I z n+1 = zn 2 + c (c ) c pd L.V. K. 2

I 2012 00-1 I : October 1, 2012 Version : 1.1 3. 10 1 10 15 10 22 1: 10 29 11 5 11 12 11 19 2: 11 26 12 3 12 10 12 17 3: 12 25 1 9 1 21 3 1

I 2012 00-2 z n+1 = zn 2 + c (c ) c http://www.math.nagoya-u.ac.jp/~kawahira/courses/12w-tenbou.html pd L.V. K. 2 http://www.math.nagoya-u.ac.jp/~kawahira/courses/mandel.pd R. Devaney 2 10 15 22 29 10 30 11 19 3 11 Cae David Cae David 12:00 13:30

I 2012 01-1 1 : October 15, 2012 Version : 1.1 (ractal) ractus (Benoit Mandelbrot, 1924 2010 (sel-similartity) 1. 2 etc 1. 1: 2 1

I 2012 01-2. I = [0, 1] R T 0, T 1 : I I T 0 (x) := x 3 T 1 (x) := 1 (x 1) + 1 3 E 0 := I E 1 := T 0 (E 0 ) T 1 (E 0 ) E 2 := T 0 (E 1 ) T 1 (E 1 ). E n+1 := T 0 (E n ) T 1 (E n ) 1-1. (1) n 0, E n+1 E n. 2 (2) n 0, E n.. C := n 0 E n I 3 (Cantor s triadic set) 1-1 3 C C C C = T 0 (C) T 1 (C). C 1 C : N C 2 C : Σ C Σ = {u 1 u 2 u k = 0 1 k N} 0 1 1.1 (a) (a) : C T 0 (C) T 1 (C) (b) : C T 0 (C) T 1 (C)

I 2012 01-3 1-2. 1-3. 1-4. (b) 1.1 1.1. 1 s = e πi/6 / 3 C 0, 1, s I T 0, T 1 : I I T 0 0, 1, s 0, s, 1/3 T 0 0, 1, s s, 1, 2/3 E 0 := I, E n+1 := T 0 (E n ) T 1 (E n ) E n K := n 0 E n (Koch s curve) 1-5. K (1) K (2) g : Σ K (3) K 2 (4) K 1-6. K [0, 1]

I 2012 02-1 2 IFS : October 22, 2012 Version : 1.1. I = [0, 1] R T 0, T 1 : I I T 0 (x) := x 3 T 1 (x) := 1 (x 1) + 1 3 E 0 := I, E n+1 := T 0 (E n ) T 1 (E n ) E n+1 E n C := E n n 0 I 3 (Cantor s triadic set). s = e πi/6 / 3 C 0, 1, s I T 0, T 1 : I I T 0 (x) := sz T 1 (x) := s(z 1) + 1 E 0 := I, E n+1 := T 0 (E n ) T 1 (E n ) E n K := E n n 0 (Koch s curve) 2-1 1-5, 1-6 K (1) K (2) g : Σ K (3) K 2 (4) K (5) K [0, 1] (6) K = T 0 (K) T 1 (K) 3. 4. 1 2 1 3

I 2012 02-2 IFS IFS Iterated Function System (IFS) X R n {F 1, F 2,..., F m } (m 2) X IFS i = 1,..., m r i (0, 1) ( ) x, y X, F i (x) F i (y) r i x y. ( ) F i X = I R X = I R 2 C IFS ( ) 2-1. IFS IFS X R n {F 1, F 2,..., F m } (m 2) X IFS A X F (A) := F 0 (A) F m (A) 2-2 IFS IFS F (E) = E E X E IFS E 0 := X E n+1 := F (E n ) (n 0) E n+1 E n E = n 0 E n. Comp (X) X A Comp (X) δ > 0 A δ- N δ (A) := {x R n a A, x a < δ} A, B Comp (X) d(a, B) := in {δ > 0 N δ (A) B N δ (B) A}

I 2012 02-3 2-3 Comp (X) d A, B, C Comp (X) (D1) d(a, B) 0 A = B (D2) d(a, B) = d(b, A) (D3) d(a, B) d(a, C) + d(c, B). 2-2. 2-3 X Comp (X) := Comp (X) 2-4 (Comp (X), d) Comp (X). 2-5 (S, d) F : S S r (0, 1) ( ) x, y S, d(f (x), F (y)) r d(x, y) F S F (x 0 ) = x 0 x 0 S 2-3. 2-5 2-2 F F : Comp (X) Comp (X) A, B Comp (X) ( ) d( F (A), F (B)) max r i d(a, B) 1 i m r := (max 1 i m r i ) < 1 F 2-5 ( ) E Comp (X) 2-4. 2-2 Hint d(e n+1, E) = d( F (E n ), E) r d(e n, E)

