph we define a condition on such a system that extends the open set condition and allows us to handle the term. 102 - , - N D 2.1. We say that oSjqj=" satisfies the weak separation condition (WSC) d if there exists x! ? and a constant a " 0 such that for QσQ, QσhQ l n, then either n Sσ(x!) l Sσh(x!)orQSσ(x!)kSσh(x!)Q & aρ . (2.1) By a suitable translation we can assume that x! l 0. Also, note that the iterated d function system is invariant on the subspace in spanned by Sσ, QσQ l n, n ? . Hence by restricting the iterated function system to the subspace, we can assume without d loss of generality that for some n large enough, oSσ(x!):QσQ l nq spans . The definition N says that after iterating x! by oSjqj=" for n times, all the points Sσ(x!), QσQ l n are either identical or separated by a distance aρn. This definition was introduced in [17] under the more general setting that the similitudes can have different contraction ratios. Since for all the practical examples considered here, the maps in the iterated function system have the same contraction ratio, we just impose it in the definition for simplicity. The following proposition is immediate from the definition.
N P 2.1. Suppose that the IFS oSjqj=" satisfies the WSC. Then any ball n Bcρ (x) contains at most % distinct Sσ(x!), σ ? Σn.
It is also easy to see that condition (2.1) is equivalent to −" −" either Sσ Sσh(x!) l x! or QSσ Sσh(x!)kx!Q & a, BQσQ l QσhQ. (2.2) N In [4] Bandt and Graf showed that oSjqj=" satisfies the open set condition if and only d −" if there exists x! ? and a " 0 such that QSσ Sσh(x!)kx!Q & a for all incomparable σ and σh. Proposition 2.2 follows.
N P 2.2. If oSjqj=" satisfies the open set condition, then it satisfies the WSC.
Our main interest is in iterated function systems that do not satisfy the open set condition. Below, we give a list of such examples with the weak separation condition.
N E 2.1. Let oSjqj=" be defined on with Sj x l 1\kxjbj where k & 2is an integer, and bj l crj with c ? and rj rationals. We take x! l 0. Then for σ l ( j",…, jn), n n rji c tji Sσ(0) l c i−" l i−" i=" k q i=" k where tj l qrj, for 1 % j % N, are integers. It is easy to see that the weak separation condition is satisfied by taking a l c\q in the definition.
Note that the case with contraction ratio ρ l 1\2 has been studied in great detail in wavelet theory in connection with the dilation equation N f(x) l cj f(2xk( jk1)). j=" The function f can be considered as the density function of the corresponding absolutely continuous self-similar measure µ in equation (1.1), with cj l 2wj.In wavelet theory the cj may be negative but cj must be 2. 103 Kenyon [12] and Rao and Wen [26] have studied the iterated function system for the contraction ratio ρ l 1\3, with N l 3. They are interested in the dimension of the invariant set K, and the tiling property related to the translation numbers rj. Hu and Lau [9] and Fan et al.[5] have also given a detailed analysis of the multifractal structure of the self-similar measure generated by the iterated function system with such ρ.
N −" E 2.2. Let oSjqj=" be defined on with Sj x l ρxjbj where β l ρ is a PV number, and bj l crj with c ? and rj rationals. (Recall that β " 1 is a PV number if it is an algebraic integer such that all its algebraic conjugates have modulus less than 1 (see [27]), for example, the golden ratio (N5j1)\2.) n i−" Similar to the above, we can write Sσ(0) l c\q i=" tji ρ , for tji ? . The weak separation condition follows from a lemma of Garsia [6, Lemma 1.51]. The self- " similar measure corresponding to S" x l ρx and S# x l ρxj(1kρ) with weights # each is the classical Bernoulli convolution [27]. The entropy dimension, Lp-spectrum, local dimension spectrum and the multifractal formalism of the Bernoulli convolution associated with the PV numbers, in particular with the golden ratio, have been studied in great detail by many authors (for example, 1, 2, 8, 14–17, 21).
