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Fractal dimension and measure of the subset of Moran set
Chaos, Solitons and Fractals 40 (2009) 190–196 www.elsevier.com/locate/chaos
Fractal dimension and measure of the subset of Moran set Meifeng Dai *, Ying Jiang Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China Accepted 16 July 2007
Communicated by Prof. Ji-Huan He
Abstract We discuss the fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. Firstly, we give the Hausdorff dimension of subset BP(x) in compatible case and incompatible case. Then we attain that there exists a subset B of the set BP(x) such that B has full lP-measure but zero Hausdorff measure in incompatible case. Finally, if the gap condition holds, we see that BP(x) and E(x) have the same Hausdorff measure and packing measure, and both of them are a-sets in compatible case. 2007 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, fractals have been widely used in many fields of science solving a great deal of problems. The subject is receiving impetus from an increasingly diverse range of applications. For example, it has been applied to physical situations and progress is being made in this direction. The Hausdorff dimension and the Euler Characteristic of Cantorian sets are used by Mohamed El Naschie in E-infinity theory [1–3]. Fangxiong Zhen has discussed the dimensions of subsets of Cantor-type sets. We further discuss fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. Let M ¼ f1; 2; . . . ; mg and for any j 2 M; i ¼ i1 i2 in 2 M 1 ; n 2 N , define dj ði; nÞ ¼ ]fk : ik ¼ j; 1 6 k 6 ng whenever there exists the limit dj ðiÞ ¼ lim
n!1
1 dj ði; nÞ; n
which is called the frequency of number j in infinite length word i. Then for a given probability vector P ¼ ðp1 ; p2 ; . . . ; pm Þ; let the Besicovitch set BP be the subset of Cantor-type set E [4] given by BP ¼ fuðiÞ : dj ðiÞ ¼ pj ; i 2 M 1 ; for any j 2 Mg: *
Corresponding author. E-mail addresses: [email protected] (M. Dai), [email protected] (Y. Jiang).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.042
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
191
Here u is a bijective between M1 and the Cantor-type set E and M1 is the set of all infinite length sequence consisted by M. In [4], the author gave the Hausdorff dimension of BP for any given probability vector P ¼ ðp1 ; p2 ; . . . ; pm Þ and the ratio rj ðj ¼ 1; 2; . . . ; mÞ; P j2M p j log p j dim BP ¼ P : j2M p j log r j In [5], Min Wu defined a class of Moran sets E(x) associated with x and discussed the multifractal spectrum of Moran measure on E(x). In [6], Meifeng Dai and Dehua Liu investigated the local dimension and Hausdorff dimension of Moran measure on E(x) satisfying the strong separation condition. In this paper, we further discuss fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. In Section 2, we give the Hausdorff dimension of BP(x) and show that there exists a subset B of BP(x) such that B has full lP-measure but zero Hausdorff measure in incompatible case. In Section 3, we see that lP, Ha and Pa are equivalent on E(x) in compatible case (see Corollary 3.5), then we can get another byproduct that the set E(x)nBP(x) has zero Ha and Pa -measure when P ai is substituted by ðcai1 ; cai2 ; . . . ; caimi Þ (see Corollary 3.6). If the gap condition (see Section 3) holds, then BP(x) and E(x) have the same Hausdorff measure and packing measure, and both of them are a-sets in the case of P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ (see Corollary 3.9). 2. Definitions and main theorems Let {nk}kP1 N be a sequence of positive integers.SFor any k 2 N, define Dk ¼ fi1 i2 ik : 1 6 ij 6 nk ; 1 6 j 6 kg, D1 ¼ fi1 i2 ik : 1 6 ij 6 nk ; j P 1g and D ¼ 1 By convention denoted by D0 = B. For k¼1 Dk . r ¼ r1 r2 rk1 2 Dk1 and ij(1 6 ij 6 nk), we define r ij ¼ r1 r2 rk1 ij . Suppose that J 2 Rd is a compact set with nonempty interior. Let k}kP1 be a sequence of positive integers. {Uk}kP1 P {n k is a sequence of positive real vectors with Uk ¼ ðck1 ; ck2 ; . . . ; cknk Þ, nj¼1 ckj 6 1, k 2 N. Suppose that F ¼ fJ r : r 2 Dg is a collection of subsets of Rd. We say that the collection F fulfills the Moran structure if it satisfies the following Moran structure conditions (MSC): (1) J; = J. (2) For any r 2 D, Jr is geometrically similar to J, that is, there exists a similarity Sr: Rd ! Rd such that Jr = Sr(J). (3) For any k P 0 and r 2 Dk1, J r1 ; J r2 ; . . . ; J rnk are subsets of Jr, and int (Jr*i) ˙ intðJ rj Þ ¼ ; for i 5 j, where int(A) denotes the interior of A. (4) For any k P 1 and r 2 Dk1, 1 6 j 6 nk, jJ rj j ¼ ckj : jJ r j S T For F Rd fulfilling the Moran structure, set Ek ¼ r2Dk J r , and E ¼ kP0 Ek . It is ready to see that E is a nonempty compact set. The set E :¼ EðFÞ is called S the Moran set associated with the collection F. Let Fk ¼ fJ r : r 2 Dk g, and F ¼ kP0 Fk . The elements of Fk are called the basic elements of order k of the Moran set E and the elements of F are called the basic elements of the Moran set E. Remark 1. If limk!1 supr2Dk jJ r j > 0, then E contains interior points. Thus the measure and dimension properties will be trivial. We assume therefore limk!1 supr2Dk jJ r j ¼ 0. T For convenience, rjk denotes the first k elements of r 2 D1. Then the set 1 k¼0 J rjk consists a single point which we shall denote by u(r): uðrÞ ¼
1 \
J rjk :
k¼0
Suppose that the set J (for convenience, we assume that the diameter of J is 1), the sequences {nk} and {Uk} are given, denoted by U ¼ UðJ ; fnk g; fUk gÞ the class of the Moran sets satisfying the MSC (1)–(4). We call UðJ ; fnk g; fUk gÞ the Moran class associated with the triplet ðJ ; fnk g; fUk gÞ. Let A ¼ fa1 ; a2 ; . . . ; am g be a finite set. For AN ¼ fx ¼ s1 s2 sk 2 D1 : sk 2 Ag, let xjk ¼ s1 s2 sk , and kxjk kai ¼ #fsj ¼ ai : sj 2 xjk g, where kxjk kai denotes the times which element ai appears in the sequence xjk. Suppose kxj k ¼ gi > 0, for every ai 2 A, we call x has frequency g ¼ ðg1 ; g2 ; . . . ; gm Þ. Obviously, we have x AN, limk!1 kk ai P P2 m m kxj k ¼ k, and k ai i¼1 i¼1 gi ¼ 1.