I 2012 02-4 X IFS {F 1,..., F m } F i : X X r i (0, 1) E r d 1 + r d 2 + + r d m = 1 d 0 d = dim S (E). 3 ( 1 d ( 3) + 1 ) d log 2 3 = 1 d = log 3 0.631. 2-5. 10 Mathematica E Comp (R n ) ν r (E) E r log ν r (E) lim r +0 log r E dim B (E) 2-6. (1) [α, β] R 1 (2) A = {0, 1, 1/2, 1/3,..., 1/n,...} 1/2 (a) U R n (diameter) diam U := sup x y x,y U (b) δ > 0 R n {U i } i N E R n δ- i N diam U i δ E i N U i (c) s 0 δ > 0 H s δ (E) := in i N(diam U i ) s [0, ] in E δ- (d) E s H s (E) := (e) E H s (E) = dim H (E) { (0 s < d) 0 (s > d) lim δ +0 Hs δ (E) [0, ] d 0. E 3 dim S (E) = dim B (E) = dim H (E).

I 2012 03-1 3 : October 29, 2012 Version : 1.1 X R n {F 1, F 2,..., F m } (m 2) X IFS F i (1 i m) r i (0, 1) E d = dim S (E) : r d 1 + rd 2 + + rd m = 1 d 0 d = dim B (E) : d log ν r (E) = lim r +0 log r, ν r(e) E r n d = dim H (E) : H s (E) = d 0 { (0 s < d ) 0 (s > d ) 3-1 IFS dim S (E) = dim B (E) = dim H (E). IFS {F 1, F 2,..., F m } (open set condition) 3 U (1) U (2) F 1 (U), F 2 (U),..., F m (U) (3) U F 1 (U) F m (U). 3-1. E 3 IFS U = X z 0 C z n+1 = z 2 n (n = 0, 1,...) z n = z 2n 0 n z 0 < 1 z n 0 z 0 > 1 z n z 0 = 1 z n = 1

I 2012 03-2 (z) z 0 C, z n+1 = (z n ) (n = 0, 1,...) n (z) = z 2 (z) 3-0. 2 m z 0 = z m z 0 z k (k = 1,, m 1) z 0 {z 0, z 1, } z 0 z 1 z 2 z 3 (1) C C C C z 0 C 0, 1, 2, 3, 4,... z 1 (z) C C C C C (dynamcal system) 1 (C, ). (1) z 0 (z 0 ) ((z 0 )) (((z 0 ))) 1 X : X X X (complex dynamics) (discrete dynamical systems)

I 2012 03-3 n {}}{ n (z) := (z) ( 0 (z) = z) z n = n (z 0 ) (C, ) z 0 C { n (z)} n 0 z 0 (orbit) (z) (C, ) (z) = z 2 z 0 z ϵ = z 0 +ϵ (ϵ C, ϵ ) ϵ z 0 < 1 z ϵ < 1 n (z ϵ ) 0 z 0 > 1 z ϵ > 1 n (z ϵ ) z 0 = 1 z ϵ > 1, z ϵ < 1, z ϵ = 1 n (z ϵ ) 0, n (z ϵ ), n (z ϵ ) = 1 2 60 X, Y C : X X, g : Y Y (X, ) (Y, g) (conjugate) ϕ : X Y ϕ = g ϕ 3-1. (X, ) (Y, g) ϕ : X Y 2 { n } n 0

I 2012 03-4 (1) x X n (x) a X (n ) g n (ϕ(x)) ϕ(a) Y (n ). (2) x X n (x) = x (n 1) g n (ϕ(x)) = ϕ(x). 3-2 2 2 (z) = az 2 + bz + c (a 0) (C, ) 2 (C, g) g(w) = w 2 + C, C = b 2 /4 + b/2 + ac C. 2 c (z) = z 2 + c (c C) 2

