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1 This is the author s version of a work that was submitted/accepted for publication in the following source: Yu, Zu-Guo (2004) Fourier Transform and Mean Quadratic Variation of Bernoulli Convolution on Homogeneous Cantor Set. Chaos, Solitons and Fractals, 2, pp This file was downloaded from: Notice: Changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this work, please refer to the published source:

2 Fourier transform and mean quadratic variation of Bernoulli convolution on homogeneous Cantor set Zu-Guo Yu,2 Department of Mathematics, Xiangtan University, Hunan 405, China. 2 Program in Statistics and Operations Research, Queensland University of Technology, GPO Box 2434, Brisbane, Q400, Australia Abstract For the Bernoulli convolution on homogeneous Cantor set, under some condition, it is proved that the mean quadratic variation and the average of Fourier transform of this measure are bounded above and below. Key Words: mean quadratic variation, Fourier transform, Bernoulli convolution, homogeneous Cantor set. AMS Classification: 28A80, 00A73. Introduction The term fractal was first introduced by Mandelbrot [7] to denote sets with highly irregular structures. The fractal methods has been used to physics, computer sciences, finance in the past 30 years. In recent years, it even used to study the biological problems (such as [8, 4, 5, 7, 8]). In order to get a general aspect of the mathematical study of fractal geometry, one can refer to the book of Falconer []. In this paper, we denote R n the n-dimensional Euclidean space, N the set of natural numbers. For given finite contractive similarities {S j (x) = ρ j R j x+b j } m of R n, where 0 < ρ j <, b j R n, R j orthogonal, Hutchinson [4] proved that there exists unique compact set E satisfying E = m S j (E). Project partially supported by Morningside Centre of Mathematics of Chinese Academy of Science and the Youth Foundation of Chinese NNSF (No ). Current address. yuzg@hotmail.com, z.yu@qut.edu.au

3 E is called self-similar set. If there exists an open set U satisfying S j (U) U and S k (U) S j (U) = (k j), we call that {S j } m satisfy open set condition. He also proved [4] that for given probability vector P = (a, a 2,, a m ) satisfying m a j =, there exists unique probability measure µ on R n satisfying µ( ) = a j µ(s j ( )) () and the support of µ is E. µ is called self-similar measure and {a j } m is called weights of µ. One of the extension of open set condition in the study of fractal theory is the weak separation condition (see Ref.[5] for deterministic case and Ref.[6] for random case). Strichartz [9-3], Lau and Wang [6] have done much study on Fourier analysis of self-similar measure. Strichartz [0] obtained that Theorem A ([0]) Assume the open set condition, ρ j = ρ for all j {, 2,, m} and the rotations R j are either all equal or generate a finite group, µ is self-similar measure, and µ(x) is the Fourier transform of µ, then sup R R n β x R µ(x) 2 dx d, where d is a positive constant and β is defined by ρ β = m a 2 j. Theorem B ([0]) Assume the open set condition, ρ j = /r for all j {, 2,, m}, where r 2 be integer, and rb j are integers, µ is self-similar measure, and µ(x) is the Fourier transform of µ, then inf µ(x) 2 dx d R R n β 2, x R where d 2 is a positive constant and β is defined same as Theorem A. Let µ be a σ-finite measure on R n, for 0 β n, let V β (t; µ) = µ(b t n+β t (x)) 2 dx, R n where B t (x) is the ball of radius t, centred at x. We will call lim sup t 0 V β (t; µ) the upper β-mean quadratic variation (m.q.v.) of µ, and simply call i-m.q.v. if the limit exists. The m.q.v. index s is defined by s = inf{β : lim sup V β (t; µ) > 0} t 0 = sup{β : lim sup V β (t; µ) < } t 0 If µ is a self-similar measure on R n, Lau and Wang [6] obtained Theorem C([6]) Suppose that {S j } m satisfy the open set condition with respect to an open set U, and suppose that µ is a self-similar measure with µ( U k U j ) = 0 for all k j, then 0 < inf V β(t; µ) sup V β (t; µ) <, 0<t< 0<t< 2