192
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196 kxj k
For any frequency g, we define ANg ¼ fx ¼ fsk gkP1 2 D1 : sk 2 A; limk!1 kk ai ¼ gi ; 1 6 i 6 mg. Suppose mi(1 6 i 6 m) be a positive integer, Ui ¼ ðci1 ; ci2 ; . . . ; cimi Þð1 6 i 6 mÞ is a sequence of positive real vectors with P mi For x 2 ANg and E 2 UðJ ; fnk g; fUk gÞ, when k P 1, sk = ai, we let nk = mi, Uk :¼ j¼1 cij 6 1. Ui ¼ ðci1 ; ci2 ; . . . ; cimi Þ. Then we obtain a class of Moran sets associated with x 2 AN, denoted by E(x). From [5], we know that dimEðxÞ ¼ DimEðxÞ ¼ a; for a satisfies the following equation: m X i¼1
gi log
mi X
caij ¼ 0;
ð2:1Þ
j¼1
where dim and Dim denote the Hausdorff dimension and packing dimension, respectively. For the Moran set E satisfying the strong separation condition, we have the map u is one-to-one. That is, for each T 1 x 2 E there exists a unique r such that uðrÞ ¼ 1 n¼0 J rjn ¼ x. Then we denote u (E) = DE and say r is the address of the point x. Similarly, for E(x), there exists DE(x) such that u(DE(x)) = E(x). For J r 2 Fk is a basic element of order k of the Moran set E(x), where r = r1 rk 2 Dk, if sn = ai 2 A (1 6 i 6 m), then rn 2 {1, 2, . . . , mi} for 1 6 n 6 k. By x = s1 sk , we define r(ai) as following: for every n (1 6 n 6 k), only when sn = ai (1 6 i 6 m), we pick rn from r and form a new sequence, denote rðai Þ ¼ ri1 ri2 rikxjk kai where rij 2 f1; 2; . . . ; mi g ð1 6 j 6 kxjk kai Þ. If kxjk kai ¼ 0, we assume r (ai) = ;, then r(a1)r(a2) r(am) is rearranged of r r1 rk. For every ai 2 A, let P ai ¼ ðpi1 ; pi2 ; . . . ; pimi Þ (1 6 i 6 m) be probability vectors, i.e. pij > 0 and P= mi (1 6 i 6 m), denote P ¼ ðP a1 ; P a2 ; . . . ; P am Þ. Define prðai Þ ¼ pri1 prikxj k (1 6 i 6 m), where j¼1 p ij ¼ 1 k ai P m prij 2 fpi1 ; pi2 ; . . . ; pimi g (If r(ai) = ; assume that prðai Þ ¼ 1). It is easy to see that r2Dk Pi¼1 prðai Þ ¼ 1Qfor any k P 1. From the above discussion, similarly we denote crðai Þ ¼ cri1 crikxjk ka (1 6 i 6 m), then we get jJ r j ¼ mi¼1 crðai Þ . i For rðai Þ ¼ ri1 ri2 rikxjk kai and j ¼ 1; 2; . . . ; mi ; we define dj ðrðai Þ; kxjk kai Þ ¼ ]ft : rit ¼ j; 1 6 t 6 kxjk kai g; whenever there exists the limit dj ðrðai ÞÞ ¼ lim
k!1
dj ðrðai Þ; kxjk kai Þ ¼ pij ; kxjk kai
for 1 6 i 6 m; 1 6 j 6 mi . Then we define the Bescovitch set BP(x) to be the subset of Moran set E(x) given by BP ðxÞ ¼ fuðrÞ : dj ðrðai ÞÞ ¼ pij ; r 2 DEðxÞ ; for 1 6 i 6 m; 1 6 j 6 mi g: A remark from [7] says that BP(x) is a Borel set. For probability vectors P a1 ; P a2 ; . . . ; P am , we denote vP(x) the corresponding product measure on DE(x), so we attain a measure Q lP = vP(x)u1 on E(x). Then for any k-th basic element J r 2 Fk ; r ¼ r1 r2 rk 2 Dk , we have lP ðJ r Þ ¼ mi¼1 prðai Þ . By the strong law of large numbers, we see that lP(BP(x)) = 1 and since BP(x) E(x), thus lP(BP(x)) = lP(E(x)) = 1. For F ¼ fJ r : r 2 Dg; we define ( ) X [ s j s j j j Hd;F ðEÞ ¼ inf jJ r j : fJ r g F; jJ r j 6 d; E Jr ; j
HsF ðEÞ ¼ lim Hsd;F ðEÞ: d!0
Similar to the definition of Hausdorff dimension, we define dimF E ¼ supfs P 0 : HsF ðEÞ ¼ 1g ¼ inffs P 0 : HsF ðEÞ ¼ 0g: Now we would like to define lsP -measure as follows: ( ) X [ s s lP ðE; dÞ ¼ inf lP ðU i Þ : lP ðU i Þ 6 d; E Ui ; i
lsP ðEÞ ¼ lim lsP ðE; dÞ; d!0
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
193
then the lP-dimension of E is given by dimlP E ¼ supfs P 0 : lsP ðEÞ ¼ 1g ¼ inffs P 0 : lsP ðEÞ ¼ 0g: We see that if lP(E) > 0, then dimlP E ¼ 1:
ð2:2Þ
Now we state the main results in this paper. Theorem 2.1. For x ¼ s1 s2 sk , denote Pm Pmi i¼1 gi j¼1 p ij log p ij P ; s¼P m mi i¼1 gi j¼1 p ij log cij
ð2:3Þ
then dimBP ðxÞ ¼ s, and s 6 a, where a is uniquely decided by (2.1). The equality is attained when P ai coincide with ðcai1 ; cai2 ; . . . ; caimi Þ. Furthermore, we show that there exists a subset B of Bescovitch set BP(x) such that B has full lP-measure but zero Hausdorff measure when P ai –ðcai1 ; cai2 ; . . . ; caimi Þ, which is described as follows: Theorem 2.2. There exists a Borel set B of BP(x) such that lP(B) = 1 and Hs ðBÞ ¼ 0 in incompatible case, that is P ai –ðcai1 ; cai2 ; . . . ; caimi Þði ¼ 1; 2; . . . ; mÞ.