In the above two examples the contraction ratios ρ are algebraic integers. It is not difficult to construct an example with a more arbitrary ρ. " E 2.3. Let 0 ! ρ ! $, S" x l ρx, S# x l ρxjρ, and S$ x l ρxj1, where # x ? . Then S" @ S$(x) l S# @ S"(x) l ρ xjρ. This implies that the open set condition is not satisfied according to the result of Bandt and Graf stated after equation (2.2). However, the iterated function system satisfies the weak separation condition: it is easy to show that n−" i Sσ(0)kSσh(0) l ρ εi, εi l 0, pρ, p1, p(1kρ) i=! n i n i can be written into the form i ! ρ ηi where ηi l 0, p1, p2. If i ! ρ ηi 0 and k = " = is the smallest index such that ηk 0, then since 0 ! ρ ! $, _ k i 2ρ k n QSσ(0)kSσh(0)Q & ρ k2 ρ & 1k ρ & aρ " 0, ) i=k+" ) 0 1kρ1 where a l (1k3ρ)\(1kρ). d −" E 2.4. In , we let Sj x l A(xjdj) where B l A is an integral d expanding similarity matrix, with dj ? and d" l 0. Then under a suitable norm on d N , A is a contraction [13]. It is easy to show that oSjqj=" satisfies the weak separation −" condition. Indeed, if we let x! l 0 and if Sσ Sσ (0) 0, then being an element in the −" h integer lattice, QSσ Sσh(0)Q & 1 and equation (2.2) applies. This class of iterated function system has been studied in detail in connection with the theory of tiles under the more general setting using self-affine maps (see, for example, [13] and the references there). In the case of tiles, it is necessary to take N l Qdet BQ.
E 2.5. Let A be as above, let Γ be a finite group of integral matrices γ with det γ lp1 and assume that Γ satisfies ΓA l AΓ. Let Sj x l Aj(xjdj) where Aj l d N γj A, γj ? Γ, dj ? . Then the above argument also implies that oSjqj=" satisfies the weak 104 - , - separation condition. This class of iterated function system was used by Bandt [3] and Xu [30] to study tiles that involve rotations and reflections. For example, let
−" 1 k1 ε" 0 0 ε" A l , Γ l , :εi lp1 , 91 1 : (90 ε#: 9ε# 0: * 1 k1 0 S" x l Ax, S# x l γAxj with γ l . Then the corresponding invariant set 90: 9 0 1: is the LeT y dragon [3].
To conclude this section we prove some useful consequences of the weak separation condition that will be needed in the following sections. In order to avoid confusion in the counting, we will identify Sσ and Sσh, QσQ l QσhQ l n, if they are equal, and denote the set of such distinct Sσ by !n. For σ ? Σn, we will use [σ] to denote the equivalence class oσh ? Σn:Sσh l Sσq. N P 2.3. Suppose that oSjqj=" satisfies the WSC. Then for any bounded d d subset D 9 , there exists γ " 0 such that for any x ? ,n? , FoS ? !n:x ? S(D)q!γ. Proof. Suppose that the proposition is false; then there exists a bounded D such that for any M " 0, FoS ? !n:x ? S(D)q"M for some n and x. By enlarging the set D we can assume that the x! in the definition N of the weak separation condition is in D and that j=" Sj(D) 9 D. It follows that n FoS ? !n:S(D) 7 Bρ QDQ(x)q"M, and hence that n FoS ? !n:S(x!) ? Bρ QDQ(x)q"M. n Since Bρ QDQ(x) contains at most l distinct Sσ(x!), QσQ l n (see Proposition 2.1), there exists s such that M FoS ? ! :S(x!) l sq" . n l
Let Es loS ? !n:S(x!) l sq. For Sσ, Sσh ? Es, we have
Sσ(x)kSσh(x) l (Aσ(xkx!)jSσ(x!))k(Aσh(xkx!)jSσh(x!)) n l (AσkAσh)(xkx!) l ρ (RσkRσh)(xkx!).
Since M can be arbitrarily large and there are at least M distinct Sσ in !n, we can choose distinct Rσ, Rσh so that RRσkRσhR is arbitrarily small (depending on M). We can assume without loss of generality that there exists n! such that d oSτ(x!):QτQ l n!q contains 0 and spans (see the remark after Definition 2.1). We claim that for any σ, σh ? Σn with Sσ Sσ, we have S(σ,τ)(x!) S(σ ,τ)(x!) for at least h −" h one of the τ. Otherwise, if we let T l Sσ Sσ, then Sτ(x!), QτQ l n! are fixed points of d h d T. Recall that oSτ(x!):QτQ l n!q spans and T is an isometry of . This forces T to be the identity map, which is a contradiction. ! Now for any a " 0, we choose σ, σh with Sσ, sσ ? Es such that n! −" RRσkRσhR % aρ minoRSτ(x!)kx!R :QτQ l n!q.
Then for τ satisfying S(σ,τ)(x!) S(σh,τ)(x!) we have
0 ! RS(σ,τ)(x!)kS(σh,τ)(x!)R l RSσ Sτ(x!)kSσh Sτ(x!)R n (n+n!) l ρ R(RσkRσh)(Sτ(x!)kx!)R % aρ . This contradicts the weak separation condition. * 105 Note that if the iterated function system satisfies the open set condition with an associated open set U, and if we take D l U, then the γ in the above proposition is 1. In later applications we actually assume that N D is closed, Dm6, x! ? Dm and j=" Sj(D) 9 D. We will call such D a basic region. It follows that the invariant set K is contained in D.
