DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2018 Fractal Sets: Dynamical, Dimensional and Topological Properties NANCY WANG KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2018 Fraktalmängder: Dynamiska, Dimensionella och Topologiska Egenskaper NANCY WANG KTH SKOLAN FÖR TEKNIKVETENSKAP
N {0, 1, 2,...} R, R + Q Z, Z + ΠC M H dim B dim B B H R R Q
F : D æ D D G(x) =2x 2 1 G(x) =2x 2 1 G(G(x)) = (2x 2 1) 2 1 G(G(G(x))) = ((2x 2 1) 2 1) 2 1 n F n (x) F 2 (x) F (F (x)) F 3 (x) F (F (F (x))) F n (x) n n x 0 œ R x 1 = F (x 0 ) x 2 = F 2 (x 0 )... x n = F n (x 0 ),... x 0 {x 0,x 1,...,x n,...} x 0 0 F (x) =x +1 {0, 1, 2,...} x 0 F (x 0 )=x 0 x 0 {x 0,x 0,...} x 0 F n (x 0 )=x 0 n x 0 = F (x 0 )=F (F (x 0 )) = = F n (x 0 ) F (x) =x 1 F (x) =x 2 { 1, 1, 1...} F n (x 0 )=x 0 n k n Æ k
G(x) = 1 +2 x 1 x +2=x x 2 2x 1=0 (x 1) 2 =2 x =1± Ô 2. F 4 (x) F (x) =2x 2 1 F 4 (x) =x G(x) x 0.41 x 2.41 y = x y = x G(x)
x 0 F (x) F : R æ R Y _] F Õ (x 0 ) < 1, x 0. F Õ (x 0 ) > 1, x 0. _[ F Õ (x 0 ) =1, x 0. F : R æ R x 0 F I x œ I F n (x) œ I n F n (x) æ x 0 n æœ x 0 F I x I x = x 0 n>0 F n (x) /œ I x 0 F F : R æ R F Õ (x 0 ) < 1 a F Õ (x 0 ) < a<1 x 0 I a =[x 0, x 0 + ] >0 F Õ (x) <a x œ I a F Õ (x) = lim xæx0 F (x) F (x 0 ) x x 0 <a ( x œ I a x = x 0 ). F (x) F (x 0 ) = F (x) x 0 <a x x 0. 0 <a<1 F (x) x 0 x x œ I a F (x) œ I a a n <a n>1 F 2 F 3... F n (x) x 0 <a n x x 0. n æœ a n æ 0 F n (x) æ x 0 x 0 F F Õ (x 0 ) > 1 b 1 < b < F Õ (x 0 ) x 0 I b =[x 0, x 0 + ] >0 F Õ (x) > b x œ I b F Õ (x) = lim xæx0 F (x) F (x 0 ) x x 0 F (x) x 0 >b x x 0. >b ( x œ I x = x 0 ). b>1 F (x) x 0 n æœ b n æœ n>0 F n (x) /œ I b
a n n F Õ (x 0 ) <a<1 F n (x) x 0 n æœ b x 0 x 0 n n F n (x 0 )=x 0 x 0 n {x 0,F(x 0 ),F 2 (x 0 ),...,F n 1 (x 0 ),x 0,F(x 0 ),F 2 (x 0 ),...} 0 G(x) =1 x 2 {0, 1, 0, 1,...} 2 0 1 2 x 0 n F mn (x 0 )=x 0 m n x 0 n F (x 0 ) = x 0 m>0 F n+i (x 0 )=F i (x 0 ) i Ø m {x 0,x 1,...,x n 1 } n F (x) F : R æ R (F n ) Õ (x 0 )=F Õ (x 0 )F Õ (x 1 ),..., F Õ (x n 1 )F Õ (x n 1 ). {x 0,x 1,...,x n 1 } n F : R æ R F k (x) k =
