THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON

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1 Anal. Theory Appl. Vol. 28, No. (202), THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON Chaoyi Zeng, Dehui Yuan (Hanhan Normal Univerity, China) Shaoyuan Xu (Gannan Normal Univerity, China) Received Jan. 9, 20 Abtract. In thi paper, we addre the problem of exact computation of the Haudorff meaure of a cla of Sierpinki carpet E the elf-imilar et generating in a unit regular pentagon on the plane. Under ome condition, we how the natural covering i the bet one, the Haudorff meaure of thoe et are euqal to E, where = dim H E. Key word: Sierpinki carpet, Haudorff meaure, upper convex denity AMS (200) ubject claification: 28A78, 28A80 Introduction The Haudorff meaure dimenion are the mot important concept in fractal geometry, their computation i very difficult. Recently, in order to tudy deeply the Haudorff meaure, the reference [] gave the notion bet covering" natural covering", poed eight open problem ix conjecture on Haudorff meaure. Uing the notion upper convex denity of a cla of elf-imilar et, the reference [2] tudied a cla of elf-imilar et-generating in a unit quare on the plane, proved that the natural covering i the bet one the Haudorff meaure of thoe et are euqal to 2. In thi paper, we addre the problem of the exact computation Partially upported by National Natural Science Foundation of China (No ). Correponding author.

2 28 C. Y. Zeng et al : Haudorff Meaure of Sierpinki Carpet Baing on Regular Pentagon of the Haudorff meaure of a cla elf imilar et-generating in a unit regular pentagon on the plane. For convenience, we firt preent ome notion that will be ued in the ret part of the paper. Definition. Suppoe E R 2, R, 0 δ > 0, the Haudorff meaure of the et E i defined a H (E) = lim inf U i : U i δ,e U i δ 0 i= where U i i= i arbitrary covering of the et E; the Haudorff dimenion of E (denoted by dim H E) i defined a dim H E = up : H (E) = = inf : H (E) = 0. Definition 2. Let δ > 0, 0, E R 2, x E. Moreover, for a convex et U x containing x, the upper convex denity of E at x i defined a D C(E,x) = lim up δ 0 0< U x <δ H (E U x ) U x The propertie of the upper convex denity are dicued in the reference []. Definition 3. (See Fig. ) Let E 0 be an unit regular pentagon A A 2 A 3 A 4 A on the plane R 2, E be the attractor generated by the iterated function ytem (IFS) f i i =,2,3,4,, where. f i (x) = λ i x+b i,0 < λ i <,i =,2,3,4,,x = (x,x 2 ) E 0 b = (( λ )in 8,0) b 2 = (( λ 2 )(in 8 + ),0) b 3 = (( λ 3 )(2in 8 + ),( λ 3 )co8 ) b 4 = (( λ 4 )co 8,( λ 4 )(in 8 + co8 )) b = (0,( λ )co 8 ) Then the elf-imilar et E i called a Sierpinki carpet generating in a unit regular pentagon, where =dim H E atifie i= λ i =. 2 Two Lemma In thi ection, we preent two lemma which will be ued in the proof of the main reult of thi paper.

3 Anal. Theory Appl., Vol. 28, No. (202) 29 Fig. From the definition 3, it i eay to ee that f i (E 0 ) E 0 i= f i (E) = E. i= Let µ be the unique probability meaure atifying the elf-imilar relation µ = i= λ i µ f i, then E i the upport of µ µ i a ma ditribution on E. For i =,2,3,4,, let E i F i be parallel to the oppoite ide of vertex A i in an unit regular pentagon interect f i (E), let d i = dit(a i,e i F i ) be the ditance between point A i line E i F i. Denote if t i = d i co8, 0 < λ i < < d i < ( + )λ i, then the line E i F i doen t interect f j (E) for i, j,2,3,4, i j. Aume that µ(t i ) i the meaure of the triangle A i E i F i. Moreover, we ue the notation: d(µ,t i ) = µ(t i) ti,d (i) = inf 0<t i 2in4 d(µ,t i),i =,2,3,4, M = (i, j) A i A j i the diagonal of E 0,i, j =,2,3,4,