4 where β is defined by m a 2 jρ β j =. In particular the m.q.v. index of µ is β. In the proofs of the above Theorems, the authors used formula () of self-similar measure. In this paper we want to extend the above Theorems to the case of Bernoulli convolution on homogeneous Cantor set [2]. Now we do not have formula () in this case. 2 Homogeneous Cantor set and Bernoulli convolution. Homogeneous Cantor sets [2] are a class of important fractal sets different from selfsimilar sets. Suppose I = [0, ], let {n k } k be a sequence of positive integers, and {c k } k be a real number sequence satisfying n k 2, 0 < n k c k (k ). Let E be the homogeneous Cantor set determined by {n k } k and {c k } k. We always assume n k = m, mc k < for all k in this paper. We use F, F 2,, F m to denote the m -ordered basic intervals from left to right. We use J = (j, j 2,, j k ) to denote the multi-index, J = k its length, and Λ the set of all such multi-indices, where j l {, 2,, m}, l =, 2,, k and k N. Define J k = {J = (j, j 2,, j k ) : j, j 2,, j k m}. We also use F j j 2 j k, F j j 2 j k 2,, F j j 2 j k m to denote the m (k + )-ordered basic intervals which are subintervals of any k-ordered basic interval from left to right. Then E k = j,j 2,,j k mf j j 2 j k, E = k 0 E k. In this paper we only consider the homogeneous Cantor set satisfies: Condition: There exist constants 0 < M M 2 < and 0 < ρ < such that M c c 2 c N ρ N M 2 (2) holds for all positive integer N. Example. Given Fibonacci sequence {s k } where s k {a, b}. We let c k = /3 when s k = a, and c k = /4 when s k = b. Let m = 2, E be the homogeneous set determined by {c k } k. In this case, we can choose ρ = (/3) λ (/4) λ where λ = ( 5 )/2. From the property of Fibonacci, we have ( + λ) ( λ)n N b + λ 3( λ) λn N a 3( λ) for all N, where N a and N b are the number of letter a and b in string s s 2 s N. If we take M 2 = 3 3( λ) 4 +λ and M = /M 2, then condition (2) holds. Denote r k = c c k ( c k ). Let {X m k } k be a sequence of random variables, X k takes values (j )r k (j =, 2,, m) with probability a j satisfying m a j =. Let N S N = X k. k= 3

5 It is easy to see S N only takes values of all left endpoints of N-ordered basic intervals. Let µ N be the probability measure induced by S N, it means that µ N (A) = P rob(s N A) for any measurable set A. When N, µ N weakly converges to a probability measure µ on E. We call µ the Bernoulli convolution on homogeneous Cantor set E. Garsia [3] discussed the case of Cantor set. If we denote ν k be the probability measure induced by X k. Then ν k (x) = e itx dν k (x) = a j e i(j )rkx, µ(x) = ν k (x) = [ a j e i(j )rkx ], (3) k= k= N N µ N (x) = ν k (x) = [ a j e i(j )rkx ] = a J e ib J x, (4) k= k= where b J is the left endpoint of F J. 3 Average of Fourier transform of Bernoulli convolution. We define β by ρ β = a 2 j. (5) Since 0 < m a 2 j <, 0 < ρ <, hence β 0. We also have β. Otherwise, if β >, then ρ β < ρ. Since mc k < for all k, hence m k c c 2 c k <, then M m k ρ k <. We have ln(m )/k + ln(mρ) < 0, hence we have mρ if we take limit k. Hence = a j = (a j ρ β/2 )ρ β/2 ( a 2 jρ β ) /2 (mρ β ) /2 = (mρ β ) /2 < (mρ) /2. This is a contradiction. Hence 0 β. Remark : If we take a j =, j =, 2,, m, from (5), m ρβ = ln m, hence β =. m ln ρ From Ref.[2], it equals to the Hausdorff dimension of the homogeneous Cantor set E. In this section, we want to prove the averages such as H(R) = µ(x) 2 dx R β x R also bounded above and below, where µ(x) is the Fourier transform of µ. First we discuss the upper bound of H(R). 4

6 Theorem Let µ be Bernoulli convolution on homogeneous Cantor set with n k = m, mc k <, and condition (2) holds, then sup R µ(x) 2 dx d < R β x R for some positive constant d, where β satisfies (5). Proof. It suffices to show for all positive integer N. Since hence for all N, For J, J J N, J J, we have x ρ N µ(x) 2 dx dρ N(β ) (6) ν k (x) = a j e i(j )rkx a j =, (7) µ(x) µ N (x) = a J e ib J x. (8) b J b J c c 2 c N M ρ N. (9) For M, from the proof of Theorem 3. of Ref.[0], after d and d 2 are chosen appropriately, we can construct a function h(x) with the following properties: i) h 0. ii) h d in x d 2. iii) ĥ(0) = and iv) ĥ(ξ) = 0 in ξ M. Hence µ(x) 2 dx d x d 2 ρ N h(ρ N x) µ(x) 2 dx x d 2 ρ N d ρ N ρ N h(ρ N x) µ(x) 2 dx R d ρ N ρ N h(ρ N x) a J e ib J x 2 dx (from (8)) R = d ρ N a J ā J ĥ(ρ N (b J b J )) J J N = d ρ N a J 2 (From (9) and iv)) which implice (6). = d ρ N ρ βn = d ρ N(β ), In order to obtain the bound below, we must the following Lemma. Lemma Let µ be Bernoulli convolution on homogeneous Cantor set with n k = m, mc k <, and condition (2) holds, then k=n+ ( m a j e i((j )r kx/(c c 2 c N )) ) is bounded away from zero on D = {x : x M (M 2 π}, where δ can be any positive small positive +δ) constant. 5