0, 1,...,n 1 (F n ) Õ (x 0 )= d dx 0 [F (F n 1 (x 0 ))] = F Õ (F n 1 (x 0 )) d dx 0 F n 1 (x 0 ) = F Õ (F n 1 (x 0 )) F Õ (F n 2 (x 0 )) d dx 0 F n 2 (x 0 ) = F Õ (F n 1 (x 0 )) F Õ (F n 2 (x 0 )) F Õ (F (x 0 )) F Õ (x 0 ) = F Õ (x n 1 ) F Õ (x n 2 ) F Õ (x 1 ) F Õ (x 0 ) 3 8 F F x F 0 I >0 F I 0 ±?? 0 I = 0 I 0?? 0 û
F = 0 I >0 F p I [ 0, 0 + ] I p 0? R 0 q 1 q 2 I F (q )= 1 q 2 0?? 0 + p æ 0 q i æ p 0 i = {1, 2}
x 0 2x 0 4x 0 f(x) =2x f(x 0 )=2x 0 f 2 (x 0 )=2 2 x 0 f n (x 0 )=2 n x 0 n f(x) n (1 x) f(x) g(x) =2x(1 x) x 0 (1 x 0 ) g(x 0 ) x 0 x 0 (1 x 0 ) g(x 0 ) x 0 x 0 g(x) x =0.1 x =0.8 0.5 0.5 0.5 0.5 0.5 0.5
g(x) =2x(1 x) x 0 =0.1 x 0 =0.8 x =0.5 x =0.5 g µ (x) =µx(1 x) µ g µ (x) µ>2 µ<2 µ =1.5 µ =3
g(x) =1.5x(1 x) x 0 =0.1 x 0 =0.8 x 0.3333 x 0.3333 g(x) =3x(1 x) x 0 =0.1 x 0 =0.8 x 0.6337 x 0.6954 µ =3 µ =1.5 µ =2
µ =3 10 7 µ =3 x 0 =0.1 x 0 =0.8 10 7 g(x) =3x(1 x) 10 7 µ =3 g µ (x) µ 3 µ =2.5 µ =3.3 g(x) =2.5x(1 x) x 0 =0.1 x 0 =0.8 x 0.600 x 0.600
g(x) =3.3x(1 x) x 0 =0.1 x 0 =0.8 {0.4794, 0.8236} µ =1.5 µ =2.5 µ =3.3 µ œ{2.5 Æ µ Æ 3.3} µ µ µ>3.3 µ =4 µ =5 g(x) =4x(1 x) x 0 =0.1 x 0 =0.8 [0, 1]
g(x) =5x(1 x) x 0 =0.1 x 0 =0.8 x 0 =0.1 Œ x 0 =0.8 µ =4 0 Æ g µ (x) Æ 1 µ =5 Œ µ>3.3 µ µ g µ (x) g µ (x) =µx(1 x) µ µ g µ (x) µ g µ : D æ D D =[0, 1] g µ g Õ µ (x) =µ(1 2x) g Õ µ (x) =0 x = 1 g 2 µ( 1)= µ 2 4 g µ : D æ D 0 Æ µ Æ 4 D µ>4 x =0 g µ (x) µ
g µ (x) x =0 µ = 0 Y ] p =0 [ p + = µ 1. µ p + µ µ =1.5 µ =2.5 x =1 x = 1 g µ µ g µ (1) = 0 g µ ( 1 µ 1 )= µ µ Y ] g Õ µ (p ) = µ [ g Õ µ (p + ) = 2 µ. F Õ (x) µ µ =1 µ>3 µ p p + µ p = p + µ µ p p + µ>0 g µ 2 (x) x =0 g µ 2 (x) =µ 2 x(1 x)(1 µx+µx 2 ) x 0 f f n (x) =x n p p + g 2 µ(x) µ = 0 µ 2 x(1 x)(1 µx + µx 2 ) x (x 0)(x µ 1 µ ) = µ[ µ 2 x 2 + µ(µ +1)x (µ +1)].