3. Proofs of main theorems Now we set to prove the results displayed above. First let us look at some lemmas. Pmi Lemma 3.1. Suppose that P ai ¼ i1 ; pi2 ; . . . ; p imi Þ are probability vectors with pij > 0 and j¼1 pij ¼ 1 for 1 6 i 6 m. Let gi Pðp m be positive real numbers with i¼1 gi ¼ 1, then for any qi1 ; qi2 ; . . . ; qimi 2 R; ! mi mi m m X X X X gi pij ð log pij þ qij Þ 6 gi log eqij : ð3:1Þ j¼1
i¼1
i¼1
The equality is attained if and only if pij ¼ eqij
j¼1
P
mi qij j¼1 e
1
for i ¼ 1; 2; . . . ; m.
Proof. From [8], we know that if P ai ¼ ðpi1 ; pi2 ; . . . ; pimi Þ are probability vectors with pij > 0 and 1 6 i 6 m,1 6 j 6 mi, then for any qi1 ; qi2 ; . . . ; qimi 2 R, mi X
pij ð log pij þ qij Þ 6 log
j¼1
mi X
eqij :
j¼1
P i q 1 The equality is attained if and only if pij ¼ eqij ð mj¼1 e ij Þ for j ¼ 1; 2; . . . ; mi . As gi are positive real numbers, thus we have gi
mi X
pij ð log pij þ qij Þ 6 gi log
mi X
j¼1
eqij ;
j¼1
for i ¼ 1; 2; . . . ; m, then we have (3.1). Lemma 3.2. [4] If E fuðrÞ : lim
jrj!1
then dimE ¼ ddimlP E.
log lP ðJ r Þ ¼ dg; log jJ r j
h
Pmi
j¼1 p ij
¼ 1 for
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M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
Proof of Theorem 2.1. For any r ¼ r1 r2 rk 2 DEðxÞ , Qmi dj ðrðai Þ;kxjk kai Þ Q log mi¼1 pij j¼1 log prjk log lP ðJ rjk Þ lim ¼ lim ¼ lim Qm Qmi dj ðrðai Þ;kxjk kai Þ k!1 log jJ rj j k!1 log crj k!1 k k log i¼1 j¼1 cij Pm kxjk kai Pmi dj ðrðai Þ;kxjk kai Þ Pm Pmi log p ij i¼1 j¼1 k kxjk kai i¼1 gi j¼1 p ij log p ij ¼P P ¼ s: ¼ lim m mi Pmi dj ðrðai Þ;kxjk kai Þ k!1 Pm kxjk ka i gi pij log cij i¼1 j¼1 log c ij i¼1 j¼1 k kxj k k ai
Since lP(BP(x)) = 1, by Lemma 3.2 and (2.2) we have dim BP(x) = s. Put qij ¼ log caij in Lemma 3.1, from (2.1) and (2.3), we get s 6 a. Furthermore, if we put !1 mi X log caij log caij pij ¼ e e ;
ð3:2Þ
j¼1
from (2.1) we know pij ¼ caij is the solution of (3.2), then we have s = a.
h
Lemma 3.3. [4] [law of iterated logarithm Hartman-Winter] Suppose that {Xn:n P 1} is a sequence of independent, identically distributed random variables (i.i.d.r.v.) satisfying E(Xn) = 0 and EðX 2n Þ ¼ a2 2 ð0; 1Þ, then Pn i¼1 X i ¼ 1; p a:e:; lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n!1 2na log log na2 Pn i¼1 X i ¼ 1; p a:e: lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n!1 2na2 log log na2 Proof of Theorem 2.2. Put p ¼ lP u; X ij ¼ log pij s log cij ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; mi ; then {Xij} is a sequence of i.i.d.r.v. with respect to p. We see that EðX ij Þ ¼
m X
gi
mi X
pij ðlog pij s log cij Þ ¼ 0:
j¼1
i¼1
Suppose that P ai –ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m; from Theorem 2.1, we get 0 < EðX 2ij Þ ¼
m X
gi
i¼1
Set
( 0
B ¼
mi X
pij ðlog pij s log cij Þ2 ¼: a2 < 1:
j¼1
) Pn l¼1 X il jl uðrÞ : lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1; for il 2 f1; 2; . . . ; mg; jl 2 f1; 2; . . . ; mil g ; n!1 2na2 log log na2
then lP ðB0 Þ ¼ 1 by Lemma 3.3. Since lP(BPP (x)) = 1, let B = B 0 ˙ BP(x), we get lP(B) = 1. With the above arguments, we see that 1 k¼1 X ilk jlk ¼ þ1 a:e., that is, Qn k¼1 p il jl log Qn s k k ! þ1; as n ! 1; where ilk 2 f1; 2; . . . ; mg; jlk 2 f1; 2; . . . ; milk g; k¼1 cil jl k
k
from which we also have lP ðJ r Þ ! 1; jJ r js
as jrj ! 1:
S k k k js > k, for r 2 Dn}, Jk ¼ 1 Let S Jkn ¼ fJ r : lP ðJ r Þ=jJ rT n¼1 Jn . For any d > 0, k > 0, choose fJ j g J such that k k k k k B j J j ; jJ j j 6 d, and J i J j ¼ ;ði–jÞ. By the definition of J , we have lP ðJ kj Þ jJ kj js
> k;
8j:
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
195
Thus Hsd ðBÞ 6
X
jJ kj js 6
1X 1 lP ðJ kj Þ 6 : k k
Letting k ! 1, we have Hsd ðBÞ ¼ 0; from which it follows that Hs ðBÞ ¼ 0.