N C 2.1. Suppose that oSjqj=" satisfies the WSC. Then there exists −dn γ " 0 such that F!n % γρ .
Proof. Let D be a basic region, and let γ be as in Proposition 2.3. Then
S?!n S(D) 9 D and m(D) " 0 where m is the Lebesgue measure. By the above proposition, we see that each x ? D is covered by at most γ of S(D) with S ? !n.It follows that m(S(D)) % γm(D). S?!n dn −dn The self-similarity implies that m(S(D)) l ρ m(D). Therefore F!n % γρ ,as claimed. *
N C 2.2. Suppose that oSjqj=" satisfies the WSC. Let D be a basic region. Then for any c " 0, there exists c" " 0 such that for any n and x, n FoS ? !n:S(D)EBcρ (x) 6q!c". n n n Proof. Let rn l cρ jQDQ ρ . Then S(D)EBcρ (x) 6 implies that S(D) 9 Brn(x). Also, by Proposition 2.3, we know that every point is covered by at most γ of the S(D), S ? !n. It follows that
n γ:m(Brn(x) & om(S(D)):S(D)EBcρ (x) 6q S?!n dn n l ρ m(D):FoS ? !n:S(D)EBcρ (x) 6q. Hence n dn −" FoS ? !n:S(D)EBcρ (x) 6q%γ(m(D) ρ ) m(Brn(x)) and we can choose a bound c" for the last expression. *
3. A sufficient condition for singularity N N Let oSjqj=" be an iterated function system with associated weights owjqj=" and let µ be the self-similar measure as defined in the last section. Our first theorem in this section gives a simple criterion for such a measure to be singular. First we will introduce some notations in the symbolic space. Let
Σ loσ` l ( j", j#,…):ji ? o1,…, Nqq, d and let Σn be the set of σ with length n. Let π be the projection of Σ to defined by
_ π(σ` ) l Sj"…jn(K), σ` l ( j", j#,…), n=" where K is the invariant set of the iterated function system. For Cσ a cylinder set with base σ ? Σn, it is clear that π(Cσ) l Sσ(K). We let P be the probability measure on Σ, 106 - , - and Pn the probability on Σn. For Λ 9 Σn, we will use the abbreviated notation P(Λ) to denote Pn(Λ); that is, P(Λ) l Pn(Λ) l P(CΛ), where CΛ is the cylinder set with base −" Λ. Note that µ l P @ π , and for σ ? Σn,
µ(π(Cσ)) l P(C[σ]) l wσh. σh?[σ] N L 3.1. Suppose that oSjqj=" satisfies the WSC. For any Λ 7 Σn, we let dn 4 ρ Λ l σ ? Λ: wσh " . ( [ ] 4γ * σh? σ EΛ " " Then P(Λ) " # implies that P(Λ4 ) " %.
−dn Proof. By Corollary 2.1 we have F!n % γρ . This implies that dn 4 4 4 ρ " P(ΛBΛ) l owσ :σ ? ΛBΛql owσh :σh ? ΛBΛq%F!n: % %, [σ] σh?[σ] 4γ and P(Λ4 ) l P(Λ)kP(ΛBΛ4 ) " 1\2k1\4 l 1\4. *
N d T 3.1. Suppose that oSjqj=" satisfies the WSC, and suppose that wj " ρ for at least one 1 % j % N. Then µ is singular.
d " Proof. Our aim is to choose, for any ε " 0, a subset E 7 such that µ(E) & # d and m(E) ! ε. Without loss of generality we assume that w" " ρ . Let D be a basic region as in the last section, and let p ? be such that ρd p 4γm(D) ! ε. 0w"1 Let Λ" l Σ" lo1,…, Nq, ρd Λ4 " l σ ? Λ":w " , ( σ 4γ* $ Λ" lo(σ,1,…,1)? Σ"+p:σ ? Λ4 "q, $ E" l oSσ(D):σ ? Λ" q.