7 Proof. We have Since for k N +, Ree iλx ( a j e i((j )r kx/(c c 2 c N )) ) = 0 (j )r k c c 2 c N c c 2 c k ( c k ) c c 2 c N a j cos( (j )r k c c 2 c N λ)x. M 2ρ k N M, we choose λ = M 2ρ k N 2M so on x M (M 2 π. Hence +δ) a j e i((j )r kx/(c c 2 c N )) = e iλx ( a j e i((j )r kx/(c c 2 c N )) ) Re[e iλx ( a j e i((j )r kx/(c c 2 c N )) )] = a j cos( (j )r k λ)x c c 2 c N a j cos M 2ρ k N 2(M 2 + δ) π = cos M 2ρ k N 2(M 2 + δ) π k=n+ = k=0 ( a j e i((j )r kx/(c c 2 c N )) ) cos M 2 2(M 2 + δ) ρk π k=n+ cos M 2ρ k N 2(M 2 + δ) π on x M (M 2 π, which gives the desired estimate, since the product converges to a +δ) positive value d. Denote D N = {x : x (c c N ) M (M 2 π}. Now we have +δ) Theorem 2 Let µ be Bernoulli convolution on homogeneous Cantor set with n k = b m, mc k <, and condition (2) holds and if J c c N is integer for all positive integer N and J J N, then inf µ(x) 2 dx d R R β 2 > 0 x R for some positive constant d 2, where β satisfies (5). Proof. It is suffices to show D N µ(x) 2 dx d 2 ρ n(β ) (0) 6

8 for all N. Because b J b J c c N is integer for all positive integer N and J J N, one can find that a J e i c x c N is a trigonometric polynomial, hence Hence a J e i b J c x c N 2 = a 2 J. () µ(x) 2 dx = (c c N ) x µ( ) 2 dx D N x M (M 2 +δ) π c c N N = (c c N ) ( a j e i (j )r k x c c 2 c N ) 2 ( a j e i (j )r k x c c 2 c N ) 2 dx D k= k=n+ = d(c c N ) a J e i b J c x c N 2 dx (From Lemma ) D J J N = d(c c N ) a 2 Jdx (From ()) D which implice (0). dm π ρ N ρ βn = (M 2 + δ) M 2 M M 2 (M 2 + δ) dπρn(β ) b J c c N Remark 2: If all c k = /l k, l k is positive integer and l k 2, m = 2, then is integer for all positive integer N and J J N. First b = 0, b 2 = c, it is obvious that 0 and /c are integers. Now we assume b J /(c c k ) is an integer for any J J k. Since for any J J k, b J is equal to either b J or b J + c c k ( c k ). For the first case, b J /(c c k ) = l k (b J /(c c k )) is an integer. For the second case, b J /(c c k ) = l k (b J /(c c k )) + l k is also an integer. b J c c N It is a problem to obtain the lower bound of H(R) without condition is integer for all positive integer N and J J N. In the case of all a j = /m, then from Ref.[2] we know µ satisfies µ(b r (x)) cr β for r and all x. If there is ɛ > 0 such that lim inf r 0 (2r) β H β (E B r (x)) > ɛ on E. Then from the fractal Plancherel theorem of Ref.[3], we have lim inf R H(R) chβ (E), where H β (E) is the β-dimensional Hausdorff measure of E. Hence if 0 < H β (E) <, we can also obtain the lower bound of H(R). 4 Mean quadratic variations of Bernoulli convolutions. In our case R n = R, hence n =, and B t (x) is a interval with length 2t, centred x. 7