µ[ µ 2 x 2 + µ(µ +1)x (µ +1)]=0. g 2 µ(x) Y p =0 _] p + = µ 1 µ apple q = µ+1 (µ+1)(µ 3) apple 2µ _[ q + = µ+1+ (µ+1)(µ 3). 2µ (µ +1)(µ 3) µ Ø 3 µ Æ 1 (µ +1)(µ 3) = 0 µ = 1 µ =3 0 <µ µ = 1 µ>3 µ =3 µ+1 2µ µ =3 F 2 (x 0 ) F Õ (x 1 )F Õ (x 0 ) {x 0,x 1 } (g µ 2 ) Õ (q )=g µ Õ (q + )g µ Õ (q )=(g µ 2 ) Õ (q + ). Y _] (g 2 µ ) Õ (p ) = µ 2 (g 2 µ ) Õ (p + ) = (2 µ) 2 _[ (g 2 µ ) Õ (q ) = (g 2 µ ) Õ (q + ) = 1 (µ +1)(µ 3). (g µ 2 ) Õ (x) µ µ Ø 3 µ
(g µ 2 ) Õ (x) µ µ 1 (µ +1)(µ 3) =1 µ =1± Ô 6 µ =1 Ô 6 µ œ (3, 4) µ =1+ Ô 6 µ>1+ Ô 6 µ p p + q ± µ p = p + =0 µ µ 1+ Ô 6 1+ Ô 6 1+ Ô 6 µ g 2 µ(x) µ =1+2 Ô 2 g µ 3 (x) µ =1+ Ô 6 g µ 4 (x) µ
g µ (x) =µx(1 x) µ g µ 0 <µ<3 µ g µ 3 <µæ 4 g µ µ>4 µ =5 Œ µ>4 D =[0, 1] g µ µ>4 g µ (x) =µx(1 x) g µ ( 1)= µ µ>4 g 2 4 µ( 1) x = 1 2 2 D D x = 1 2 S 1 = {x œ D g µ (x) > 1} S 1
g µ (x) =1 x = 1 2 ± Ò 1 4 1 µ S 1 S 1 = A Û 1 1 2 4 1 µ, 1 Û 1 2 + 4 1 B. µ µ>4 S 1 Œ g µ : D æ D S 1 D 1 1 = C 0, 1 2 Û 1 4 1 µ D C Û 1 1 2 + 4 1 D µ, 1. 1 g µ g µ 1 D gµ 2 S 2 = {x œ D gµ(x) 2 > 1} S 2 S 1 S 2 1 2 = 1 S 2 2 2 2 =4 S n = {x œ D gµ(x) n > 1} n D n n = {x œ D gµ(x) n œ D n.} g µ Œ n=1 n
g µ D =[0, 1] 1 1, 2 2 3 3 D 5 C 1 = 0, 1 6 5 6 2 fi 3 3, 1. 1 1, 2 2 9 9 1 7, 2 8 9 9 5 C 2 = 0, 1 6 5 2 fi 9 9, 3 5 6 fi 96 9, 7 5 6 8 fi 96 9, 1. Œ C C n.
a k = 1 a < 1. 1 a k=0 0.a 1 a 2 a 3... x a i x = a 3 i i œ{0, 1, 2}. 0.012012012... = 0 3 + 1 3 + 2 2 3 + 0 3 3 + 1 4 3 + 2 5 3 + 0 6 3 + 1 7 3 + 2 8 3 +... 9 = 1 5 1+ 1 3 2 3 + 1 6 3 3 +... + 2 5 1+ 1 6 3 3 3 + 1 6 3 3 +... 6 = 1 C 1 3 2 (3 3 ) + 1 0 (3 3 ) + 1 D 1 (3 3 ) +... 2 = 1 ÿ Œ 1 3 2 i=0 (3 3 ) + 2 i 3 3 i=0 3 1 = 9 274 + 2 1 1 1 3 3 = 5 26. 1 (3 3 ) i + 2 C 1 3 3 (3 3 ) + 1 0 (3 3 ) + 1 D 1 (3 3 ) +... 2 0.21000... 0.20222... 0.21000 = 2 3 + 1 3 2 = 7 9. 0.20222 = 2 3 + Œ ÿ i=3 2 3 i = 2 3 + 2 27 i=0 1 3 i = 7 9. x œ D =[0, 1] 0.a 1 a 2 a 3... a i... x D D
[0, 1], [ 1, 2], [ 2, 1] a 3 3 3 3 1 x a 1 =0 x œ [0, 1] a 3 1 =1 x œ [ 1, 2 ] 3 3 x œ [ 2, 1] a 3 1 =2 x a 2 x D a i œ{0, 1, 2} 0 Æ a i Æ 2 i =1, 2,... a 1 0= i=2 0 3 i Æ Œ ÿ i=2 a i 3 i Æ Œ ÿ i=2 2 3 i = 1 3. a 2 0 Æ q Œ a i i=3 Æ 1 a 3 i 9 n 1 3 n 1 x =0.a 1 a 2 a 3... a i,i = 1, 2, 3,... Y _] 0, x. a i = 1, x. _[ 2, x. i a i =1 x =0.a 1 a 2 a 3 = a i 3 i a i œ{0, 2}.
