1. Introduction (Newton method) --- fig1 H0a, a0 id a1A, a1AP, a1AC z(z-1)(z-i) : resize a1B (z+1)(z-1)^2 : double root, subwindow Ex1.1 a1A*, a1z z(z-1)(z-i) 1 <- -0.2426101492 1.3274644056, ( 1.3999638729 0.1737144654 ), ( -0.1573537238 -0.5011788710 ) i <- ( -0.5115310157 0.9606227978 ), ( 0.3814885471 -0.5970050144 ) ( 1.1300424687 0.6363822166 ) b,a3: Nf = (2z^3 + (-1-i)z^2)/(3z^2+ (-2-2i)z+i) Ex 1.2 c, c1, c2 [Iterate] z^2 + z + 1 [Show] f = z^2 + z + 1; f[1] N_f(z) = (z^2 -1)/(2z+1) upper half plane lower half plane Ex 1.3 d f = z^3 - 2z - 5; f[1] r = Nf = (2z^3 + 5)/(3z^2 - 2) color region plug in f Ex 1.4 e, e1 not work f=z^4 + 3z + 3; f[1] 0: s.a. 2-cycle Ex 1.5 z^3 - 0.4493762319 z + 0.4493762319 z(z+3)^3 + 2(z+2)^3 0....: s.a. 3-cycle Ex 1.6 z^3 - 2z + 2 0: s.a. 2-cycle Ex 1.7 f;f1,f2,f3,f4 -- 587 not evenly distributed 5 z^6 - 6 z r = (5 z^6 - 6 z)/(4z^6 - 5); r[1] Nf ---- 644 lines ----- (2) Fixed Points Ex 2.1 a1 2z Ex 2.2 a2a1, a2b1 (z-i)/(z+i) r=(z-i)/(z+i); r[1] [Show] r=(z-i)/(z+i); g=r[3](z); g[1] [Show] Ex 2.3 a3b, a3d q = z^2 + (1/4 + i/2); q[4] a4a, a4b: z^2 +(-1-i/8) orbit Ex 2.4 a5a, a5b, a5b8, a5b6, a5c Blaschke product: ((z-0.5)/(1-z/2))^2 : 2 a.f.p. 1 r.f.p. at 1 Ex 2.5 a60a2 z^2 + (-i/2) Ex 2.6 a6a, a6a4, a6a5, a6a1, a6a2, a6b, a6b1 h(z)= ((z-0.5)/(1-z/2))^2 - z Nh Please input a new color center Rez Imz : 1 0 New color center = (1.000000 0.000000) in function_buttons, Nf is pressed. Ex 2.7 a7b z^2 + (1/4 + i/2) - z Nh Ex 2.8 a8c (z^2 - 1)(z^2 + 2) periodic point n=1 z=( infinity ) |(f[1])'(z)|=0.000000 periodic point n=1 z=( 1.1604969072 0.0000000000 ) |(f[1])'(z)|=8.572605 periodic point n=1 z=( -0.8313154850 0.0000000000 ) |(f[1])'(z)|=3.960671 periodic point n=1 z=( -0.1645907111 1.4303878340 ) |(f[1])'(z)|=9.158602 periodic point n=1 z=( -0.1645907111 -1.4303878340 ) |(f[1])'(z)|=9.158602 (z^2 - 1)(z^2 + 2) - z Nh Ex 2.9 a9a2 z + z^2 Ex 2.10 aaa,aaa1,aab1 z - z^3 Ex 2.11 afd1,afa,afa1, afb1,afc1 z^2 + (1/4 + i/2) iz + z^2 g=iz + z^2; h=g[4](z); h Ex 2.12 ag exp(1.1994i)z + z^2 : 0:irrat.n.f.p. s disk (3) Julia Set Ex 3.2 J=Cantor set: 2(z-1/z) Ex 3.3 a9a4 z + z^2 flow pattern and cardioid-shaped region Ex 3.4 b2a3 z^2 Ex 3.5 b3a,b3b r = z + z^2; rrr -1.3076266489 0.5709939970 : rpp of 3 r_3 = z + 3z^2 + 6z^3 + 9z^4 + 10z^5 + 8z^6 + 4z^7 + z^8 Ex 3.6 b4a, b4b r = z^2 + i r_2 = z^4 + 2iz^2 + (-1+i) r = z^2 + i; rr Ex 3.7 b5a, b5a1, b5a2, b5a4, b5b z^3 (1-z/4) / (z-0.25) : 4 a. comps f=z^3 (1-z/4) / (z-0.25); ff Ex 3.8 b6a,b6c,b6d,b6e,b6f,b6g f=(5 z^6 - 6 z)/(4z^6 - 5); f[45] f=(5 z^6 - 6 z)/(4z^6 - 5); ff Ex 3.9 