4 30 C. Y. Zeng et al : Haudorff Meaure of Sierpinki Carpet Baing on Regular Pentagon That i M 2 = (i, j) A i A j i the ide of E 0,i, j =,2,3,4,. M = (,3),(3,),(,4),(4,),(2,4),(4,2),(2,),(,2),(3,),(,3) M 2 = (,2),(2,),(2,3),(3,2),(3,4),(4,3),(4,),(,4),(,),(,). Lemma 4. Let 0 < λ i < 7 2, 0 < <, i =,2,3,4,. 9 Aume that K i a nonnegative integer. Then d(µ,t i ) attain it infimum d (i) only at the following value of t i : () t i = λi K λ p 2in8,(i, p) M, or (2) t i = λi K 2in i q,p q 8 ( λ q ),(i,q) M 2. Proof. Since E i elf-imilar, then we need only prove that the reult i true when 2λ in4 < t 2in 4 for i =. Denote r = λ in4,r 2 = 2( λ 2 )in 8,2( λ )in 8, r 3 = 2in 4 2( λ 2 )in 8,2( λ )in8, r 3 = 2in 4 max2λ 3 in4,2λ 4 in4. Cae. d(µ,t ) attain it infimum d () at the interval (r 2,r 3 ]. In thi cae, the line E F interect f 2 (E) or f (E). Therefore d(µ,t ) = µ(t ) t = λ + µ(t 2( λ 2 )in8 )+ µ(t 2( λ )in8 ) (r 2 +t r 2 ) λ + maxµ(t 2( λ 2 )in8 ), µ(t 2( λ )in 8 ) (r 2 +t r 2 ) λ + maxµ(t ( λ 2 )2in 8 ), µ(t ( λ )2in 8 ) r2 +(t r 2 ), λ r 2 τ, (t r 2 ) where τ = maxµ(t ( λ 2 )2in 8 ), µ(t ( λ )2in 8 ).

5 Anal. Theory Appl., Vol. 28, No. (202) 3 Thi contradict the aumption that d(µ,t ) attain it infimum d () at the interval (r 2,r 3 ]. Cae 2. d(µ,t ) attain it infimum d () at (r 4,2in 4 ]. In thi cae, the line E F interect f 3 (E) or f 4 (E), then d(µ,t ) = µ(t ) t = λ + λ 2 + λ 3 + maxµ(t 2λ 3 in4 ), µ(t 2λ 4 in 4 ) (r 4 +t r 4 ) λ + λ 2 + λ 3 + maxµ(t 2λ 3 in8 ), µ(t 2λ 4 in 8 ) (r 4 +t r 4 ) λ + λ 2 + λ 3 + maxµ(t 2λ 3 in8 ), µ(t 2λ 4 in 8 ) r4 +(t r 4 ), λ r 4, λ 2 r 4, λ 3 r 4 τ, (t r 4 ) where τ = maxµ(t 2λ 3 in4 ), µ(t 2λ 4 in 8 ). Cae 3. d(µ,t ) attain it infimum d () at (2λ in4,r 2 ]. In thi cae, we have d(µ,t ) = λ t λ r2. Thi mean that t = 2( λ 2 )in 8 or t = 2( λ )in8, K = 0. Cae 4. d(µ,t ) attain it infimum d () at the interval(r 3,r 4 ]. We have d(µ,t ) = λ + λ 2 + λ 3 t λ + λ 2 + λ 3 r4. Therefore, d(µ,t ) = λ + λ 2 + λ 3 r4. Thi mean that t = λ 3 2in 8 or t = λ 4, K = 0. 2in 8 Similarly, if K > 0 2λ k+ in4 < t < λ k in 4, we can prove that d(µ,t ) attain it infimum d () only at t = λ k λ 3 2in8, λ k λ 4 2in 8, 2λ k( λ 2)in 8 2λ k( λ )in 8. Lemma [3]. Let 0 < α <, p p 0, a a 0, y λx α. If then p y (a x) α < p a α. ( a0 λ 0 < x p 0 ) α,