D =[0, 1] ( Œ, 0) fi (1, +Œ) D C c ( Œ, 0) fi (1, +Œ) fi ( 1, 2) fi ( 1, 2) fi ( 7, 8) fi ( 1, 2 ) fi... 3 3 9 9 9 9 27 27 Cc C N S R Y ] N [ C f : C æ N f f f 1 : N æc f 1 f 1 (1) = 0. a 11 a 12 a 13 a 14... f 1 (2) = 0.a 21 a 22 a 23 a 24... f 1 (3) = 0.a 31 a 32 a 33 a 34... f 1 (4) = 0.a 41 a 42 a 43 a 44... f 1 (n) =0.a n1 a n2 a n3 a n4... a nn...
a ij œ{0, 2} b =0.b 1 b 2 b 3 b 4... Y ] 0 a ij =2 b j = [ 2 a ij =0 b 0 2 b j b = f 1 (k) k =1, 2,...,n 1 b = f 1 (n) b n a nn n f 1 1 3 1 3 3 1 3 +21 1 3 2 +22 3 + = ÿ Œ 2 i 1 3 i=0 3 = 1 3 4 2 i =1. i+1 3 i=0 3 C I œc I L(I) I I L(I) > 0 I œc L(I) Æ L(C) L(I) > 0 L(I) Æ L(C) L(C) > 0
>0 x œc y œc y = x x y < x y C x>y a =0.a 1 a 2 a 3 = q Œ a i a 3 i i œ{0, 2} x, y x y = x = y = x i 3 i x i œ{0, 2} y i 3 i y i œ{0, 2} x i y i 3 i x i y i œ{0, 2}. x œc y œc, y = x y i = x i k k œ{1, 2, 3,...} x y = i=k x i y i 3 i = 1 3 k i=0 x i y i 3 i Æ 1 3 k i=0 2 3 = 1 i 3. k 1 k k 1 3 k 1 <
S R n F : S æ S S c 0 <c<1 F (x) F (y) Æc x y, x, y œ S F F c F M = 1 c 1 3 2n n 1 3 n 3 n D =[0, 1] n S R n {F 1,...,F m } U S F i m U = F i (U). {F 1,...,F m },F i : R n æ R n
C F 1 (x) = 1x F 3 2(x) = 1x + 2 F 3 3 1(C) C 1 F 2 (C) C 1 C = F 1 (C)fiF 2 (C) 2 n C n {F1 1,...,F2 1 n } C n D n F1 1 (x) =3 n x C n D F2 1 (x) n =3n x (3 n 1) C n D n 3 n n 3 n D =[0, 1] D ( Œ, 0) A Û 1 1 2 4 1 µ, 1 Û 1 2 + 4 1 B (1, Œ), µ D m 2 Æ m < Œ D 2m 1 3 4 1 2m 1, 2 3 2m 1 1 = 5 0, 6 1 5 2m 1 3 2m 1, 4 2m 1 2 2m 1, 3 2m 1 4 3 4 2m 3 2m 1, 2. 2m 1 6 5 2m 1 2m 1, 1 6.