h
Lemma 3.4. [9] Suppose that l is a finite Borel measure such that 0 < lim
jrj!1
lðJ r Þ < 1; jJ r ja
0 < lim jrj!1
lðJ r Þ < 1; jJ r ja
for any r 2 D;
then l Ha Pa on a Moran set E. Corollary 3.5. lP Ha Pa on E(x) if P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m, where P ¼ ðP a1 ; P a2 ; . . . ; P am Þ. Proof. For r = r1 rk 2 Dk and P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m; we have Qm Qmi dj ðrðai Þ;kxjk kai Þ lP ðJ r Þ i¼1 j¼1 p ij ¼ 1; ¼Q Q m mi jJ r ja ðca Þdj ðrðai Þ;kxjk kai Þ i¼1
j¼1
ij
so by Lemma 3.4, we get lP Ha Pa on E(x).
h
Corollary 3.6. Ha ðEðxÞ n BP ðxÞÞ ¼ Pa ðEðxÞ n BP ðxÞÞ ¼ 0; Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ and Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ in case of P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þði ¼ 1; 2; . . . ; mÞ. Proof. By the definition of equivalence, lP Ha on E () for any F E, lP(E) = 0 if and only if Ha ðF Þ ¼ 0. It is clear that BP(x) is the subset of E(x) with lP(E(x)) = lP(BP(x)) = 1, then we get immediately lP(E(x)nBP(x)) = 0, which further implies Ha ðEðxÞ n BP ðxÞÞ ¼ 0 by Corollary 3.5. Furthermore, Ha ðBP ðxÞÞ 6 Ha ðEðxÞÞ 6 Ha ðBP ðxÞÞ þ Ha ðEðxÞ n BP ðxÞÞ ¼ Ha ðBP ðxÞÞ; we have Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ. Similar arguments will prove that Pa ðEðxÞ n BP ðxÞÞ ¼ 0 and Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ. h For the simplicity of proof of the following results, we would like to make a hypothesis here. We say that the Moran set E(x) satisfies the gap condition if there exists a constant d > 0 such that distðJ r1 r2 rn ; J r1 r2 r0n Þ P d maxfjJ r1 r2 rn j; jJ r1 r2 r0n jg for any n 2 N and rn –r0n . Lemma 3.7. [9] Suppose that E is a Moran set satisfying the gap condition, then for any u(r) 2 E and finite Borel measure l, supp(l) E, the following inequalities hold: ðcdÞa lim jrj!1
ðcdÞa lim
jrj!1
jJ r ja ra jJ r ja 6 lim 6 ca lim ; lðJ r Þ r!0 lðBðuðrÞ; rÞÞ jrj!1 lðJ r Þ jJ r ja ra jJ r ja 6 lim 6 ca lim ; jrj!1 lðJ r Þ lðJ r Þ r!0 lðBðuðrÞ; rÞÞ
where c = min{cij}, for i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; mi . Lemma 3.8. [10] For finite Borel measure l, there exists four positive constants b1 ; b2 ; b3 ; b4 such that ra ra b1 lðEÞ inf lim 6 Ha ðEÞ 6 b2 lðRd Þ sup lim ; x2E x2E r!0 lðBðx; rÞÞ r!0 lðBðx; rÞÞ ra ra b3 lðEÞ inf lim 6 Pa ðEÞ 6 b4 lðRd Þ sup lim : x2E r!0 lðBðx; rÞÞ r!0 lðBðx; rÞÞ x2E Corollary 3.9. If E(x) is the Moran set defined above satisfying gap condition and BP(x) is the Bescovitch set defined in Section 2, then 0 < Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ < 1 and 0 < Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ < 1; when P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ.
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M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
Proof. From Lemma 3.8, we see that the density of balls is comparable with that of Jr, so we have ( ) ( ) jJ r ja jJ r ja 6 Ha ðEðxÞÞ 6 b02 lðRd Þ sup lim lim b01 lðEðxÞÞ inf x2EðxÞ jrj!1 lðJ r Þ x2EðxÞ jrj!1 lðJ r Þ
ð3:3Þ
together with Lemmas 3.7 and 3.8. Since jJ r ja ¼ 1; in case ofP ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; lðJ r Þ we get 0 < b01 6 Ha ðEðxÞÞ 6 b02 by substituting l with lP in (3.3), which implies that 0 < Ha ðEðxÞÞ < 1. Since we have proved in Corollary 3.6 that Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ; we have 0 < Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ < 1. Analogously, we can prove 0 < Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ < 1. h
4. Conclusion The physical relevance of the present paper is due to El Naschie’s E-infinity theory. We first recall the definition of the Moran sets E(x) associated with x and some known results about fractal dimension and measure in physics and mathematical fields. Secondly, we give the Hausdorff dimension of the subset BP(x) of E(x) in Rd satisfying the strong separation condition. Finally, we attain some results about lP ; Ha and Pa -measures in compatible case and incompatible case.
Acknowledgements This research is supported the National Science Foundation of China (10671180) and Jiangsu University 05JDG041.
References [1] El Naschie MS. The concepts of E-infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495–511. [2] El Naschie MS. A guide to the mathematics of E-infinity Cantorian space-time theory. Chaos, Solitons & Fractals 2005;25:955–64. [3] El Naschie MS. Modular qroups in Cantorian E-infinity high energy physics. Chaos, Solitons & Fractals 2003;16:353–66. [4] Zhen Fangxiong. Dimensions of subsets of Cantor-type sets. Int J Math Math Sci 2006:1–8. Artical ID 26359. [5] Wu Min. The multifractal spectrum of a class of Moran measures. Sci China Ser A 2005;35(2):147–61. [6] Meifeng Dai, Dehua Liu. The local dimension of Moran measures satisfying the strong separation condition. Chaos, Solitons & Fractals 2008;38(4):1025–30. [7] Moran M, Rey J-M. Singularity of self-similar measures with respect to Hausdorff measures. Trans Am Math Soc 1998;350(6):2297–310. [8] Falconer K. Techniques in fractal geometry. Chichester: John Wiley; 1997. [9] Meifeng Dai. The equivalence of measures on Moran set in general metric space. Chaos, Solitons & Fractals 2006;29(1):55–64. [10] Tayor SJ, Tricot C. Packing measure and its evaluation for a Brownian path. Trans Am Math Soc 1985;288(2):679–99.