" $ " p $ Then Lemma 3.1 implies that P(Λ4 ") & %, so that P(Λ" ) & %ω". Moreover, for σ ? Λ" we have d d("+p) p d("+p) ρ p ρ w" ρ w " w" l " m(D). σ 4γ 4γ 0ρd1 ε It follows that " $ $ m(E ) ρd( +p)m(D):F S :σ Λ ε w εP(Λ ). (3.1) " % o σ ? " q% $ σ % " σ?Λ" $ $ Suppose that we have chosen Λi 7 Σ"+ip, and Ei l oSσ(D):σ ? Λi q, for i l 1,…, kk1, such that $ $ $ (i) Λi 7 Σ"+ip and CΛ ECΛ l 6 for i j; $ " p i j (ii) P(Λ ) & %w" ; i $ (iii) m(Ei) % εP(Λi ). 107 k−" $ If i=" P(Λi ) & 1\2, we stop the construction. Otherwise we let σQn denote the first n coordinates of σ and define $ Λk loσ ? Σ"+(k−")p:σQ"+ip ^ Λi ,1% i % kk1q, d 4 ρ (1j(kk1)p) Λk l σ ? Λk: wσh " , ( [ ] 4γ * σh? σ EΛ $ Λk lo(σ,1,…,1)? Σ"+kp:σ ? Λ4 kq, $ Ek l oSσ(D):σ ? Λk q.
" $ $ $ Then it is clear that P(Λk) " # and Λk ? Σ"+kp. That CΛi ECΛk l 6 for 0 % i ! k follows from the choice of Λ . Condition (ii) is a direct consequence of the $ k construction of Λ and Λk and an application of Lemma 3.1. The proof of condition (iii) is similar to that of inequality (3.1). In view of condition (ii), the process must stop at some finite step, say at k. Let k N E l i=" Ei. We recall that D satisfies j=" Sj(D) 7 D, and the closedness of D implies that K 7 D. Hence π(C $) S (K) S (D) E . Λi l $ σ 7 $ σ l i σ?Λi σ?Λi k $ This implies that π(i=" CΛi ) 7 E, and it follows that k k −" $ $ " µ(E) l P(π E) & P CΛi l P(CΛi ) & #. 0i=" 1 i=" On the other hand, condition (iii) implies that k k $ m(E) % m(Ei) % ε P(CΛi ) ! ε. i=" i=" The singularity of µ is proved. *
We have an interesting consequence of the above theorem.
N T 3.2. Suppose that oSjqj=" satisfies the WSC. If µ is absolutely continuous, d then the density function f l Dµ will be bounded; that is, f ? L_( ). Moreoer, f satisfies N −" d f(x) l cj f @ Sj (x), x ? , j=" −" where cj l Qdet AjQ wj.
d Proof. We need only to show that f ? L_( ). Suppose otherwise; then for any n large M " 0, the Lebesgue density theorem implies that there exists a ball Bρ (x) such that n µ(Bρ (x)) 1 dn l dn f(t) dt " M. (3.2) ρ ρ &B n(x) ρ N If we iterate oSjqj=" for n times, we obtain the family !n, which we will use to form a new iterated function system. For each S l Sσ ? !n, the associated weight is
wS l owσh :σh ? [σ]q. 108 - , - −" The corresponding self-similar identity is µ l S?!n wS µ @ S . By equation (3.2) we have −" n dn wS µ @ S (Bρ (x)) " Mρ . (3.3) S?!n n Now we fix a basic region D; then µ is supported by D. Hence S(D) Bρ (x) l 6 −" n −" n implies that D S (Bρ (x)) l 6, and thus µ(S (Bρ (x))) l 0. In view of Corollary n 2.2, FoS ? !n:S(D)EBρ (x) 6q!c", so we see that the sum on the left of inequality (3.3) has at most c" nonzero terms, and there exists at least one wS such that wS " dn (M\c") ρ . We choose M such that M\c" " 1. Then Theorem 3.1 implies that µ is singular. This is a contradiction. *
N C 3.1. Suppose that oSjqj " satisfies the WSC. Then the self-similar = # measure µ is absolutely continuous if and only if the L -density of µ exists.
4. The transition matrix Hardy and Littlewood [7] proved that for a probability measure µ, 1 # lim #d Qµ(Bh(x))Q dx ! _ h ! h & d ! # if and only if µ is absolutely continuous and Dµ ? L . There is no similar criterion for " the absolute continuity of µ (with Dµ ? L ). However, by using Corollary 3.1, we have the following proposition.
N P 4.1. Suppose that oSjqj=" satisfies the WSC. Let β be such that 1 # 0 ! lim #β Qµ(Bh(x))Q dx ! _. (4.1) h ! h & d ! Then µ is absolutely continuous if and only if β l d.