m m m n 1 n (2m 1) n n n = C 0, D C 1 (2m 1) n 2 (2m 1), 3 n (2m 1) n D C (2m 1) n D 1, 1. (2m 1) n = Œ n=1 n. D ( Œ, 0) fi (1, Œ) ( Œ, 0) ( ) (1, Œ). 1 m 1 2m 1 1 m (m 1) 1 m 1 (2m 1) 2 m(m 1) m n 1 1 (m 1) n (2m 1) n m n 1 1 (m 1) n=1 (2m 1) = m 1 3 4 m n =1. n 2m 1 n=0 2m 1 D
>0 x œ y œ y = x x y < x y x>y a =0.a 1 a 2 a 3 = q Œ a i a (2m 1) i i œ {0, 2, 4,...,2m} x, y x = x i (2m 1) i, x i œ{0, 2,...,2m 2, 2m}. y = x y = y i (2m 1) i, y i œ{0, 2,...,2m 2, 2m}. x i y i (2m 1) i x i y i œ{0, 2,...,2m 2, 2m}. x œ y œ, y = x y i = x i k k œ{1, 2, 3,...} x y = i=k x i y i (2m 1) = 1 i (2m 1) k i=0 x i y i (2m 1) i x i,y i œ{0, 2,...,2m 2, 2m} 1 (2m 1) k i=0 x i y i (2m 1) Æ 1 i (2m 1) k k k i=0 2m (2m 1) = m i (m 1)(2m 1). k 1 m (m 1)(2m 1) k 1 <
1 n 2 n D =[0, 1] 1 ( 1, 3) 2 4 4 5 1 = 0, 1 6 5 6 3 4 4, 1. 1 4 5 2 = 0, 1 6 5 3 16 16, 4 6 5 12 16 16, 13 6 5 6 15 16 16, 1. 1 8 3 = 5 0, 1 6 5 3 64 64, 4 6 5 12 64 64, 13 6 5 15 64 64, 16 6 5 48 64 64, 49 6 5 51 64 64, 52 6 5 60 64 64, 61 6 5 6 63 64 64, 1. = Œ i. 1 2 1 ú 1 4 4 1 ú 1 8 64 2 i 1 1 1 4 i 1 2 i 2 i 4 1 = 2 4 i 2 i 1 =2 4 i = 1 1 2 i=0 4 i = 2 3 1 3
1 3
1 2 3 32 2 2 2 n ( 3 2 )n Œ n æœ
[0, 1] [0, 1] [0, 1] ( 1, 2) [ 2, 1] 3 3 3 3 8 9 ( 8 9 )2 ( 8 9 )n n n æœ M [0, 1] [0, 1] [0, 1] 1 3
( 1 3 )3 20 27 202 ( 1 9 )3 ( 20 27 )2 n ( 20 27 )n 2( 20 9 )n +4( 8 9 )n n n æœ
S S S k S S k 1 k L p S p p S L S L L P p P P P P M M p S p M M
R R n R d R =[a 1,b 1 ] [a 2,b 2 ] [a n,b n ]. a i Æ b i,i = {1, 2,,...,n} R = {(x 1,x 2,...,x n ) œ R n : a i Æ x i Æ b i, i =1, 2,...,n}. b 1 a 1,b 2 a 2,...,b n a n R R =(b 1 a 1 )(b 2 a 2 )...(b n a n ). n =1 n =2 (a 1,b 1 ) (a 2,b 2 ) (a n,b n ). Q b 1 a 1 = b 2 a 2 = = b n a n = l. Q l n π 1 1 1 2 n 1 n N( ) = 1 n = 3 1 4 n, n n n V n N( ) =V n 3 1 4 n,
V n n n = lnn( ) lnv n ln 1 2. 1 lnn( ) lnv n n = lim æ0 ln 1 lnn( ) 2 = {V 1 n } = lim æ0 ln 1 2. 1 S R n S S S µ R n dim B (S) = lim sup lnn( ) æ0 ln 1 2 1. dim B (S) = lim inf lnn( ) æ0 ln 1 2 1. dim B (S) = dim B (S) dim B (S) S lnn( ) dim B (S) = lim æ0 " R + R R n! 1 ln
ln2 ln3. N =2 n n =(1/3) n dim B (C) = ln2n ln 1 = ln2n ln3 = nln2 (1/3) n nln3 = ln2 ln3 n 0.6309... ln3 ln2 n 3 n 1 2 n dim B = ln3n ln 1 = ln3n ln2 = nln3 (1/2) n nln2 = ln3 ln2 n 1.5849... ln4 ln3. n 4 n (1/3) n dim B = ln4n ln 1 = ln4n ln3 = nln4 (1/3) n nln3 = ln4 ln3 n 1.261... ln8 ln3 8 n n 1 3 n dim B = ln8n ln 1 = ln8n ln3 = nln8 (1/3) n nln3 = ln8 ln3 n 1.8927...