Fractal dimension and measure of the subset of Moran set Meifeng Dai *, Ying Jiang Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China Accepted 16 July 2007
Communicated by Prof. Ji-Huan He
Abstract We discuss the fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. Firstly, we give the Hausdorff dimension of subset BP(x) in compatible case and incompatible case. Then we attain that there exists a subset B of the set BP(x) such that B has full lP-measure but zero Hausdorff measure in incompatible case. Finally, if the gap condition holds, we see that BP(x) and E(x) have the same Hausdorff measure and packing measure, and both of them are a-sets in compatible case. 2007 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, fractals have been widely used in many fields of science solving a great deal of problems. The subject is receiving impetus from an increasingly diverse range of applications. For example, it has been applied to physical situations and progress is being made in this direction. The Hausdorff dimension and the Euler Characteristic of Cantorian sets are used by Mohamed El Naschie in E-infinity theory [1–3]. Fangxiong Zhen has discussed the dimensions of subsets of Cantor-type sets. We further discuss fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. Let M ¼ f1; 2; . . . ; mg and for any j 2 M; i ¼ i1 i2 in 2 M 1 ; n 2 N , define dj ði; nÞ ¼ ]fk : ik ¼ j; 1 6 k 6 ng whenever there exists the limit dj ðiÞ ¼ lim
n!1
1 dj ði; nÞ; n
which is called the frequency of number j in infinite length word i. Then for a given probability vector P ¼ ðp1 ; p2 ; . . . ; pm Þ; let the Besicovitch set BP be the subset of Cantor-type set E [4] given by BP ¼ fuðiÞ : dj ðiÞ ¼ pj ; i 2 M 1 ; for any j 2 Mg: *
Corresponding author. E-mail addresses: [email protected] (M. Dai), [email protected] (Y. Jiang).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.042
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
191
Here u is a bijective between M1 and the Cantor-type set E and M1 is the set of all infinite length sequence consisted by M. In [4], the author gave the Hausdorff dimension of BP for any given probability vector P ¼ ðp1 ; p2 ; . . . ; pm Þ and the ratio rj ðj ¼ 1; 2; . . . ; mÞ; P j2M p j log p j dim BP ¼ P : j2M p j log r j In [5], Min Wu defined a class of Moran sets E(x) associated with x and discussed the multifractal spectrum of Moran measure on E(x). In [6], Meifeng Dai and Dehua Liu investigated the local dimension and Hausdorff dimension of Moran measure on E(x) satisfying the strong separation condition. In this paper, we further discuss fractal dimension and measure for the subset BP(x) of Moran set E(x) in Rd satisfying the strong separation condition. In Section 2, we give the Hausdorff dimension of BP(x) and show that there exists a subset B of BP(x) such that B has full lP-measure but zero Hausdorff measure in incompatible case. In Section 3, we see that lP, Ha and Pa are equivalent on E(x) in compatible case (see Corollary 3.5), then we can get another byproduct that the set E(x)nBP(x) has zero Ha and Pa -measure when P ai is substituted by ðcai1 ; cai2 ; . . . ; caimi Þ (see Corollary 3.6). If the gap condition (see Section 3) holds, then BP(x) and E(x) have the same Hausdorff measure and packing measure, and both of them are a-sets in the case of P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ (see Corollary 3.9). 2. Definitions and main theorems Let {nk}kP1 N be a sequence of positive integers.SFor any k 2 N, define Dk ¼ fi1 i2 ik : 1 6 ij 6 nk ; 1 6 j 6 kg, D1 ¼ fi1 i2 ik : 1 6 ij 6 nk ; j P 1g and D ¼ 1 By convention denoted by D0 = B. For k¼1 Dk . r ¼ r1 r2 rk1 2 Dk1 and ij(1 6 ij 6 nk), we define r ij ¼ r1 r2 rk1 ij . Suppose that J 2 Rd is a compact set with nonempty interior. Let k}kP1 be a sequence of positive integers. {Uk}kP1 P {n k is a sequence of positive real vectors with Uk ¼ ðck1 ; ck2 ; . . . ; cknk Þ, nj¼1 ckj 6 1, k 2 N. Suppose that F ¼ fJ r : r 2 Dg is a collection of subsets of Rd. We say that the collection F fulfills the Moran structure if it satisfies the following Moran structure conditions (MSC): (1) J; = J. (2) For any r 2 D, Jr is geometrically similar to J, that is, there exists a similarity Sr: Rd ! Rd such that Jr = Sr(J). (3) For any k P 0 and r 2 Dk1, J r1 ; J r2 ; . . . ; J rnk are subsets of Jr, and int (Jr*i) ˙ intðJ rj Þ ¼ ; for i 5 j, where int(A) denotes the interior of A. (4) For any k P 1 and r 2 Dk1, 1 6 j 6 nk, jJ rj j ¼ ckj : jJ r j S T For F Rd fulfilling the Moran structure, set Ek ¼ r2Dk J r , and E ¼ kP0 Ek . It is ready to see that E is a nonempty compact set. The set E :¼ EðFÞ is called S the Moran set associated with the collection F. Let Fk ¼ fJ r : r 2 Dk g, and F ¼ kP0 Fk . The elements of Fk are called the basic elements of order k of the Moran set E and the elements of F are called the basic elements of the Moran set E. Remark 1. If limk!1 supr2Dk jJ r j > 0, then E contains interior points. Thus the measure and dimension properties will be trivial. We assume therefore limk!1 supr2Dk jJ r j ¼ 0. T For convenience, rjk denotes the first k elements of r 2 D1. Then the set 1 k¼0 J rjk consists a single point which we shall denote by u(r): uðrÞ ¼
1 \
J rjk :
k¼0
Suppose that the set J (for convenience, we assume that the diameter of J is 1), the sequences {nk} and {Uk} are given, denoted by U ¼ UðJ ; fnk g; fUk gÞ the class of the Moran sets satisfying the MSC (1)–(4). We call UðJ ; fnk g; fUk gÞ the Moran class associated with the triplet ðJ ; fnk g; fUk gÞ. Let A ¼ fa1 ; a2 ; . . . ; am g be a finite set. For AN ¼ fx ¼ s1 s2 sk 2 D1 : sk 2 Ag, let xjk ¼ s1 s2 sk , and kxjk kai ¼ #fsj ¼ ai : sj 2 xjk g, where kxjk kai denotes the times which element ai appears in the sequence xjk. Suppose kxj k ¼ gi > 0, for every ai 2 A, we call x has frequency g ¼ ðg1 ; g2 ; . . . ; gm Þ. Obviously, we have x AN, limk!1 kk ai P P2 m m kxj k ¼ k, and k ai i¼1 i¼1 gi ¼ 1.