6 32 C. Y. Zeng et al : Haudorff Meaure of Sierpinki Carpet Baing on Regular Pentagon 3 The Main Reult Theorem 6. Let E be a elf-imilar et defined by Definition 3, 0 < λ i < 7 2 (i = 9,2,3,4,), =dim H (E) 0 < <. Moreover, aume the following two condition λi () + λ j ( ) ( λ i λ j ) 2in 4, for (i, j) M 2 ; (2) 2(λ i + λ j )in 4 2d (i) in4,2d ( j) in 4, for (i, j) M are atified. Then for any x E, if the cloed convex et U x containing x i the cloure E 0 of E 0, then Proof. H D C(E,x) (E U x ) = up 0< U x U x = H (E E 0 ) (2in 4 ) =. Let V R 2, V E V E 0 (if not, replacing V by V E 0 ). Denote d(v) = µ(v) V,d max = up d(v),v E 0, where µ i the ma ditribution of E defined a above. We now prove that if V = E 0 then Cae. 0< v d max = µ(v) V = µ( E 0 ) E 0. V f i (E) for all i (See Fig. 2). In thi cae, we can elect fine tangent Fig. 2 line of V, denoted by E i F i, uch that E i F i i parallel to the oppoite ide of the vertex A i for i =,2,3,4,. Moreover, denote t i = d i co8,

7 Anal. Theory Appl., Vol. 28, No. (202) 33 where d i i the ditance between the vertex A i E i F i, then Therefore V 2in 4 t i t j, for (i, j) M ; µ(v) i= (λi µ(t i)) = i= µ(t i ) µ(t i ) µ(v) i= V (2in 4 t i t j ) (µ(t i)+ µ(t j )) (2in 4 t i t j ). Replacing α by, a a 0 by 2in 4, p p 0 by, repectively, in Lemma, employing Lemma 4, we have Notice the condition (2), we have µ(t i )+ µ(t j ) (t i +t j ) µ(t i)+ µ(t j ) ti +t j 0 < λ i + λ j d (i) j),d(. λ 2d (i) in4,2d ( j) in4 2in4 = [(2in 4 ) ] d (i) j),d( µ(t i ) ti, µ(t j) t j [(2in 4 )] d (i) j),d( = ( a0 λ p 0 ). Thi mean that the condition of Lemma are atified. Denote w = λ i + λ j, then Therefore, That i Cae 2. y = µ(t i )+ µ(t j ) λ(t i +t j) λ(t i +t j ) = λw. µ(v) V p y (a w) α (2in 4 ). d max = µ(v) V = µ( E 0 ) E 0. There exit only four of five et f i (E) uch that V f i (E). For convenience, let f (E), f 2 (E), f 3 (E) f 4 (E) be thee four et. Then V 2in4 t t 3, V 2in4 t t 4, V 2in4 t 2 t 4,;

8 34 C. Y. Zeng et al : Haudorff Meaure of Sierpinki Carpet Baing on Regular Pentagon µ(v) µ(t ) µ(t 3 ), µ(v) µ(t ) µ(t 4 ), µ(v) µ(t 2 ) µ(t 4 ). So, V 2in 4 t i t j µ(v) µ(t i ) µ(t j ), for (i, j) M 3, where M 3 M \(2,),(,2),(3,),(,3). Therefore, Employing Lemma, we get That mean that the reult i till true. Cae 3. µ(v) V (µ(t i)+ µ(t j )) (2in 4 t i t j ),(i, j) M 3. µ(v) V (2in 4 ). There exit only three of five et f i (E) uch that V f i (E). In thi cae, there exit (i 0, j 0 ) M uch that Similar to the proof of Cae 2, we deduce that Thi combining with Lemma follow that Cae 4. V f i0 (E) V f j0 (E). µ(v) V (µ(t i 0 )+ µ(t j0 )) (2in 4 t i0 t j0 ). µ(v) V (2in 4 ). There exit only two of five et f i (E) uch that V f i (E). Therefore, we can aume that there exit (i, j) uch that V f i (E) V f j (E). If (i, j) M, then we get the required reult by Cae 3. If (i, j) M 2, aume (i, j) = (,2), then µ(v) λ + λ 2 µ(t ) µ(t 2 ) λ + λ 2