In the following we will study the expression (4.1) and the exponent β through # certain transition matrix, to be defined. Note that the L -dimension (also called the correlation dimension)ofµ is defined as # log ! dQµ(Bh(x))Q dx dim#(µ) l lim kd h!! log h 1 # l sup α:lim d+α Qµ(Bh(x))Q dx ! _ , ( h ! h & d * ! and it follows that dim#(µ) l 2βkd (see [29]). The limit expression in expression (4.1) corresponding to the β is called the upper mean quadratic ariation of µ (see [19]). Similarly, we can define dim#(µ) and dim#(µ). −" _ Let denote the set of maps S l Sσ Sσh for (σ, σh) ? n="(ΣniΣn). We will consider as a state space and define an (infinite) transition matrix on as follows. For S ? , let
T(S) l w(S,Sh)Sh Sh? where −" w(S,Sh) l owi wj:Si @ S @ Sj l Shq. i,j 109
This amounts to saying that the transition from S to Sh has weight w S S . Also note # ( , h) that Sh w(S,Sh) l (i wi) l 1. It follows that T defines a Markov matrix on . For a fixed β and for any S ? , we define 1 ΦS(h) l µ(Bh(Sx)) µ(Bh(x)) dx. #β d h &
We use Φ(h) to denote the vector oΦS(h)qS? and let fg denote the linear space spanned by . Then ΦS(h) l fΦ(h), Sg and for any ? fg,
fΦ(h), g l S ΦS(h). S
For convenience we also write the above expression as Φv(h). The following is the main purpose for defining the Markov matrix T and the consideration of ΦS(h).
d−#β P 4.2. For S ? , ΦS(h) l ρ ΦTS(h\ρ).
−" Proof. By substituting µ l j wj µ @ Sj into ΦS(h), we have
1 −" −" ΦS(h) l #β wi wj µ(Bh/ρ(Si S(x))) µ(Bh/ρ(Sj (x))) dx h i,j & d ρ −" l #β wi wj µ(Bh/ρ(Si SSj(x))) µ(Bh/ρ(x)) dx h i,j & ρd l #β w(S,Sh) µ(Bh/ρ(Sh(x))) µ(Bh/ρ(x)) dx h S & h
d−# h l ρ βΦ . * TS 0 ρ1
We recall that supp µ is contained in a ball of radius (ρ\(1kρ)) maxQdjQ. We also recall that in the definition of the weak separation condition we can take x! l 0 without loss of generality.
D 4.2. Let Sj(x) l Aj(xjdj), where j l 1,…, N, with Aj l ρRj as before. Let Cg l (2ρ\(1kρ)) maxjQdjQ, and let g loS ? :QS(0)Q % Cg q. N We say that oSjqj=" satisfies the weak separation condition* (WSC*) if g is a finite set.
We remark that from the definition of the WSC*, the set oS(0):S ? g q is a finite set, so there exists a " 0 such that for S, Sh ? g with S(0) Sh(0), then QS(0)kSh(0)Q & a " 0. This makes the WSC* stronger than the WSC, which only requires that QS(0)Q & a for any S ? g with S(0) 0 (see equation (2.2)). We also remark that all the examples in Section 2 satisfy the WSC*. We will discuss this in more detail at the end of the section.
N P 4.3. Suppose that oSjqj=" satisfies the WSC*. Then (i) T maps Bg to itself; (ii) There exists h! " 0 such that for 0 ! h ! h! and S ? Bg , ΦS(h) l 0. 110 - , - −" g g Proof. (i) We let S l Sσ Sσh ? B, then QS(0)Q " Cjδ for some δ " 0. It follows that −" −" −" −" QSi SSj(0)Q & ρ QSSj(0)QkQdiQ & ρ QS(0)jρRσ Rσh(Rj dj)QkmaxQdkQ k 2ρ δ δ & maxQdkQj l Cg j . (1kρ) k ρ ρ −" −" (ii) In view of the above fact that QS(0)Q " Cg jδ implies that QSi SSj(0)Q " Cg jρ δ and the hypothesis that g is a finite set, we can choose h! small enough so that for S ^ g , QS(0)Q " Cg j2h!. We claim that for such S, µ(Bh(Sx)) µ(Bh(x)) l 0 for all x. Otherwise, there exist x and h such that µ(Bh(Sx)) 0. By observing that the support of µ is continuous in C4 \2, we have QxQ % Cg \2jh. It follows that
−" Cg Cg QS(x)Q l QS(0)jR R (x)Q " (Cg j2h!)k jh! " jh! σ σh 02 1 2 and µ(Bh(Sx)) l 0 for h % h!. This is a contradiction, and part (ii) follows from the claim. *