192
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196 kxj k
For any frequency g, we define ANg ¼ fx ¼ fsk gkP1 2 D1 : sk 2 A; limk!1 kk ai ¼ gi ; 1 6 i 6 mg. Suppose mi(1 6 i 6 m) be a positive integer, Ui ¼ ðci1 ; ci2 ; . . . ; cimi Þð1 6 i 6 mÞ is a sequence of positive real vectors with P mi For x 2 ANg and E 2 UðJ ; fnk g; fUk gÞ, when k P 1, sk = ai, we let nk = mi, Uk :¼ j¼1 cij 6 1. Ui ¼ ðci1 ; ci2 ; . . . ; cimi Þ. Then we obtain a class of Moran sets associated with x 2 AN, denoted by E(x). From [5], we know that dimEðxÞ ¼ DimEðxÞ ¼ a; for a satisfies the following equation: m X i¼1
gi log
mi X
caij ¼ 0;
ð2:1Þ
j¼1
where dim and Dim denote the Hausdorff dimension and packing dimension, respectively. For the Moran set E satisfying the strong separation condition, we have the map u is one-to-one. That is, for each T 1 x 2 E there exists a unique r such that uðrÞ ¼ 1 n¼0 J rjn ¼ x. Then we denote u (E) = DE and say r is the address of the point x. Similarly, for E(x), there exists DE(x) such that u(DE(x)) = E(x). For J r 2 Fk is a basic element of order k of the Moran set E(x), where r = r1 rk 2 Dk, if sn = ai 2 A (1 6 i 6 m), then rn 2 {1, 2, . . . , mi} for 1 6 n 6 k. By x = s1 sk , we define r(ai) as following: for every n (1 6 n 6 k), only when sn = ai (1 6 i 6 m), we pick rn from r and form a new sequence, denote rðai Þ ¼ ri1 ri2 rikxjk kai where rij 2 f1; 2; . . . ; mi g ð1 6 j 6 kxjk kai Þ. If kxjk kai ¼ 0, we assume r (ai) = ;, then r(a1)r(a2) r(am) is rearranged of r r1 rk. For every ai 2 A, let P ai ¼ ðpi1 ; pi2 ; . . . ; pimi Þ (1 6 i 6 m) be probability vectors, i.e. pij > 0 and P= mi (1 6 i 6 m), denote P ¼ ðP a1 ; P a2 ; . . . ; P am Þ. Define prðai Þ ¼ pri1 prikxj k (1 6 i 6 m), where j¼1 p ij ¼ 1 k ai P m prij 2 fpi1 ; pi2 ; . . . ; pimi g (If r(ai) = ; assume that prðai Þ ¼ 1). It is easy to see that r2Dk Pi¼1 prðai Þ ¼ 1Qfor any k P 1. From the above discussion, similarly we denote crðai Þ ¼ cri1 crikxjk ka (1 6 i 6 m), then we get jJ r j ¼ mi¼1 crðai Þ . i For rðai Þ ¼ ri1 ri2 rikxjk kai and j ¼ 1; 2; . . . ; mi ; we define dj ðrðai Þ; kxjk kai Þ ¼ ]ft : rit ¼ j; 1 6 t 6 kxjk kai g; whenever there exists the limit dj ðrðai ÞÞ ¼ lim
k!1
dj ðrðai Þ; kxjk kai Þ ¼ pij ; kxjk kai
for 1 6 i 6 m; 1 6 j 6 mi . Then we define the Bescovitch set BP(x) to be the subset of Moran set E(x) given by BP ðxÞ ¼ fuðrÞ : dj ðrðai ÞÞ ¼ pij ; r 2 DEðxÞ ; for 1 6 i 6 m; 1 6 j 6 mi g: A remark from [7] says that BP(x) is a Borel set. For probability vectors P a1 ; P a2 ; . . . ; P am , we denote vP(x) the corresponding product measure on DE(x), so we attain a measure Q lP = vP(x)u1 on E(x). Then for any k-th basic element J r 2 Fk ; r ¼ r1 r2 rk 2 Dk , we have lP ðJ r Þ ¼ mi¼1 prðai Þ . By the strong law of large numbers, we see that lP(BP(x)) = 1 and since BP(x) E(x), thus lP(BP(x)) = lP(E(x)) = 1. For F ¼ fJ r : r 2 Dg; we define ( ) X [ s j s j j j Hd;F ðEÞ ¼ inf jJ r j : fJ r g F; jJ r j 6 d; E Jr ; j
HsF ðEÞ ¼ lim Hsd;F ðEÞ: d!0
Similar to the definition of Hausdorff dimension, we define dimF E ¼ supfs P 0 : HsF ðEÞ ¼ 1g ¼ inffs P 0 : HsF ðEÞ ¼ 0g: Now we would like to define lsP -measure as follows: ( ) X [ s s lP ðE; dÞ ¼ inf lP ðU i Þ : lP ðU i Þ 6 d; E Ui ; i
lsP ðEÞ ¼ lim lsP ðE; dÞ; d!0
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
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then the lP-dimension of E is given by dimlP E ¼ supfs P 0 : lsP ðEÞ ¼ 1g ¼ inffs P 0 : lsP ðEÞ ¼ 0g: We see that if lP(E) > 0, then dimlP E ¼ 1:
ð2:2Þ
Now we state the main results in this paper. Theorem 2.1. For x ¼ s1 s2 sk , denote Pm Pmi i¼1 gi j¼1 p ij log p ij P ; s¼P m mi i¼1 gi j¼1 p ij log cij
ð2:3Þ
then dimBP ðxÞ ¼ s, and s 6 a, where a is uniquely decided by (2.1). The equality is attained when P ai coincide with ðcai1 ; cai2 ; . . . ; caimi Þ. Furthermore, we show that there exists a subset B of Bescovitch set BP(x) such that B has full lP-measure but zero Hausdorff measure when P ai –ðcai1 ; cai2 ; . . . ; caimi Þ, which is described as follows: Theorem 2.2. There exists a Borel set B of BP(x) such that lP(B) = 1 and Hs ðBÞ ¼ 0 in incompatible case, that is P ai –ðcai1 ; cai2 ; . . . ; caimi Þði ¼ 1; 2; . . . ; mÞ.