b7a, b7a5 z/(1-z)^2 0:rnfp f=z/(1-z)^2; g=(z-1)/(z+1); h=(1+z)/(1-z); hfg f=z/(1-z)^2; ff Ex 3.10 b8 J=C* (z^2 + 1)^2/(4z(z^2-1)) (4) the Structure of Fatou Set Ex 4.1 a rationally neutral cycles (z-1)/(z^2 + z) r = (z-1)/(z^2 + z); rr r = (z-1)/(z^2 + z); g = r(r(z)); g g = (-z^4 -z^3-z^2 -z)/(z^3+z^2-3z+1) all r=(z-1)/(z^2 + z); g=(z-1)/(z+1); h=(1+z)/(1-z); hrg (z-i)/(z^2 + z) (z + 0.70710678 + 0.70710678i)/(z^2 + z) Siegel cycle of length 2 Ex 4.2 b 1.+(.3419079 + .25169029 i)z -(1.3419079 + .25169029 i) z^2 0 <--> 1. Ex 4.3 c Attracting periodic components of period 6: 1.4142136*((2+i)z^2+(-1+i)z+(-1+i))/(z^4+(-2+2i)z^3-4iz^2+(2+2i)z-1) Ex 4.4 d attr. cycle of 5, 2 a.f.p. exp(1.1994 i) z^2 (z+1) (z-5i)/((1-1.5z)(1+5iz)) Ex 4.5 e superattr. 3-cycle: 0->-i->infty->0 1/((1-z)(z+i)) f=1/((1-z)(z+i)); g=(z-1)/(z+1); h=(1+z)/(1-z); hfg Ex 4.6 f -z + i z^4 pp_f[2](z) = ( ((0.000000e+00) + (1.000000e+00)i)z^16 + ((-4.000000e+00) + (0.000000e+00)i) z^13 + ((-0.000000e+00) + (-6.000000e+00)i)z^10 + ((4.000000e+00) + (0.000000e+00)i)z^7 + ( (1.000000e+00) + (0.000000e+00)i)z^1 ) z + 4z^7 -6i z^10 -4z^13 +i z^16 Ex 4.7 g z + z^2 - 0.5 z^3 = 2 - (z-2) -2 (z-2)^2 - 0.5 (z-2)^3 : 2:nfp--2 pedals 0:nfp--1 pedal g = -z-2z^2-0.5z^3; gg Ex 4.8 h Steinm p.103 exp(7.0248147 i) z^2 (z-4) /(4z-1) : H. ring (5) critical points Thm 5.1 Ex 5.1 a 1/z+z+0.0001z^7 : 5 a.f.p. [crit pt] --> 7 crit pts Thm 5.2 Ex 5.2 (Ex 4.1) b (z-1)/(z^2 + z) f = (z-1)/(z^2 + z); g=(z-1)/(z+1); h=(1+z)/(1-z); hfg Ex 5.3 c -z + i z^4 r = -z + i z^4; rr Ex 5.4 d z + z^4 Cor 5.3 Ex 5.5 e 1/(z^2 -3) - (z-1)/(1-5z) : 1 attr. 6-cycle 1 para. 2-cycle r = 1/(z^2 -3) - (z-1)/(1-5z);g=(z-1)/(z+1); h=(1+z)/(1-z); hrg 2 a.f.p. N_a + N_rn = 3 + 1 = 4 = 2d -2, where d=3; Cor 5.3 sharp Ex 5.6 f z^2/(1-z) f=z^2/(1-z);g=(z-1)/(z+1); h=(1+z)/(1-z); hfg Ex 5.7 g z(z-1)(z-i) 1 r.n. cycle, 2 a. cycles 3 crit. pts. Ex 5.8 (Ex 4.7) h z + z^2 - 0.5 z^3 : 2:nfp--2 pedals 0:nfp--1 pedal 2 crit pts in immed. petals corresp. to 0 and 2 Ex 5.9 (Ex 2.11) i iz + z^2 Ex 5.10 j z+z^2+iz^3 Thm 5.4 Ex 5.11 k p.83 0:rnfp 1/3:afp z/(z^2 - 0.5 z + 1) Ex 5.12 l p.86 exp(3.883222078 i) (z - z^2/2) 1: crit pt. Ex 5.13 m exp(1.1994i) z^2 (z + 2)/((1-z)(z+i)) s disks 0:s.a.f.p. r=exp(1.1994i) z^2 (z + 2)/((1-z)(z+i)); g=(z-1)/(z+1); h=(1+z)/(1-z); hrg Ex 5.14 n exp(3.883222078 i)z - 0.1 z^4 Siegel disk around 0 3 crit pts are on the bdry of the Siegel disk. all Ex 5.15 o (z^2 - z)/(1+ (1+i)z) Petal, S. disk f=(z^2 - z)/(1+ (1+i)z); g=(z-1)/(z+1); h=(1+z)/(1-z); hfg Ex 5.16 (Ex 4.2) p Siegel cycle of length 2 1.