From the above proposition we can write T as Tg 0 T l 0Q Th1 where Tg is a sub-Markov matrix on the states g (since the sum of each column of T is 1, the sum of each column of Tg is % 1). Tg is a finite matrix by the WSC*. Let λ be an eigenvalue of Tg and a corresponding eigenvector. By Propositions 4.2 and 4.3, we have
d−#β h d−#β h h Φ (h) l ρ Φ l ρ Φ g jΦ v Tv 0ρ1 0 Tv 0ρ1 Qv 0ρ11
d−#β h d−#β h d−#β h l ρ Φ g j0 l ρ Φ l λρ Φ . 0 Tv 0ρ1 1 λv 0ρ1 v 0ρ1 If the matrix Tg is irreducible, the Perron–Frobenius theorem implies that the maximal eigenvalue λmax is positive, and that all the coordinates of the eigenvector are d−#β positive. If we take β such that λmax ρ l 1, then Φv(h) l Φv(h\ρ). It is easy to show that Φv(h) " 0 and 0 ! limh ! Φv(h) % limh ! Φv(h) ! _. From Proposition 4.1 we ! ! d see that the absolute continuity criterion is λmax l ρ . However the matrix Tg is not always irreducible. Hence we cannot guarantee that Φv(h) " 0, and we will need a little more elaborate work. We need to pick up the essential part of Tg first. Let I be the identity map in g . Let I be the Tg -irreducible component of g that contains I; that is, (m) (n) S ? I if and only if there exist m, n & 1 such that w(I,S), w(S,I) " 0 (n) g n where w(S,Sh) denotes the (S, Sh) entry of T . Let TI be the truncated square matrix of Tg on I; then TI is irreducible and is a finite matrix by the WSC*. N T 4.1. Suppose that oSjqj=" satisfies the WSC*. Let λmax be the maximal eigenalue of TI and let 1 log λmax β l dj . 2 0 ) log ρ )1 111 Then the mean quadratic ariation satisfies 1 # 1 # 0 ! lim #β Qµ(Bh(x))Q dx % lim #β Qµ(Bh(x))Q dx ! _. h ! h & d h ! h & d ! ! # The L -dimension of µ is hence gien by dim#(µ) l Qlog λmax\log ρQ.
As a direct consequence we have the following corollary.
N C 4.1. Suppose that oSjqj=" satisfies the hypotheses of Theorem 4.1. Then d µ is absolutely continuous if and only if λmax l ρ .
The theorem is a direct consequence of the following Lemmas 4.1 and 4.2. We first n observe that the (I, I) entry of TI is (n) −" w(I,I) l owσ wσh :Sσ Sσh l Iq σ,σh?Σn (n) # and w(I,I) l S?!n wS. Since TI is a nonnegative irreducible matrix, it is well known that there exist a", a# " 0 such that n (n) n a" λmax % w(I,I) % a# λmax. (4.2) d P 4.4. Under the assumption of Theorem 4.1,wehae λmax & ρ . −dn Proof. By Corollary 2.1, we have F!n % γρ . Since S ? !n wS l 1, the Cauchy– Schwartz inequality implies that (n) # −" dn w(I,I) l wS & γ ρ . S?!n d Hence by inequality (4.2) we conclude that λmax & ρ . *
L 4.1. Under the assumption of Theorem 4.1 and letting D be a basic region, there exists c " 0 such that
# nd n Qµ(B D n(x))Q dx & cρ λmax. d Q Qρ &
n Proof. Let h l QDQ ρ . For S ? !n,ifx ? S(D), then S(D) 9 Bh(x). It follows that
# # # dn # Qµ(Bh(x))Q dx & Qµ(S(D))Q dx & wS dx l m(D) ρ wS. &S(D) &S(D) &S(D) Since the WSC* implies the WSC, we see that each point is covered by at most γ of the S(D) (by Proposition 2.3). Hence
# 1 # m(D) dn # nd (n) Qµ(Bh(x))Q dx & Qµ(S(D))Q dx & ρ wS l cρ w(I,I) d & γS?!n &S(D) γ S?!n where c l m(D)\γ. This, together with inequality (4.2), implies the lemma. * n+" n L 4.2. Under the same assumption as in Theorem 4.1 and for ρ % h ! ρ , we hae # dn n Qµ(Bh(x))Q dx ! cρ λmax, d & for some c " 0 independent of n. 112 - , - n Proof. Let Qj be a ρ -mesh cube and let Qg j l x Q Bh(x). Let D(8 K) be a basic −"? j region as before. Then by using µ l S?!n wS µ @ S , we have for x ? Qj, −" µ(Bh(x)) % owS:DES (Bh(x)) 6q% owS:S(D)EQg j 6q. S?!n S?!n By Corollary 2.2, FoS ? !n:S(D)EQg