Tricorn

Triorn 016 3

1 Mandelbrot Triorn Mandelbrot Robert L DevaneyAn introdution to haoti dynamial Systems Addison-Wesley, 1989 Triorn 1 W.D.Crowe, R.Hasson, P.J.Rippon, P.E.D.Strain- Clark, On the struture of the Mandelbar set, Nonlinearity, (1989), 541-553 Mandelbrot Triorn Julia Julia 3 Julia Julia z z + ( ) Julia Julia F ig.1f ig.f ig.3 F ig.4 F ig1 F ig 1

F ig3 F ig4 ( F ig.f ig.4 Cantor F ig.1 : = 0.36084 + 0.100376iF ig. : = 0.5F ig.3 : = 1F ig.4 : = 0.7688 + 0.163i ) Julia Cantor 0 0 Julia 0 Cantor Julia 0 Mandelbrot Mandelbrot P (z) = z + f (z) = z + Mandelbrot 0 f 0 Triorn T M T F ig.5f ig.6

F ig.5 Mandelbrot F ig.6 Triorn { C : } Mandelbrot Triorn 3 Triorn 3 4 4.4 ([1]Theorem5) Ω 1, Ω D 1, D P (z) = z + P n P P P P n Ω 1 {P n (0)} n N 1 () Ω {P n (0)} n N f (z) = z + D 1 {f n (0)} n N 1 () 3

D {f n (0)} n N Ω 1 Ω = 3/4 1 D 1 D 4.5 4.6 4.4 4.6 f (z) = z [] 4 Mandelbrot Mandelbrot Julia 3 Triorn 4 Mandelbrot Mandelbrot Julia Mandelbrot n N Q n Q n = Q Q Q P (z) = z + (z C C).1 () n N P n z 0 P n (z 0 ) = z 0 (P n ) (z 0 ) > 1.1 Julia. (Julia ) C P Julia P.3 ( Julia ) C P Julia K 4

C n N K = {z C : P n (z) (n )}.4 (Proposition8.) Julia Julia. Mandelbrot.5 (Mandelbrot ) Mandelbrot M = { C : P n (0) (n )} Mandelbrot F ig.1. F ig.1 Mandelbrot ( Re(z) Im(z) ) F ig.1 5

M.6 M { C : }.7 M.6 (i) (ii) (i) { C : > } z C z P (z) = z + z z z = z ( 1) > 1 > 1 λ = 1 P (z) λ z n P n (z) λ n z λ = 1 > 1 n P n (z) > M (ii) M M M P n (0) (n ) P n 1 () (n ) N N P N 1 () > P n 1 () P N 1 ( ) > 6

< P N 1 ( ) P N ( ) = {P N 1 ( )} + P N 1 ( ) > P N 1 ( ) P N 1 ( ) = P N 1 λ = P N 1 ( ) 1 > 1 k ( ) ( P N 1 ( ) 1) P N ( ) λ P N 1 ( ) P N+k ( ) λ k P N 1 ( ) k P N+k ( ) > (i) M M M M M (i)(ii) M { C : } : = P ( ) = P () = M Q.E.D. { C : > } M M.7 : C n N P n (0) = P n (0) z C P n (z) = P n (z) 7

(i) n = 1 P (z) = z + P (z) = z + P (z) = P (z) P (z) = P (z) (ii) n = k P k (z) = P k (z) P (z) = P k (P (z)) = P k+1 (z) = P k (P (z)) = P k (P (z)) = P k+1 (z) P k+1 (z) = P k+1 (z) n = k + 1 n 1 P n (z) = P n (z) P n (0) = P n (0) M = { C : P n (0) (n )} = { C : P n (0) (n )} M Q.E.D. 8

3 Triorn Mandelbrot P (z) = z + f (z) = z + 3.1 (Triorn) Triorn T T = { C : f n (0) (n )} Mandelbrot P (z) = z + Triorn f (z) = z + f (z) = z 4 +z + + Triorn F ig3.1 F ig3.1 Triorn ( Re(z) Im(z) ) T 9

T 3. σ : z e πi/3 z σ(t ) = T () 3.3 T { C : } 3.4 T 3. σ : z e πi/3 z σ(t ) = T z C, C ω = e πi/3 f ω (ωz) = ωf (z) f ω (ωz) = (ωz) + ω = ω z + ω = ωz + ω = ω(z + ) = ωf (z) f ω (ωz) = ωf (z) n f n ω(ωz) = ωf n (z) f n ω(0) = ωf n (0) f n ω(0) = f n (0) T = { C : f n (0) (n )} = { C : f n ω(0) (n )} σ : z e πi/3 z σ(t ) = T 10 Q.E.D.