9 Anal. Theory Appl., Vol. 28, No. (202) 3 From the condition (), we have Cae. V d co 4 d 2 co4 λ λ 2. µ(v) λ V + λ 2 ( λ λ 2 ) (2in 4 ). There exit only one of five et f i (E) uch that V f i (E), for example, V f (E). Notice the function of amplification of f, Denote V = f (V f (E)), we can aume µ(v f (E)) V µ( f ((V f (E))) f (E) = λ V. f (E) V f i (E), V f j (E) for ome (i, j) the denity i invariant, if not, then take f (V ) a V. Similar to the proof of above cae, we get the required reult. Therefore, we finih the proof. d max = µ(v) V = µ( E 0 ) E 0, By the definition of probability meaure, we know that there exit a contant C uch that µ = CH. So attain the upremum at the et E 0. H D C (E U x ) (E,x) = up 0< u x U x Combining Theorem 2.3 in reference [3] Propoition 2 in reference [4], we get H D C (E U x ) (E,x) = up 0< u x U x = H (E E 0 ) (2in 4 ) =. Employing Theorem 6, we have the following corollary. Corollary 7. If the aumption in Theorem 6 are atified, then E 0 i the bet covering" of E. That i H (E) = E 0 = (2in 4 ), where =dim H (E) atifie i= λ i =.

10 36 C. Y. Zeng et al : Haudorff Meaure of Sierpinki Carpet Baing on Regular Pentagon 4 Example Example 8. Let λ = λ 2 = λ 3 = λ 4 = λ = 2, then =. Moreover, we have 2 λi () + λ j ( λ i λ j ) = 2 ( ) in 4 = , for (i, j) M 2 ; (2) for(i, j) M,2in 4 (λ i + λ j ) 0.294, ( d (i) 2in 4 = 24 ) 0.296, 2in 4 d (i),2in 4 d ( j) ( (2in 4 ) + = 24 Hence, the aumption of Theorem 6 are atified. Therefore, 3 ( 62) =. Denote x = ( 2 H (E) = (2in 4 ).272. ) = Example 9. Let λ = λ 3 = λ = 62, λ 2 = λ 4 = 2. Since ) i=, then 3x 2 + 2x = = 2 log 3. Then, ( ) ( () 2in4 = 2in 4 or or λ i + λ j ( λ i λ j ) = λ i + λ j ( λ i λ j ) = λ i + λ j ( λ i λ j ) = ) 2 log 3 ( = , for (i, j) M 2, ) ( 2) + 2 ( , 2 ) ( ) ( 2 + ) 62 ( ( ) ) + ( 62 ( ) , ) (2) for (i, j) M,2in 4 (λ i + λ j ) or 0.002, ( d () 2in 4 = d(3) = d() = 624 ( d (2) 2in4 = d(4) = 24 ) 2 log 3 ) 2 log 3 2d (i) in 4,2d ( j) in , 0.309, = ((2in 4 ) 0.309) = λ i =, then 2 ( 2) +

11 Anal. Theory Appl., Vol. 28, No. (202) 37 So the aumption of Theorem 6 are atified. Therefore, H (E) = (2in 4 ) = (2in 4 ) log Reference [] Zhou, Z. L. Feng, L., Twelve Open Problem on the Exact Value of the Haudorff Meaure on Topological Entropy: a Brief Survey of Recent Reult, Nonlinearity, 7(2004), [2] Zhu, Z. W., Zhou, Z. L. Jia, B. G., The Haudorff Meaure Upper Convex Denity of a Cla of Self-imilar Set on the Plane, Acta Mathematica Sinica, Chinee Serie, 48:3(200), [3] Ayer, E. Strichartz, R. S., Exact Haudorff Meaure Interval of Maximum Denity for Cantor Set, Tran. Amer. Math. Soc., 3:9 (999), [4] Zhou, Z. L., Haudorff Meaure of Self-imilar Set-koch cure, Science of China (Ser A), 28:2(998), [] Falconer, K. J., The Geometry of Fractal et, Cambridge Univerity pre, Cambridge (98). Department of Mathematic Information Technology Hanhan Normal Univerity Chaozhou, 204 P. R. China C. Y. Zeng zcy@htc.ecu.cn D. H. Yuan ydhlxl@htc.edu.cn S. Y. Xu School of Mathematic Computer Gannan Normal Univerity Ganzhou, xuhaoyuan@26.com