3. Proofs of main theorems Now we set to prove the results displayed above. First let us look at some lemmas. Pmi Lemma 3.1. Suppose that P ai ¼ i1 ; pi2 ; . . . ; p imi Þ are probability vectors with pij > 0 and j¼1 pij ¼ 1 for 1 6 i 6 m. Let gi Pðp m be positive real numbers with i¼1 gi ¼ 1, then for any qi1 ; qi2 ; . . . ; qimi 2 R; ! mi mi m m X X X X gi pij ð log pij þ qij Þ 6 gi log eqij : ð3:1Þ j¼1
i¼1
i¼1
The equality is attained if and only if pij ¼ eqij
j¼1
P
mi qij j¼1 e
1
for i ¼ 1; 2; . . . ; m.
Proof. From [8], we know that if P ai ¼ ðpi1 ; pi2 ; . . . ; pimi Þ are probability vectors with pij > 0 and 1 6 i 6 m,1 6 j 6 mi, then for any qi1 ; qi2 ; . . . ; qimi 2 R, mi X
pij ð log pij þ qij Þ 6 log
j¼1
mi X
eqij :
j¼1
P i q 1 The equality is attained if and only if pij ¼ eqij ð mj¼1 e ij Þ for j ¼ 1; 2; . . . ; mi . As gi are positive real numbers, thus we have gi
mi X
pij ð log pij þ qij Þ 6 gi log
mi X
j¼1
eqij ;
j¼1
for i ¼ 1; 2; . . . ; m, then we have (3.1). Lemma 3.2. [4] If E fuðrÞ : lim
jrj!1
then dimE ¼ ddimlP E.
log lP ðJ r Þ ¼ dg; log jJ r j
h
Pmi
j¼1 p ij
¼ 1 for
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M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
Proof of Theorem 2.1. For any r ¼ r1 r2 rk 2 DEðxÞ , Qmi dj ðrðai Þ;kxjk kai Þ Q log mi¼1 pij j¼1 log prjk log lP ðJ rjk Þ lim ¼ lim ¼ lim Qm Qmi dj ðrðai Þ;kxjk kai Þ k!1 log jJ rj j k!1 log crj k!1 k k log i¼1 j¼1 cij Pm kxjk kai Pmi dj ðrðai Þ;kxjk kai Þ Pm Pmi log p ij i¼1 j¼1 k kxjk kai i¼1 gi j¼1 p ij log p ij ¼P P ¼ s: ¼ lim m mi Pmi dj ðrðai Þ;kxjk kai Þ k!1 Pm kxjk ka i gi pij log cij i¼1 j¼1 log c ij i¼1 j¼1 k kxj k k ai
Since lP(BP(x)) = 1, by Lemma 3.2 and (2.2) we have dim BP(x) = s. Put qij ¼ log caij in Lemma 3.1, from (2.1) and (2.3), we get s 6 a. Furthermore, if we put !1 mi X log caij log caij pij ¼ e e ;
ð3:2Þ
j¼1
from (2.1) we know pij ¼ caij is the solution of (3.2), then we have s = a.
h
Lemma 3.3. [4] [law of iterated logarithm Hartman-Winter] Suppose that {Xn:n P 1} is a sequence of independent, identically distributed random variables (i.i.d.r.v.) satisfying E(Xn) = 0 and EðX 2n Þ ¼ a2 2 ð0; 1Þ, then Pn i¼1 X i ¼ 1; p a:e:; lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n!1 2na log log na2 Pn i¼1 X i ¼ 1; p a:e: lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n!1 2na2 log log na2 Proof of Theorem 2.2. Put p ¼ lP u; X ij ¼ log pij s log cij ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; mi ; then {Xij} is a sequence of i.i.d.r.v. with respect to p. We see that EðX ij Þ ¼
m X
gi
mi X
pij ðlog pij s log cij Þ ¼ 0:
j¼1
i¼1
Suppose that P ai –ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m; from Theorem 2.1, we get 0 < EðX 2ij Þ ¼
m X
gi
i¼1
Set
( 0
B ¼
mi X
pij ðlog pij s log cij Þ2 ¼: a2 < 1:
j¼1
) Pn l¼1 X il jl uðrÞ : lim pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1; for il 2 f1; 2; . . . ; mg; jl 2 f1; 2; . . . ; mil g ; n!1 2na2 log log na2
then lP ðB0 Þ ¼ 1 by Lemma 3.3. Since lP(BPP (x)) = 1, let B = B 0 ˙ BP(x), we get lP(B) = 1. With the above arguments, we see that 1 k¼1 X ilk jlk ¼ þ1 a:e., that is, Qn k¼1 p il jl log Qn s k k ! þ1; as n ! 1; where ilk 2 f1; 2; . . . ; mg; jlk 2 f1; 2; . . . ; milk g; k¼1 cil jl k
k
from which we also have lP ðJ r Þ ! 1; jJ r js
as jrj ! 1:
S k k k js > k, for r 2 Dn}, Jk ¼ 1 Let S Jkn ¼ fJ r : lP ðJ r Þ=jJ rT n¼1 Jn . For any d > 0, k > 0, choose fJ j g J such that k k k k k B j J j ; jJ j j 6 d, and J i J j ¼ ;ði–jÞ. By the definition of J , we have lP ðJ kj Þ jJ kj js
> k;
8j:
M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
195
Thus Hsd ðBÞ 6
X
jJ kj js 6
1X 1 lP ðJ kj Þ 6 : k k
Letting k ! 1, we have Hsd ðBÞ ¼ 0; from which it follows that Hs ðBÞ ¼ 0.