j 6q!c". It follows that # Qµ(Bh(x))Q dx &Qj # % owS:S(D)EQg j 6q :m(Qj) 0S?!n 1 # % c" owS:S(D)EQg j 6q:m(Qj) (Cauchy–Schwartz inequality) S?!n dn # l c# ρ owS:S(D)EQg j 6q. S?!n For each S ? !n, it is clear that FoQj:S(D)EQg j 6q!c$ for some fixed c$. Summing both sides of the above expressions, we obtain
# dn # dn (n) Qµ(Bh(x))Q dx % c# c$ ρ wS l c% ρ w(I,I). d & S?!n The lemma now follows from expression (4.2). *
To conclude this section we will discuss the examples in Section 2 in regard to the WSC*. We show that the state space g and the matrix Tg can easily be implemented after some concrete identifications. N Let oSjqj=" be an iterated function system on with Sj(x) l ρ(xjdj), where 0 ! ρ ! 1. Without loss of generality we assume that 0 l d" ! d#…! dN. We can prove −" by induction that the state S l Sσ Sσh ? has the form Sx l xjs, x ? for some s ? . We can represent the map S by the translation number s. We construct the set inductively, starting from s l 0, by letting −" shlρ sjdjkdi,1% i, j % N. (4.3) The set g can be obtained by keeping those sh with QshQ % Cg l (2ρ\(1kρ)) dN. The matrix T will send s into the states sh in equation (4.3) with weight −" w(s,sh) l owi wj:ρ sjdjkdi l shq. It can be checked from this that Examples 2.1–2.3 have the WSC* (for Example 2.2, we need to use Garsia’s lemma again). For the numerical examples the reader can check on [5, 15, 18]. In all those cases Tg l TI and Tg can be reduced further to smaller size by the symmetry of the g . d For the iterated function system in Example 2.4, Sj(x) l A(xjdj)in , the maps S ? still have the form Sx l xjs. The WSC* is clear and the construction of the set g and the map Tg is the same as the above one-dimensional case. For the iterated function system in Example 2.5, we can prove by induction that −" each S l Sσ Sσh ? can be represented as S(x) l γ(x)js, −" −" d where γ l Aσ Aσh ? Γ and s l Sσ Sσh(0) ? . Hence each S ? can be represented by 113 (γ, s). It is immediate to see that the system satisfies the WSC* since Γ is a finite set d and s ? . For the construction of g and Tg , we first define γF by γA l AγF and then observe that −" −" Si SSj(0) l Ai (γAj djjs)kdi −" F −" l (γi γγj) djjAi skdi −" F −" l (γi γγj) djjSi (s). Hence the set can be constructed inductively as follows, starting from (γ, s) l (I, 0), −" F −" F −" (γh, sh) l ((γi γγj) ,(γi γγj) djjSi (s)), 1 % i, j % N. −" F −" The set g is obtained by choosing the (γ, s) ? such that (γi γγj) djjSi (s) % Cg in the construction. The map T will send (γ, s) into the above states with weight
−" F −" F −" w((γ,s),(γh,sh)) l owi wj:(γi γγj) l γh,(γi γγj) djjSi (s) l shq. i,j
5. The case of PV numbers
_ n _ Erdo$ s proved that if X l i=" ρ Xn where oXnqn=" are independent identically −" distributed Bernoulli random variables and 1 ! ρ ! 2 is a PV number, then the distributional measure µ is singular [27]. In our present notation, µ is the self-similar measure " −" " −" µ l # µ @ S" j# µ @ S# where S" x l ρx, and S# x l ρxj1. In the following we will extend Erdo$ s’ theorem −" to Sj(x) l ρxjbj,1% j % N, where ρ " 1 is a PV number and bj are rationals, and the probability weights are arbitrary (even negative). We need two technical lemmas. For fixed (k!, n!) ? i, consider the following relation: n n kβ jk! β ! ? . (5.1) n n If k, n, p ? satisfy kβ jk! β ! l p, we say that (k, n, p) is a solution of relation (5.1); we also say that (k, n) is a solution of relation (5.1) if (k, n, p) is a solution for some p ? .
