bertalan Ágnes - a mandelbrot halmaz.pdf

7/23/2019 Bertalan gnes - A Mandelbrot halmaz.pdf

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z (az+ b)/(cz+ d) d/c a/c

C

S

A

B

N

A

B


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zn z

zn z

z= zn =

zn

|zn|

D C

F

g: D C

F gn F

D C F

f :D S2 S2

S2 ={x,y,z:x2+y2+z2 = 1}

F

gkn

F

f(z) = zn

f

zn

F

gn gkn


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g

gn gkn gkn g g

gkn

g F

|g |K F

f0(z) =id

f1(z) =f(z)

f2(z) =f f(z) fn(z) =

f f .... f n

(z)

fn

f : C C

z0

fn

z0 F(f) z0

J(f) = C\F(f)

f(z) = z2

f

f

fn(B(0, r)) =B(0, r2

n

)

f

B(0, 1)

F(f)

C

=J(f) F(f) f


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f(z) = z2 +c

c C

c =

0.122565+0.744864i

z0 = g(z) = 1f( 1

z)

J

J(f)

f(J(f)) =f1(J(f)) =J(f)

z0 f

f(z0) z0=

z0 = g(z) =

1f( 1

z)

z0

f

f(z0) f z0

z0


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||< 1

||> 1

z0 f

(fk)(z0) z0=

z0

||< 1

= 0

||> 1

D0 = B(z0, ) > 0 D1 f

1(B(z0, ))

z0 Dn

f1(Dn1) z0

D0 D1.... Dn1Dn

D=

n=0

Dn D z0

D

D F(f) D J(f)

z0 z0 = f(z0) =

f(z0) = 0

z0

z0


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fn(z0) = z0

fn

f

n

f(z) =z+ a2z2 + a3z

3 + ...

|| ={0, 1}

: S C

(0) = 0 w =(z)

f 1 :w w

f(z) =n=kanzn k 2 ak = 0

: S C

(0) = 0

w = (z)

f1 : w wn

f(z)

d 2 K(f)

K(f)

z

K(f)

f(z) =zk +ak1z

k1 +...+a1z+a0

C:= 2 + 2(|ak1| + |ak2| +...+ |a1| + |a0|) f

{fn(z0)} z0 K(f)

{fn(z0)}

fn

n

n N {0}

|fn(z0)| C

n N {0} |fn(z0)|> C


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fn C n

|fn|

C

C =C

|z0|> C

|f(z0)|= |zn0 +an1z

n10 +...+a1z0+a0|

|zn

0 | |zn10 |(|an1| + |an2| +...+ |a1| + |a0|)

|zn10 |{|z0| (|an1| + |an2| +...+ |a1| + |a0|)}

2|zn10 | 2|z0|

|fk(z0)| 2k|z0| 2

kC

|fl(z0)| > C |fl+k(z0)| > 2

kC

fn

n

z0 C

z0

K(f)

f(z) =zn +an1z

n1 +...+a1z+a0

C := 2 + 2(|an1|+ |an2|+ ... + |a1|+ |a0|) f

z0 K(f)

z0

C

K=J(f)

J(f) K

int(K) F(f) C \ K=ext(K)

z0 ext(K) fn


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ext(K) F(f)

F(f) J(f) = C

J(F) K

KJ(f)

z0 K z0 z w

|fn(z)| K fn(w) K

F(f)

KJ(f)

J(f) K

KJ(f)

K=J(f)

f

J(f)

K(f)

f


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M C M={c C|xn= x

2n1+ c }

M

xn = x

2n1+ c

fc : C C f(z) = z

2 +c

c C


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fc(z) = z

2 +c

c C

fc

fc

z3

, z4

...


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f(x, y) =x2 y2 +c1 g(x, y) = 2xy+c2 x,y,c1, c2 R

f(x, y) =x2 y2

g(x, y) = 2xy+ 0.9x2


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z20+ c

0,

0

f(x, y) +

g(x, y) +

, R

f(x, y) = x2 y2

g(x, y) = 2xy


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g(x, y)

f(x, y)

g(x, y) = 2xy +x2+

0

f(x, y)

g(x, y)

2x 2y

2y+ 2x 2x

4x2+4y2+4xy= 4(x2+y2+xy) = 4(x+

2y)2(

2

41)y2

= 0

= 2

>2

(0, 2)

= 0.9

2x 2y

2y+ 1.8x 2x

4((x + 0.45y)2 + (1 0.452)y2)

x= y = 0

(0, 0)

(, )

(, )


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= 0

= 0.3

= 0.5 = 0.6

= 0.9

= 1.2

= 1.5 = 2


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bertalan Ágnes - a mandelbrot halmaz.pdf

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