h
Lemma 3.4. [9] Suppose that l is a finite Borel measure such that 0 < lim
jrj!1
lðJ r Þ < 1; jJ r ja
0 < lim jrj!1
lðJ r Þ < 1; jJ r ja
for any r 2 D;
then l Ha Pa on a Moran set E. Corollary 3.5. lP Ha Pa on E(x) if P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m, where P ¼ ðP a1 ; P a2 ; . . . ; P am Þ. Proof. For r = r1 rk 2 Dk and P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; i ¼ 1; 2; . . . ; m; we have Qm Qmi dj ðrðai Þ;kxjk kai Þ lP ðJ r Þ i¼1 j¼1 p ij ¼ 1; ¼Q Q m mi jJ r ja ðca Þdj ðrðai Þ;kxjk kai Þ i¼1
j¼1
ij
so by Lemma 3.4, we get lP Ha Pa on E(x).
h
Corollary 3.6. Ha ðEðxÞ n BP ðxÞÞ ¼ Pa ðEðxÞ n BP ðxÞÞ ¼ 0; Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ and Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ in case of P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þði ¼ 1; 2; . . . ; mÞ. Proof. By the definition of equivalence, lP Ha on E () for any F E, lP(E) = 0 if and only if Ha ðF Þ ¼ 0. It is clear that BP(x) is the subset of E(x) with lP(E(x)) = lP(BP(x)) = 1, then we get immediately lP(E(x)nBP(x)) = 0, which further implies Ha ðEðxÞ n BP ðxÞÞ ¼ 0 by Corollary 3.5. Furthermore, Ha ðBP ðxÞÞ 6 Ha ðEðxÞÞ 6 Ha ðBP ðxÞÞ þ Ha ðEðxÞ n BP ðxÞÞ ¼ Ha ðBP ðxÞÞ; we have Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ. Similar arguments will prove that Pa ðEðxÞ n BP ðxÞÞ ¼ 0 and Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ. h For the simplicity of proof of the following results, we would like to make a hypothesis here. We say that the Moran set E(x) satisfies the gap condition if there exists a constant d > 0 such that distðJ r1 r2 rn ; J r1 r2 r0n Þ P d maxfjJ r1 r2 rn j; jJ r1 r2 r0n jg for any n 2 N and rn –r0n . Lemma 3.7. [9] Suppose that E is a Moran set satisfying the gap condition, then for any u(r) 2 E and finite Borel measure l, supp(l) E, the following inequalities hold: ðcdÞa lim jrj!1
ðcdÞa lim
jrj!1
jJ r ja ra jJ r ja 6 lim 6 ca lim ; lðJ r Þ r!0 lðBðuðrÞ; rÞÞ jrj!1 lðJ r Þ jJ r ja ra jJ r ja 6 lim 6 ca lim ; jrj!1 lðJ r Þ lðJ r Þ r!0 lðBðuðrÞ; rÞÞ
where c = min{cij}, for i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; mi . Lemma 3.8. [10] For finite Borel measure l, there exists four positive constants b1 ; b2 ; b3 ; b4 such that ra ra b1 lðEÞ inf lim 6 Ha ðEÞ 6 b2 lðRd Þ sup lim ; x2E x2E r!0 lðBðx; rÞÞ r!0 lðBðx; rÞÞ ra ra b3 lðEÞ inf lim 6 Pa ðEÞ 6 b4 lðRd Þ sup lim : x2E r!0 lðBðx; rÞÞ r!0 lðBðx; rÞÞ x2E Corollary 3.9. If E(x) is the Moran set defined above satisfying gap condition and BP(x) is the Bescovitch set defined in Section 2, then 0 < Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ < 1 and 0 < Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ < 1; when P ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ.
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M. Dai, Y. Jiang / Chaos, Solitons and Fractals 40 (2009) 190–196
Proof. From Lemma 3.8, we see that the density of balls is comparable with that of Jr, so we have ( ) ( ) jJ r ja jJ r ja 6 Ha ðEðxÞÞ 6 b02 lðRd Þ sup lim lim b01 lðEðxÞÞ inf x2EðxÞ jrj!1 lðJ r Þ x2EðxÞ jrj!1 lðJ r Þ
ð3:3Þ
together with Lemmas 3.7 and 3.8. Since jJ r ja ¼ 1; in case ofP ai ¼ ðcai1 ; cai2 ; . . . ; caimi Þ; lðJ r Þ we get 0 < b01 6 Ha ðEðxÞÞ 6 b02 by substituting l with lP in (3.3), which implies that 0 < Ha ðEðxÞÞ < 1. Since we have proved in Corollary 3.6 that Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ; we have 0 < Ha ðEðxÞÞ ¼ Ha ðBP ðxÞÞ < 1. Analogously, we can prove 0 < Pa ðEðxÞÞ ¼ Pa ðBP ðxÞÞ < 1. h
4. Conclusion The physical relevance of the present paper is due to El Naschie’s E-infinity theory. We first recall the definition of the Moran sets E(x) associated with x and some known results about fractal dimension and measure in physics and mathematical fields. Secondly, we give the Hausdorff dimension of the subset BP(x) of E(x) in Rd satisfying the strong separation condition. Finally, we attain some results about lP ; Ha and Pa -measures in compatible case and incompatible case.
Acknowledgements This research is supported the National Science Foundation of China (10671180) and Jiangsu University 05JDG041.
References [1] El Naschie MS. The concepts of E-infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495–511. [2] El Naschie MS. A guide to the mathematics of E-infinity Cantorian space-time theory. Chaos, Solitons & Fractals 2005;25:955–64. [3] El Naschie MS. Modular qroups in Cantorian E-infinity high energy physics. Chaos, Solitons & Fractals 2003;16:353–66. [4] Zhen Fangxiong. Dimensions of subsets of Cantor-type sets. Int J Math Math Sci 2006:1–8. Artical ID 26359. [5] Wu Min. The multifractal spectrum of a class of Moran measures. Sci China Ser A 2005;35(2):147–61. [6] Meifeng Dai, Dehua Liu. The local dimension of Moran measures satisfying the strong separation condition. Chaos, Solitons & Fractals 2008;38(4):1025–30. [7] Moran M, Rey J-M. Singularity of self-similar measures with respect to Hausdorff measures. Trans Am Math Soc 1998;350(6):2297–310. [8] Falconer K. Techniques in fractal geometry. Chichester: John Wiley; 1997. [9] Meifeng Dai. The equivalence of measures on Moran set in general metric space. Chaos, Solitons & Fractals 2006;29(1):55–64. [10] Tayor SJ, Tricot C. Packing measure and its evaluation for a Brownian path. Trans Am Math Soc 1985;288(2):679–99.