3.3 : M (i) (ii) (i) { C : > } z C z f (z) = z + z z z z z = z ( 1) 1 > 1 λ = 1 f (z) λ z n f n (z) λ n z n f n (z) > T T { C : } (ii) T T T f n (0) (n ) f n 1 () (n ) N N f N 1 () > f n 1 () f N 1 ( ) > 11

< f N 1 ( ) f N ( ) = f N 1 ( ) + f N 1 ( ) > f N 1 ( ) f N 1 ( ) = f N 1 λ = f N 1 ( ) 1 > 1 k ( ) ( f N 1 ( ) 1) f N ( ) λ f N 1 ( ) f N+k ( ) λ k f N 1 ( ) k f N+k ( ) > (i) T T T T T (i)(ii) M { C : } 3.4 C n N f n (0) = f n (0) z C f n (z) = f n (z) Q.E.D. (i) n = 1 f (z) = z + f (z) = z + = z + 1

f (z) = f (z) f (z) = f (z) (ii) n = k f k (z) = f k (z) f (z) = f k (f (z)) = f k+1 (z) = f k (f (z)) = f k (f (z)) = f k+1 (z) (z) = f k+1 (z) f k+1 n = k + 1 f n (z) = f n (z) f n (0) = f n (0) T = { C : f n (0) (n )} = { C : f n (0) (n )} T Q.E.D. 4 Mandelbrot Triorn 4.1 () P (z) 13

n z 0 P n (z 0 ) = z 0 (n ) P n (z 0 ) = z 0 (P n ) (z 0 ) 1 z 0 n z 0 P (z 0 ) = z 0 P (z 0 ) 1 f(z) z f(z) f (z) ( 1 ) z 0 f (z 0 ) = z 0 f (z 0 ) = z 0 (f ) (z 0 ) < 1 z 0 f (z 0 ) = z 0 (f (z 0 ) z 0 ) f (z 0 ) = z 0 (f ) (z 0 ) < 1 z 0 P P (z 0 ) = z 0 P (z 0 ) 1 z 0 f f (z 0 ) = z 0 f (z 0 ) 1 f f (z) f (z) = z 4 + z + + f (z 0 ) = z 0 (f ) (z 0 ) < 1 Mandelbrot Ω 1, Ω P (z) = z + Ω 1 {P n (0)} n N 1 () Ω {P n (0)} n N 4. Ω 1 Ω Ω 1 Ω Ω 1 = { C : = w w, w C, w < 1/} Ω = { C : + 1 < 1/4} 14

F ig.4.1 Ω 1 Ω 4. (i) Ω 1 P (z) w P (w) = w P (w) < 1 w P (w) = w w + = w = w w P (w) < 1 P (w) = w w < 1/ Ω 1 = { C : = w w w < 1/} (ii) Ω P (z) w 1, w P (w 1 ) = w P (w ) = w 1 ( P (w 1 ) = w 1 ) (P ) (w 1 ) < 1 P (w 1 ) = w P (w ) = w 1 P (P (w )) = P (w ) = w 15

(P ) (w ) = P (P (w ))P (w ) = P (w 1 )P (w ) = 4w 1 w (P ) (w ) < 1 (P ) (w ) = 4 w 1 w < 1 w 1 w = + 1 + 1 < 1 4 Ω = { C : + 1 < 1/4} Q.E.D. Triorn D 1, D f (z) = z + D 1 {f n (0)} n N 1 () D {f n (0)} n N 4.3 D 1 D 1 D 1 = { C : = w w, w < 1/, w C} F ig.4. D 1 D 16

