é• h ÷Œ³¹â

«ª¹—Õ¡È¦h¸åê¢—ÝsË•äK'f'fbCpÑf

vÑ

2007.3.24 NHK‡ »ó¿ü

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

1/1

¹¹’ ãOko’ãM_Dpfgo¹

I x2 − 2 = 0kˆOú( ⇐⇒F‹Hp&x2 = 2)I 2x3 − x2 + 2x − 5 = 0
I x100 − 12x2 − 1 = 0

SŒ‰n¹,knã’hfB•_Dm = 2, 3, 4, . . . k_DW
am xm + am−1 xm−1 + · · · + a1 x + a0 = 0 (am 6= 0)

hDFbn¹’

Tomoki Kawahira (Nagoya University)

Hf•ˆF

Cosmos & Chaos

NHK Culture Center 20070324

2/1

¹’ ãOkopfn‹ û¬¦¹kˆŒp&¬¦¹ Carl Friedrich Gauss, 1777-1855 o!nš ’<W_ãpfnú,š
m!¹am xm + am−1 xm−1 + · · · + a1 x + a0 = 0 (am 6= 0)oÍ¦¼•g mnpã α1 , α2 , . . . , αm ’ÅZd=⇒ go α1 , α2 , . . . , αm ’wS„kB•‹koiFYŒpˆDK
Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

3/1

¹’ ãOkoãnl¹nã’H‹‚nk ãnlLB‹

ax2 + bx + c = 0 nãnl o√√−b + b2 − 4ac−b − b2 − 4ac
α1 =h α2 =2a2aI m = 2 nhM

2!¹I m = 3 nhM

3!¹o

ëÀÎnl

ax3 + bx2 + cx + d = 0 nãnlaαk = ω k A+ + ω 2k A− − .
3√−1 + 3i 2_`W ω =(i = −1, ω 3 = 1)
2D•†‹«(k = 1, 2, 3)

v
s„u
«2 „
«3u
327c + 2a3 − 9ab
27c + 2a3 − 9ab3b − a2t
A± =±−+54
54
9Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

4/1¹ãnl’ ãOkodeM

I m = 4 nhM‹l4!¹k_DWf‚ Õ§éênl h|pŒhDFˆŠoãÕ LX(Y‹I m ≥ 5 nhMhL<I èI¢üÙëUŒfD‹¬í¢kˆŠ ãnlãnX(êSoãpfnú,š,m!¹×íüÁLÅ•=⇒Tomoki Kawahira (Nagoya University)’ãOkˆŠÝ<UŒfD‹koCosmos & ChaosoX(WjDSãnlån¢NHK Culture Center 20070324

5/1

¹’ ãOko%n¢×íüÁ

I ]‚]‚ãnloykËdngB•FK√
I _WKklˆŠ x = ± 2 ’—‹L þŸ√„ko 2 = 1.41421356 . . . hDF$h]n¾¦LÍ• ]Sg
I¹’ãO ho ]nãnhf’û n _hHp
¹pW150 A jin ¾¦g H‹ShgB‹hãÈWf•ˆFx2 − 2 = 0 kþWI ågo]n¹Õ’ Hf•‹

Tomoki Kawahira (Nagoya University)ŸoCosmos & Chaos

jºiL{4NHK Culture Center 20070324

6/1

¹

› n’ ãOkoBnºgYI ¢¤¶Ã¯ûËåüÈó

Issac Newton

1643–1727
I 1667t±óÖêÃ¸'fYˆk1

û!P

I ]n,n-gH‹KÕ

Tomoki Kawahira (Nagoya University)

Cosmos & Chaospnã•’W•Y¹
nã’û n¾¦gËåüÈóÕ ’zh

NHK Culture Center 20070324

7/1

ËåüÈóÕËåüÈóÕ
IH‰Œ_ m!¹f (x) := am xm + am−1 xm−1 + · · · + a1 x + a0 = 0
k_DW ~Z¢pn°éÕ y = f (x) ’ H‹
I °éÕn y øhn¤¹’‹Œp 'SnãnMnL•K‹

yy = f (x)α1α2OTomoki Kawahira (Nagoya University)

α3x

Cosmos & Chaos

NHK Culture Center 20070324

8/1

ËåüÈóÕËåüÈóÕn¢ë´êºà(x0 , f (x0 ))

I ãα

Dv

I °éÕ

n¹ (x0 , f (x0 )) K‰¥Ú’ M y øhn¤¹’(x1 , 0) hY‹(x1 , f (x1 ))
αx2

å~`cºj$o•K‰jnB‹ ¦ÑDp x0 ’xx1x0 xø I °éÕ n¹ (x1 , f (x1 )) K‰¥Ú’ M y øhn¤¹’(x2 , 0) hY‹.