+(.3419079 + .25169029 i)z -(1.3419079 + .25169029 i) z^2 0 <--> 1. Ex 5.17 q exp(1.1994i)z(z+(-0.5+0.8660254i))(z-i)/((1-z)(z+i)) s disk r=exp(1.1994i)z(z+(-0.5+0.8660254i))(z-i)/((1-z)(z+i)); g=(z-1)/(z+1); h=(1+z)/(1-z); hrg [125] critical point : z=( 0.5902846372 0.6698067766 ) -> bdry of s.disk at 1 critical point : z=( -0.4592491785 -0.7565440929 ) in preimage of s.disk at 1 critical point : z=( 0.8853701320 -1.5321477133 ) in a. 6-cycle critical point : z=( -3.0164055906 -0.3811149704 )->bdry of s.disk at -1 2 s. disks; a. 6-cycle Ex 5.18 r exp(1.1994 i) z^2 (z-5i)/(1+5iz) (Herman ring) crit pt : 2.3797958971 i, 0.4202041029 i, 0, infty Ex 5.19 s exp(3.883222078 i)z((z+5)/(1+5z))((1+5iz)/(z-5i)) r=exp(3.883222078 i)z((z+5)/(1+5z))((1+5iz)/(z-5i)); g=(z-1)/(z+1); h=(1+z)/(1-z); hrg conj.; 1 H. ring, 2 s. disks critical point : z=( 1.0204024695 0.1896548616 ) --> S.disk at 1 critical point : z=( 0.3622971089 -0.0876140453 )--> r.bdary of H.ring critical point : z=( -0.3622971089 -0.0876140453 )-> l.bdary of H.ring critical point : z=( -1.0204024695 0.1896548616 )--> S.disk at -1 Ex 5.20 t g = exp(1.1994 i) z^2 (z-5i)/(1+5iz);f=exp(7.0248147 i) z^2 (z-4) /(4z-1); fg critical point : z=( -0.3555234765 0.2947748409 ) -> inner bdry of H. ring critical point : z=( -1.6668615254 1.3820433061 ) -> outer bdry of H. ring Ex 5.21 u g = exp(1.1994 i) z^2 (z-5i)/(1+5iz); f=exp(i)z((z-4)/(1-4z))((1+4iz)/(z-4i)); gf critical point : z=( 0.2403329734 0.0298262007 ) -> onter bdry critical point : z=( 0.1805284172 0.0891936667 ) critical point : z=( -0.5053466850 0.0058462856 ) inner bdry critical point : z=( 0.6348894321 -0.1245084484 ) -> inner curl critical point : z=( 1.5167444663 -0.2974494306 ) -> curl Ex 5.22 v g = exp(3.883222078 i)z((z+8)/(1+8z))((1+8iz)/(z-8i)); f=exp(6.7382901 i) z^2 (z-4) /(4z-1); gf critical point : z=( 0.5470655771 0.00 ) -> inner bdry of H. ring critical point : z=( 1.8279344229 0.00 ) -> outer bdry of H. ring Thm 5.5 Ex 5.23 w J=C* 1-2/z^2 F -- 0 compos (6) the Program \fbox{Iterate} z^3 (1-z/4) / (z-0.25) \fbox{Show} z^3 (1-z/4) / (z-0.25) \fbox{Iterate} f = i(z-1) + (z-1)^2; g = z + 1; fg \fbox{Iterate} g = iz + z^2; h=g[4](z); h \fbox{Iterate} f = z^2; g = 3 f(2z); h = g(f(z))/4; k = z + 1; kh \fbox{Show} f = ((z-0.5)/(1-z/2))^2; f[12] \fbox{Show} f = z^2 + z + 1; f[1] \fbox{Show} z^2 + z + 1 \fbox{Iterate} f=(z-1)/(z^2 + z); g=(z-1)/(z+1); h=(1+z)/(1-z); hfg \fbox{Iterate} f = -z + i z^4; ff \fbox{Show} f = -z + i z^4; f[2] \fbox{Iterate} g=(z-0.5)/(1-z/2); f=exp(0)(g(z))^2-(cos(1)^2+sin(1)^2)z; f \fbox{Iterate} ((z+1)/(z-1))((z-1)/(z+2)^2)