F ig4. 4.3 z 0 f (z 0 ) = z 0 z 0 + = z 0 = z 0 z 0 z 0 (f ) (z 0 ) < 1 f (z 0 ) = z 0 + = z 0 (f ) (z 0 ) = 4z0 3 + 4z 0 = 4 z 0 z0 + = 4 z 0 z 0 = 4 z 0 z 0 < 1/ D 1 = { = w w : w < 1/} Q.E.D. Ω 1 Ω 3/4 1 D 1 D Triorn 3.3.4 17

= 3/4 D 1, D Triorn D 1, D = 3/4 f (z) 3/4 D 1 D Triorn D 1 D = a + bi 1 < a < 00 < b < 1 D 1 D 4.4 (1Theorem5) ε > 0 D( 3/4, ε) D 1 {f n (0)} n N D( 3/4, ε) 3/4 ε F ig4.3 4.1 4.5 f (z) = z + 4 18

4.5 : ( 1 ) w f (w) = w f (w) w = 0 f (w) w = w 4 + w w + + = 0 4 4 Q.E.D. 4.6 (1,Lemma 1) = a + bi a, b 1 < a < 00 < b < 1 (a)() (a) (b) D 1 f z 1 > 1/ z > 1/ z 1, z D 1 f z 1 > 1/ z > 1/ w 1 1/, w 1/ 4 z 1, z, w 1, w () D 1 f z 1 > 1/ z > 1/ z 1, z w = 1/ w 3. 4.6 f (z) = z + = z z = x+iy, = a+bi H 1, H. ( 1 < a < 00 < b < 1) H 1, H H 1 : ( x 1 ) y = 1 4 a H : ( x + 1 ) y = b ( y = ± x 1 ) 1 4 + a y = b ( x + 1 ) 19

( y = x 1 ( y = x 1 y = b ( x + 1 ) ) 1 4 + a ) 1 4 + a H + 1 H 1, H ( H 1 + : y = x 1 ) 1 ( 4 + a x < 1 1 4a ( H1 : y = x 1 ) 1 ( 4 + a x < 1 1 4a b H : y = ( ) x + 1 ( 1 ) 1 + 1 4a 1 + 1 4a ) < x ) < x F ig.4.4 H + 1 F ig4.5 H 1 0

F ig.4.6 H x < 1/ H 1 H ( x 1 ) 1 4 + a = b ( ) x + 1 ϕ 1 ( ϕ 1 (x) = x 1 ) 1 4 + a b ( ) x + 1 ( ϕ 1(x) = 1 x 1 + a + (x 1)x ) 4b (x + 1) x < 1/ ϕ 1(x) > 0 ϕ 1 (x) (x 1/ 0) ϕ 1 (x) (x ) x < 1/ ϕ 1 (x) = 0 1 x 1 H b (x 1 + 1 ) y 1 H 1, H (x 1, y 1 ) x 1 < 1/ x 1 + y1 > 1/ x < 1/ H 1 H 1/ 1

x > 1/ H + 1 H a 1 < 1+ 1 4a H 1 + 1 < x < 1+ 1 4a 1+ 1 4a < x ( x 1 ) 1 4 + a = b ( ) x + 1 ϕ ϕ (x) = ( x 1 ) 1 4 + a b ( ) x + 1 ( ϕ (x) = 1 x 1 + a + (x 1)x ) 4b (x + 1) x < 1/ ϕ (x) > 0 ( ) 1 + 1 4a ϕ b = < 0 1 4a + ϕ (x) (x ) x > 1+ 1 4a ϕ (x) 1 x > 1/ ϕ (x) = 0 1 x H b (x + 1 ) y H 1, H (x, y ) x > 1/ x + y > 1/ x > 1/ H 1 H 1/ 1/ < x < 1/ H + 1 1 1 4a < x < 1+ 1 4a H 1, H 1 < a 3/4 1 1 4a < 1 1 < 1+ 1 4a 1/ < x < 1/ H + 1 H