I 2012 04-1 4 2 : October 29, 2012 Version : 1.1 3-2 2 2 (z) = az 2 + bz + c (a 0) (C, ) 2 (C, g) g(w) = w 2 + C, C = b 2 /4 + b/2 + ac C. w = ϕ(z) = az + b/2 C ϕ = g ϕ (z) = c (z) = z 2 + c (c C) 2 4-1. 2 (z) = az 2 + bz + c (a 0) λ C (C, ) (C, λw + w 2 ) B c K c c C (C, c ) 4-1 z C (1) c n (z) (n ) (2) {c n (z)} n 0. 1, 1, 2, 1/2, 3, 1/3,..., n, 1/n,... 4-2 z 3 + c c (z) 2 z 4-2. (Hint: ) 4-1. (2) m N c m (z) 3 + c 4-2 k N c k+m (z) = c k (c m (z)) 2 k c m (z) 2 k (3 + c ) (k ). z (1) 4-1 (1) (2) (1) B c := {z C c n (z) (n )} { } (2) K c := z C {c n (z)} n 0. J c := K c. B c K c = C (disjoint union)

I 2012 04-2. c = 0 B 0 = { z > 1}, K 0 = { z 1}. B c, K c. c = 2 B 2 = C [ 2, 2], K 2 = [ 2, 2] R.. B 0 = { w > 1} ϕ : B 0 ϕ(b 0 ) C ϕ(w) = w + 1/w ϕ(b 0 ) = C [ 2, 2] ϕ 2 (ϕ(w)) = ϕ(w 2 ) 4-3. (B 0, 0 ) (ϕ(b 0 ), 2 ) z = ϕ(w) ϕ(b 0 ) = C [ 2, 2] w > 1 n 2(z) = n 2(ϕ(w)) = ϕ(w 2n ) ϕ(w 2n ) = w 2n + 1/w 2n w 2n 1/ w 2n w 2n 1 (n ). z B 2 C [ 2, 2] B 2 2 ([ 2, 2]) = [ 2, 2] z [ 2, 2] 2 n (z) [ 2, 2] n [ 2, 2] K 2 B 2 = C [ 2, 2] K 2 = [ 2, 2] B c K c 4-3 (1) B c (2) K c (3) c (B c ) = B c = 1 c (B c ) c (K c ) = K c = 1 (K c ) (3) (B c, c ) (K c, c ) ( 4-3). (1) z 0 B c m N m c (z 0 ) > 4 + c z m c (z) δ > 0 z z 0 < δ m c (z) m c (z 0 ) < 1 m c (z) 3 + c 2-4 z B c B c c

I 2012 04-3 (2) K c = C B c (1) K c 2-4 K c { z < 3 + c } n n c (z) = z C K c K c (3) 4-4. 4-3 (3). 4-4 B c B c E g : E C E E z 0 E z E g(z) g(z 0 ) z 0 E 4-4. B c 2 K c 4-3 U z 0 U c m (z 0 ) 3 + c m N E = Ū = U U g = c m E E = U K c c m ( E) K c z < 3 + c K c Kc IFS (revisited) 4-5 c > 2 z c (> 2) z B c.. c (z) z 2 c z 2 z = ( z 1) z ( c 1) z λ = c 1 > 1 c n (z) λ n z λ n c (n ). 0 0 c c c c 2 + c c (c 2 + c) 2 + c 8 1. z = c C := { z = c } c 1 (C) z = 0 8 U := { z < c } U 0 c 1 (U) U 1 U 1 U 2 8 U U U 0 U 1 I I 0 I 1 c : I 0 I c : I 1 I F 0 := ( c I 0 ) 1 : I I 0, F 1 := ( c I 1 ) 1 : I I 1 {F 0, F 1 } IFS

I 2012 04-4 r = r c (0, 1) i = 0, 1 x, y U d c (x, y) = c x y c 2 xy. d c (F i (x), F i (y)) rd c (x, y). 4-6. c > 2 (1) K c (2) Σ = {u 0 u 1 u i = 0 1} σ : Σ Σ σ(u 0 u 1 ) := u 1 u 2 ϕ c : Σ K c ϕ c σ = c ϕ c (Σ, σ) (K c, c ) Σ ϕ c Σ ϕ c 1 K c 0 B c M := {c C K c } 1: c = 2 + 2i IFS M 1 Σ u = u 1u 2..., v = v 1v 2..., d Σ(u, v) := n 1 Σ u n v n 2 n