ln20 ln3 n 20 n (1/3) n dim B = ln20n ln 1 = ln20n ln3 (1/3) n n = nln20 nln3 = ln20 ln3 2.7268... ln(m) ln(2m 1). m n n 1 (2m 1) n dim B ( ) = ln(mn ) 1 = ln(mn ) ln ln(2m 1) = (1/(2m 1)) n n n ln(m) n ln(2m 1) = m æœ dim B ( ) æ 1 ln(m) ln(2m 1). V n g µ (x) =µx(1 x) n g µ 2 n
g µ Q = fi N Q i Nÿ Q = Q i. Q Q i,i =1, 2,...,N Q Q = fi N Q i Q i Q Q i Q = q N Q i. S R n S m ú (S) = inf Q i. S = fi Œ Q i Q i S
S 1 µ S 2 m ú (S 1 ) Æ m ú (S 2 ) S = fi Œ S i m ú (S) Æ q Œ m ú (S i ) S µ R n m ú (S) = inf m ú (O) O S d(s 1,S 2 ) > 0 m ú (S 1 fi S 2 )=m ú (S 1 )+m ú (S 2 ). S S = fi Œ Q i m ú (S) = q Œ Q i S 1 µ S 2 S 1 S 2 m ú (S) m ú (S) < Œ >0 Q ij S i Œ S i Q ij j=1 Q ij Æm ú (S i )+ j=1 2 j t Œ i,j=1 Q i,j t Œ S i = S >0 Œ m ú (S) =m ú ( S i ) Æ = Æ = i,j=1 j=1 Q i,j Q i,j 3m ú (S i )+ 2 j 4 m ú (S i )+. m ú (S) Æ inf m ú (O) Æ m ú (S) m ú (S) = inf m ú (O) Y ] m ú (S) Æ inf m ú (O). [ inf m ú (O) Æ m ú (S)
O S S µo m ú (S) Æ m ú (O) Q i S >0 Q i Æm ú (S)+ 2. Q i Q Õ i O = t Œ Q Õ i Q i = Q Õ i + 2 i+1. m ú (O) Æ (1) m ú (Q Õ i) = Q Õ i Æ 3 Q Õ i + 4 2 i+1 Æ ÿ Œ Q i + 2 Æ m ú (S)+. >0 inf m ú (O) Æ m ú (S) d(s 1,S 2 ) > 0 Y ] m ú (S 1 fi S 2 ) Æ m ú (S 1 )+m ú (S 2 ) [ m ú (S 1 )+m ú (S 2 ) Æ m ú (S 1 fi S 2 ). Q i S 1 fi S 2 >0 Q i Æm ú (S 1 fi S 2 )+.
d(s 1,S 2 ) > >0 Q i Q i diam Q i = sup{ x y : x, y œ Q i } Œ S 1 = Q i, S 2 = iœe 1 Œ iœe 2 Q i E 1 S 1 E 2 S 2 E 1 fle 2 =? Q i S 1 S 2 m ú (S 1 )+m ú (S 2 ) Æ ÿ Q i + ÿ Q i iœe 1 iœe 2 Æ Q i Æ m ú (S 1 fi S 2 )+. >0 m ú (S 1 )+m ú (S 2 ) Æ m ú (S 1 fi S 2 ) m ú (S 1 fi S 2 )= m ú (S 1 )+m ú (S 2 ) S Y ] m ú (S) Æ q Œ Q i [ q Œ Q i Æm ú (S). S = fi Œ Q i Q i Q Õ i Q i Q Õ i µ Q i, i Q i Æ Q Õ i + 2 i, >0 i S Õ = t Œ Q Õ i S Õ µ S {Q Õ i} d(q Õ i,q Õ j) > 0 i = j {Q Õ i} A Œ B m ú Q Õ i = m ú (Q Õ i)= Q Õ i Ø ( Q i 2 )= ÿ Œ Q i i.