1 < x < 1 1 4a H 1 H 3/4 < a < 0 b 0 < b < 1 A = 1 4 ab = b X = x 1 1 4 < A < 1B > 0 1 < X < A A, B, X ( x 1 ) 1 4 + a = b ( ) x + 1 X A = B... () X + 1 X 1 < X A (X + 1) > 0 X A = B (X + 1) B = (X + 1) (X A) ϕ(x) = (X + 1) (X A) X ϕ(x) = (X + 1)(X A) + X(X + 1) 1 < X A X 1... 1 1+8A 4... A ϕ (X) 0 + 0 - - ϕ(x) 0 M(A) 0 3

( ) 1 1 + 8A M(A) = ϕ 4 F ig4.7 () B > M(A) 0 B = M(A) 1 0 < B < M(A). B = M(A) a, b 8a 4a 3 8a(8a 3) + 8b + 9 = 0... () a, b () = a + bi D 1 = a + bi { C : = w w, w C, w = 1/} w = p + qi = w w { a = p p + q b = q + pq { p = 1 os θ q = 1 sin θ θ θ < θ < π 4

θ 0 < θ < π os θ = 1 3 a, b 3 4 < a < 00 < b < 1 θ 1 a, b θ { 3 4 < 1 os θ 1 4 os θ + 1 4 sin θ < 0 0 < 1 sin θ + 1 os θ1 sin θ os θsin θ { 1 < os θ < 1 3 sin θ > 0 θ θ < θ < π ( θ 0 < θ < π os θ = 1 3 3 4 < a < 00 < b < 1 θ a, b () = a + bi D 1 B = M(A) = a + bi D 1 () 1 () X = 1 os θ + 1 () H 1 H x 1 os θ H H 1 H y 1 sin θ H 1 H (x, y) = ( 1 os θ, 1 ) sin θ (θ ) D 1 1/ < x < 1/ H 1 H 1/ () b = b = Φ(a) = 1 8 (8a 4a 3 8a(8a 3) + 9 ) ) F ig4.8 5

a, b = a + bi D 1 a, b D 1 0 B < M(A) (F ig.4.8 () ) b < Φ(a) = 1 8 (8a 4a 3 8a(8a 3) + 9 ) Y = ϕ(x) Y = B () = a + bi D 1 B > M(A) (F ig.4.8 () ) b > Φ(a) = 1 8 (8a 4a 3 8a(8a 3) + 9 ) Y = ϕ(x) Y = B () = a + bi D 1 0 < B < M(A) Y = ϕ(x) Y = B X 1, X 1 < X 1 < 1 1 + 8A 4 < X < A X = x 1 X 1, X x u 1, u u 1, u v 1, v H 1 < u 1 < 1 1 + 8A 4 1 1 + 8A < u < 1 4 A v 1 = B u 1 + 1 v = B u 1 + 1 (u 1, v 1 )(u, v ) H 1, H u 1 + v 1 > 1 4 u 1 + v 1 1 4 = u 1 + 4B (u 1 + 1) 1 4 6

0 < B u 1 α = 1 1+8A 4 u 1 + v 1 1 4 > u 1 1 4 > α 1 4 > 0 u 1 + v 1 > 1 4 u + v < 1 4 u + v 1 4 = u + 4B (u + 1) 1 4 B < M(A) u α = 1 1+8A 4 u + v 1 4 < α + 4M(A) (α + 1) 1 4 = 4A( 8A 1 3) + 8A 1 ( 8A 1 1) = 0 u + v < 1 4. B < M(A) (F ig.4.8 () ) H 1, H 1 1/ 1/ H 1 H D 1 4 z 1 Re(z 1 ) < 1 z Re(z ) > 1. w w < 1 w 1 1 < Re(w 1) < 1 w 1 > 1 D 1 3 z 1 Re(z 1 ) < 1 7

z Re(z ) > 1. w w = 1 D 1 z 1 Re(z 1 ) < 1 z Re(z ) > 1. D 1 z 1, z, w 1, w z 1 z w 1 > 1/ w < 1/ D 1 z 1, z, w z 1 z > 1/ w = 1/ D 1 z 1, z w 1, w z 1 z > 1/ ()() f (z) z = 0 () f (z) z = 0 () 4.5 4 () 4 () () () () z 1 = x 1 + iy 1 z = x + iy w 1 = u 1 + iv 1 w = u + iv w = 1 os θ + i 1 sin θ 1 < a < 0 0 < b < 1 3 D 1 z 1, z () 4 z 1, z () () () f D 1 z 1, z, w 1, w 4 4 () () 4 8