9 For any 0 < t <, let Λ t = {J = (j, j 2,, j Nt ) : c c Nt t < c c jnt } and a J = a j a j2 a jnt. Now we have Theorem 3 Let µ be Bernoulli convolution on homogeneous Cantor set with n k = m, c k m < for all k, and condition (2) holds, then 0 < d inf V β(t; µ) sup V β (t; µ) d 2 < (2) 0<t< 0<t< for two positive constants d and d 2, where β defined by (5). In particular the m.q.v. index of µ is β. Proof. If β = 0, then a j = for some j, and a l = 0 if l j, so µ is a point mass measure and V β (t; µ) = c 0. Hence (2) holds. Now we assume 0 < β and let L be the -dimensional Lebesgue measure. For any ξ, η F J, J Λ t, we have B t (ξ) B t (η) contains a interval with length t. Hence V β (t; µ) = µ(b t +β t (x)) 2 dx = [ χ Bt(ξ)(x)χ Bt(η)(x)dµ(ξ)dµ(η)]dx t +β = χ Bt(ξ)(x)χ Bt(η)(x)dxdµ(ξ)dµ(η) t +β = L(B t +β t (ξ) B t (η))dµ(ξ)dµ(η) L(B t +β t (ξ) B t (η))dµ(ξ)dµ(η) J Λ ξ,η F J t dµ(ξ)dµ(η) = a 2 ξ,η F J J = ρntβ ρ Nt ( c c Nt ) β ρ β ρβ M β 2 = d > 0 Denote Λ(t, J) = {J Λ t : d(f J, F J ) 2t}. Similar to proof of Lemma 2.9 of Ref.[6], one can easily prove Λ(t, J) can have at most K members, where K is a positive integer which is independent of t and J. From Lemma 3. of Ref.[6], we have V β (t; µ) c 2 = c 2 c 2 c 2 ξ η 2t J Λ ξ F J η R t J Λ(t,J) d(f J,F J ) 2t dµ(ξ)dµ(η) χ { ξ η 2t} dµ(ξ)dµ(η) ξ F J,η F J a J a J c dµ(ξ)dµ(η) d(f J,F J ) 2t (a 2 J + a 2 j )

10 c 2 2 ( J Λ(t,J) a 2 J + J Λ t J Λ(t,J ) a 2 J ) c 2K a 2 J = c 2K (ρnt ) β c ρ Nt 2K( ) β c c c 2K(/M ) β = d 2 < Nt Remark 3: Since the m.q.v. index of µ is β, from Ref.[6], if lim R H(R) exists, then lim t 0 V β (t; µ) exists, and reverse is also true. Hence β is the critical value of H(R) from 0 to. Acknowledgements The author would like to express his thanks to Prof. Zhi-Ying Wen of Tsinghua University for useful comments and suggestions, and encouragement. He also would like to thank Prof. Hui Rao and Dr. Ji-Hua Ma of Wuhan University for many helpful discussions. In particular, the example in this paper is given by Prof. Hui Rao. References [] K.J. Falconer, Techniques in Fractal Geometry, Wiley, London, 997. [2] De-Jun Feng, Hui Rao and Jun Wu, The net measure property of homogeneous Cantor set and its application (in Chinese), Progress In Natural Science 6(6) (996) [3] A. M. Garsia, Arithonetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 02 (962) [4] J.E.Hutchinson, Fractals and Self-similarity, Indiana University Math.Journal, 30(5) (98), [5] K. S. Lau and S. M. Ngai, Multifractal measures and a weak separation condition, Adv. Math. 4 (999), [6] Ka-sing Lau and Jian-rong Wang, Mean Quadratic Variations and Fourier Asymptotics of Self-similar Measures, Monatshefte Fur Mathematik, 5 (993), [7] B. B. Mandelbrot, The Fractal Geometry of Nature, in Bifurcation theory and applications in Scientific disciplines, Ann. New York Acad. Sci. 36 (Freeman and Co., New York, 982) PP [8] C.K. Peng, S. Buldyrev, A.L. Goldberg, S. Havlin, F. Sciortino, M. Simons, and H. E. Stanley, Long-range correlations in nucleotide sequences. Nature 356 (992) [9] R.S.Strichartz, Fourier asymptotics of fractal measures, J.Functional Anal. 89 (990), [0] R.S.Strichartz, Self-similar measures and their Fourier transforms I, Indiana University Math.Journal, 39 (990), [] R.S.Strichartz, Self-similar measures and their Fourier transforms II, Trans. Amer. Math. Soc. 39 (990), [2] R.S.Strichartz, Self-similar measures and their Fourier transforms III, Indiana University Math. Journal, 42 (993),

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