S Õ µ S m ú (S) Ø m ú (S Õ ) æ 0 Q i Æm ú (S). R + fi {Œ} R n m (S) S >0 m (S) S S S S m (S) =0 S S m (S) =Œ S S U R n U diam U sup{ x y : x, y œ U} S R n {U i } S S µ fi Œ i U i 0 < diam U i Æ i œ Z + {U i } S S R n, Ø 0 >0 S H (S) m ú (S) = lim æ0 H (S), I J ÿ H (S) inf (diam U i ) : {U i } S i >0 S U i,i =1, 2,... q k(diam U i ) æ 0 H (S) H (S) ÆH (S) >0
S 1 µ S 2 H (S 1 ) ÆH (S 2 ) H (fi Œ S i ) Æ q Œ H (S i ) {S i } R d d(s 1,S 2 ) Ø 0 H (S 1 fi S 2 )=H (S 1 )+H (S 2 ) H (S) < Œ > H (S) =0 H (S) > 0 < H (S) =Œ H (S) = lim æ0 H (S) = lim æ0 inf < 0 Æ diam U i Æ I J ÿ (diam U i ) : {U i } S. i (diam U i ) = (diam U i ) (diam U i ) Æ (diam U i ). H (S) Æ H (S). H (S) ÆH (S) >0 H (S) < Œ H (S) Æ H (S) = æ0 0 H (S) =Œ H (S) > 0 <. B W BœW, S œw B\S = S c œw, S i œw,, 2,... t S i œw. B ii) iii) B B (A fi B) c = A c fl B c R n H R n H
{S i } S = fi Œ S i Œ H ( S i )= H (S i ). H ( S) = H (S), >0. S diam S = sup{ x y : x, y œ S} x y x, y œ S diam S = diam S, >0 R n B R n iv) Y ] H Œ if <, (S) = [ 0 if <. 0 ÆH (S) ÆŒ S
R n S = sup{ Ø 0:H (S) =Œ} = inf{ Ø 0:H (S) =0} = dim H (S) S ln2 ln3 D C1 L = D fl [0, 1] 3 CR 1 = D fl [ 2, 1] 3 1 C 3 1 = C1 L fic1 R iii) 3 4 1 d 3 4 1 d 3 4 1 d H d (C 1 )=H d (C1 L )+H d (C1 R )= H d (C 1 )+ H d (C 1 )=2 H d (C 1 ) 3 3 3 d 0 < H d (C 1 ) < Œ d = dim H (C) d = ln2 ln3 ln2 ln3 S µ R n dim H (S) Æ dim B (S) Æ dim B (S). Q fl [0, 1] dim B (A) =1 dim H (A) =0. cl(s) S
cl(a) =[0, 1] cl(a) dim B (A) = dim B (cl(a)) = 1. A i H 0 (A i )=1 dim H (A i )=0. fi Œ A i dim H (A) =0 {F 1,...,F m },F i : R n æ R n F i V m F i (V ) µ V {F 1,...,F m },F i : R n æ R n c i œ (0, 1) (1 Æ i Æ m) S S = fi m F i (S) mÿ dim H (S) = dim B (S) =d, d c d i =1. d 0 < H d (S) < Œ d {F 1,...,F m },F i : R n æ R n S R n F i (x) F i (y) Æc i x y, c i œ (0, 1), x, y œ S, mÿ dim H (S) Æ d, d c d i =1. {F 1,...,F m },F i : R n æ R n S R n c i x y Æ F i (x) F i (y), c i œ (0, 1), x, y œ S. U U = fi m F i (U) mÿ d Æ dim H (S), d c d i =1.
fl Œ n {S i } D =[0, 1] S n = {x œ D g n µ(x) > 1, µ>4} S 1 g µ (x) =1 S 1 = A Û 1 1 2 4 1 µ, 1 Û 1 2 + 4 1 B =(,1 ). µ = 1 2 Ò 1 4 1 µ 1 = 1 2 + Ò 1 4 1 µ F 1 F 2 D [0, ] [1,1] Y ] F 1 (x) = 1 Ò 1 x 2 4 µ [ F 2 (x) = 1 + Ò 1 x. 2 4 µ x, y œ D x = y Õ F (z i )= F i(x) F i (y) i =1, 2. z i œd x y i inf F i Õ (x) Æ F i(x) F i (y) xœd x y Æ sup Fi Õ (x). xœd Fi Õ (x) = 1 A 1 2µ 4 x B 1 2,, 2, µ 1 µ x y Æ F i(x) F i (y) Æ 1 A 1 2µ 4 1 B 1 2 x y, i =1, 2. µ F 1 F 2 0 < 1 A 1 2µ 4 1 B 1 2 Ô < 1, µ>2+ 5. µ F 1,F 2 1 ( 1 1 2µ 4 µ ) 1 2 < 1 dim H ( ) Æ d d 2( 1 ( 1 1 2µ 4 µ ) 1 2 ) d =1 d dim H ( ) Æ ln2 3 ln µ 1 1 4 µ 2 1 4 2.
F 1,F 2 s Æ dim H ( ) s 2( 1 2µ )s =1 ln2 lnµ Æ dim H( ). ln2 lnµ Æ dim H( ) Æ ln2 3 ln µ 1 2 1 4 1 4 2. µ dim H ( ) ln2 µ dim lnµ H( ) æ 0 µ æœ. g µ f(x) = xsin( x)
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