L 5.1. Suppose that β " 1 is an algebraic number and that at least one of its conjugate roots has modulus ! 1. Then for any (k!, n!) ? i with k!, n! 0, the number of solutions of equation (5.1) is finite.
n Proof. We first observe that n 0, for otherwise β ! l (pkk)\k! will imply that β and its conjugates are multiples of unity. This contradicts that β " 1 and one of its conjugates has modulus ! 1. Next, if (k, n, p) and (kh, nh, p) are two distinct solutions n−n of relation (5.1), then β h l kh\k which is impossible for the same reason. Let λ be a conjugate of β such that QλQ ! 1. If relation (5.1) has infinitely many solutions (for different p), then we have solutions (k, n, p) with QpQ arbitrarily large. Hence n n n β Qkβ Q Qpkk! β !Q l n l n ! 1, )λ) Qkλ Q Qpkk! λ !Q n when Q pQ!_. This contradicts the fact that Qβ\λQ " Qβ\λQ " 1 when n " 0, and n −" Qβ\λQ ! Qβ\λQ ! 1 when n ! 0. * 114 - , -
N #πib ξ L 5.2. Let Q(ξ) l j=" cj e j be a trigonometric polynomial with N j=" cj 0,cj ? and bj rationals. Let B be such that Bj l Bbj, where 1 % j % N, are integers, and let β be an algebraic number as in Lemma 5.1. Then there exists m ? k such that Q(mBβ ) 0 for all k ? .
Proof. Let Qg (x) N c xBj and let x ,…, x be the roots of Qg (x) with l j=" j n o " sq #πik β N modulus 1 that can be written as e % %, where 1 s. Then c 0 implies % % % j=" j k k #πimβ that kl and n% 0 for all 1 % l % s. Note that Q(mBβ ) l 0 if and only if e is a root of Qg . It follows that k n % mβ kk% β ? (5.2) for some 1 % % % s. By Lemma 5.1 there are only finitely many m for equation (5.2) k to hold. Hence we can choose m ? such that Q(mBβ ) 0 for all k ? . *
−" T 5.1. Let ρ be an irrational PV number, let bj be rationals, and let
Sj x l ρxjbj, j l 1,…, N. N Then for any set of probability weights owjqj=" the self-similar measure µ is singular.
Proof. By equation (1.1), the Fourier transformation Φ(ξ) l µV (ξ) satisfies
_ j Φ(ξ) l Q(ξ) Φ(ρξ) l Q(ρ ξ), j=!
N #πib ξ where Q(ξ) l j=" wj e j . Note that Φ(0) l 1 and the product converges to Φ uniformly on compact subsets of . We will show that QΦ(ξ)Q does not converge to 0 as QξQ!_. The Riemann–Lebesgue lemma will imply that µ is singular. −" Let β l ρ . We assume without loss of generality that there exists B such that k _ −n Bj l Bbj are integers and Q(Bβ ) 0 for all k ? (Lemma 5.2). Let C l n=! Q(Bβ ). Then C 0. For k " 0,
_ k _ k k−n n n QΦ(Bβ )Q l QQ(Bβ )Q l C QQ(Bβ )Q & C QQ(Bβ )Q. (5.3) n=! n=" n=" Now we recall a well-known property of the PV number [27]: there exists 0 ! θ ! 1 n n such that Q[β ]Q ! θ for large n, where [x] denotes the distance from x to the nearest n " n n integer. We can choose n! such that for n " n!,2Bjθ ! #. Let Chln=! " QQ(Bβ )Q " 0. Then _ _ n n QQ(Bβ )Q l Ch QQ(Bβ )Q n=" n=n!+"
_ N n #πiB [β ] l Ch wj e j n=n!+" )j=" ) _ # iB n & Ch min QRe e π jθ Q n=n!+" "%j%N _ n & Ch cos(2πBj θ ). "%j%N n=n!+" The product is a positive constant and if we put it back into expression (5.3), we have Φ(Bβk) 0ask !_. * 115 Note that in the proof we have not used the property of the positive weights. The proof can hence be applied directly to conclude the following corollary.
N −" C 5.1. Let the IFS be as in Theorem 5.1, and let j=" cj l ρ . Then the functional equation N −" f l cj f @ Sj j=" " has no L -solution.
Acknowledgements. The authors would like to thank Dr Lifeng Xi for the simplification of Lemma 5.1 from the original, more complicated, argument. The second and third authors acknowledge their gratitude to the Department of Mathematics, Chinese University of Hong Kong, for postdoctoral support. The second author also acknowledges the support received from the Southeast Applied Analysis Center in the School of Mathematics, Georgia Institute of Technology.
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K. S. Lau S. M. Ngai Department of Mathematics School of Mathematics Chinese Uniersity of Hong Kong Georgia Institute of Technology Hong Kong Atlanta GA 30332 kslau!math.cuhk.edu.hk USA
H. Rao Current address of S. M. Ngai: Department of Mathematics Department of Mathematics Wuhan Uniersity Georgia Southern Uniersity Wuhan 430072 Statesboro China GA 30460 USA raohui!nlsc.whu.edu.cn ngai!gsu.cs.gasou.edu