D 1 z 1, z, w 3 3 () () 4 z 1, z, w 3 () 1 () 3 w () () 4 z 1, z, w 1, w w 1 = w = w 4.6 (a)() Q.E.D. 4.4 4.4 w 1, w w 4.6 () w 1, w D 1 w 1 w = w = 1/4 D 1 D 1 w 1 w (f ) (w 1 ) = 4w 1 f (w 1 ) = 4w 1 w w 1, w (f ) (w 1 ) < 1 w 1 w < 1/4 (1, 1) D 1 w 1 w < 1/4 f F ig4.9 9

f (z) = z z 1 z w 1 w = + w 1 w = + z 1 z D 1, D w 1 w + z 1 z n +, n = a b n z 1, n z (i) n + = a + bi n + = a(a + 1)(a + 1) + b (4a 6) b(a 6a + 1) 4b 3 a + bi = 3/4 3/16 a b 3/4 n + > 0 n + > 0 3/4 ε > 0 (ii) n z 1 z 1 = x + iy H 1, H x a = x + 1 y a = y 30

x b = y y b = x 1 = 4(x + y ) 1 b > 0 x n = x a x b y n = y a y b = (x + 1) y = y (x 1) z 1 z 1 > 1/ > 0H 1 H z 1 x < 1/ y < 0 x < 1/y < 0 (x + 1) y > 0 a < 1/4 1/4 a > 0 ( x 1 ) y = 1 4 a > 0 y (x 1) > 0 x n = x a x b y n = y a y b = (x + 1) y = y (x 1) x, y (1, 1) x 1/ y 0 z 1 (iii) z n z 1 < 0 z 1 z = x + iy = 4(x + y ) 1 b > 0 x n = x a x b 31 > 0 > 0 = (x + 1) y

y n = y a y b = y (x 1) z z > 1/ > 0H 1 H z H 1, H x > 1/ y > 0 x > 1/y > 0 (x + 1) y < 0 a < 1/4 1/4 a > 0 ( x 1 ) y = 1 4 a > 0 x n = x a x b y n = y a y b y (x 1) < 0 = (x + 1) y = y (x 1) < 0 < 0 x, y (1, 1) x 1/ y 0 z (i)(ii)(iii) n z < 0 n + > 0 n z 1 < 0 n z < 0 + n z 1 z = n w 1w > 0 w 1 w (1, 1) 3

4.7 R Ω = R D = ( 5/4, 3/4) 4.7 R, z R P (z) = f (z) P (0) = f (0) P n (0) = f n (0) R {P n (0)} n N {f n (0)} n N R Ω = R D Ω 4.1 Ω R = ( 5/4, 3/4) R Ω = R D = ( 5/4, 3/4) Q.E.D. 4.3 ( 5/4, 3/4) w 1 w < 1/4 n w 1w > 0 D 1 w 1 w = 1/4 D 1 (1, 1) w 1 w < 1/4 ( 5/4, 3/4) (1. 1) w 1 w < 1/4 F ig.4.10 w 1 w < 1/4 F ig4.10 4.4 (i) n + > 0 3/4 ε D( 3/4, ε) D 1 f 33

4.8 (Theorem4.6) Q z 0 Q Q (z) = 0 z Q Q 1 z 0 Q(z) = f (z) (f ) (z) = 4z 3 + 4z = 4z(z + ) (f ) (z) = 4 z z + f (z) z = 0 z = z = f 0 z = f 0 {f n (0)} n N F ig4.10 D D 1 D Q.E.D. 34

1W.D.Crowe, R.Hasson, P.J.Rippon, P.E.D.Strain-Clark, On the struture of the Mandelbar set, Nonlinearity, (1989), 541-553 Robert L DevaneyAn introdution to haoti dynamial Systems Addison- Wesley, 1989 35