ØkšQŒpx0 7→ x1 7→ x2 7→ x3 7→ x4 7→ · · · 7→ · · · → α

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

i“i“ÑeONHK Culture Center 20070324)

9/1

ËåüÈóÕËåüÈóÕn¢ë´êºà deMI ËåüÈóÕg—‰Œ‹px0 7→ x1 7→ x2 7→ x3 7→ x4 7→ · · · 7→ · · · → αi“i“ÑeO)f (xn )’€_Y
f 0 (xn )I “cf Snl k$’Í©WfãeYŒpËåüÈóÕn¢ë´êºàhj‹
I ‹f (x) = x2 − 2 nhMool

xn+1 = xn −xn+1 = xn −I ‹x2n − 2x2 + 2= n2xn2xn

f (x) = x3 − 1 nhMo
xn+1 = xn −Tomoki Kawahira (Nagoya University)x3n − 1
2x3n + 1
=3x2n
3x2n

Cosmos & Chaos

NHK Culture Center 20070324

10 / 1ËåüÈóÕËåüÈóÕo

D_hHp f (x) = x2 − 2 = 0 k(Wf•‹

l¹ÆÃ×

0
1
2
3
4
5
6$’ x0 = 2 hWfËåüÈóÕ’i
2 +2x2n −2nxn+1 = xn − 2xn = x2x
ˆŠ&nxn2.0000000000000000000
1.5000000000000000000
1.4166666666666666667
1.4142156862745098039
1.4142135623746899106
1.4142135623730950488
1.4142135623730950488

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos√xn h 2 n¤î
0.586
0.08580.00245
2.12 × 10−6
1.59 × 10−12
8.99 × 10−25
2.86 × 10−49

NHK Culture Center 20070324

11 / 1ËåüÈóÕWKW ‘O¹L&I ËåüÈóÕg—‰Œ‹px0 7→ x1 7→ x2 7→ x3 7→ x4 7→ · · · 7→ · · ·L
IDdg‚ã α kÑeOhoP‰jD³óÔåü¿ügªD‹’rhd‹f•~W‡FI go

α k_DW in ¦ÑOk x0 ’hŒp—‰Œ‹n`•FKI=⇒ ¶Áo

Tomoki Kawahira (Nagoya University)nˆFjp Lpg‹_{FL•KŠ„YD

Cosmos & Chaos

NHK Culture Center 20070324

12 / 1ËåüÈóÕa‡ch`QŸpp’©ÒW~W‡FpôÚ =⇒Tomoki Kawahira (Nagoya University)

ppsb
Cosmos & Chaos

i2 = −1
NHK Culture Center 20070324

13 / 1ËåüÈóÕ2!¹f (x) = x2 − 2 = 0 n4√

I ±2 ’ãk‚d¹f (x) = x2 − 2 = (x −√2)(x +√2) = 0

x2n − 2f (x)=x−nf 0 (xn )2xn√√
I − 2 + 2kÎ_Y‹ ß: Ÿpn4
pn4I ]nËåüÈóÕ

xn+1 = xn −に収束する部分に収束する部分どちらにも収束しない

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

14 / 1ËåüÈóÕ,2!¹±üêünšn4.

±üêünšp α, β ’ãkに収束する部分‚d2!¹f (x) = x2 − px + qに収束する部分= (x − α)(x − β) = 0kdDf
どちらにも収束しないËåüÈóÕL
Ïα kÎ_Y‹o α h β n‚ô2I
Tomoki Kawahira (Nagoya University)ß D(α) h Ïβ kÎ_Y‹ ß D(β)ÚgIrUŒ‹Cosmos & Chaos

NHK Culture Center 20070324

15 / 1ËåüÈóÕ

3!¹n4‚!X]Fj‹I 3!¹f (x) = x3 − 1= (x − 1)(x − ω)(x − ω 2 ) = 0
I ão 1, ω, ω 2_`W√−1 − 3 i−1 + 3 i2, ω =.
ω=22
√I ËåüÈóÕI goxn+1 = xn −x3n − 1f (xn )=x−nf 0 (xn )3x2n

D(1), D(ω), D(ω 2 ) nbo

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

16 / 1ËåüÈóÕ

D(1), D(ω), D(ω 2 ) nb
2!¹K‰^¨Y‹h&WKW Ÿ›ohjcfW~FI³óÔåü¿g0è~gsWO‹f•~W‡FI Snþa’¬Y‹ko i“jTomoki Kawahira (Nagoya University)ÖLÅ•`•FK

Cosmos & Chaos

NHK Culture Center 20070324

17 / 1›fûn ÖËåüÈóÕ

¢pnÍ©

I m!¹f (x) = am xm + am−1 xm−1 + · · · + a1 x + a0 = 0f (xn )f 0 (xn )I ]nËåüÈóÕ

xn+1 = xn −I °WD¢p’š©

Nf (x) := x −I xn+1 = Nf (xn ) ˆŠf (x)f 0 (x)ËåüÈóÕoNfNfNf

Nf

Nf

x0 7−→ x1 7−→ x2 7−→ x3 7−→ x4 7−→ · · ·h¢p Nf (x) ’Í©Wf

Tomoki Kawahira (Nagoya University)Y‹ShkþÜ

Cosmos & Chaos

NHK Culture Center 20070324

18 / 1›fûn Ö, Y‹h&I m!

f (x) kþWNf (x) := x −f (x)xf 0 (x) − f (x)[m! ]==00f (x)f (x)[(m − 1)! ]am xm + · · · a1 x + a0
hDFbn¢pbm xm + · · · b1 x + b0
¢phˆpŒ‹ ’ H‹
WfF (x) =I SŒ’Í©

Y‹hI,FFF

Fx0 7−→ x1 7−→ x2 7−→ x3 7−→ · · ·hDFp
I Snp

LO+o•K‰jD)

L—‰Œ‹’Tomoki Kawahira (Nagoya University)$ x0 n F kˆ‹ÌShˆv

Cosmos & Chaos

NHK Culture Center 20070324

19 / 1›fûn Ö›fûn

L³FFFFx0 7−→ x1 7−→ x2 7−→ x3 7−→ · · ·LO+o•K‰jD)

SFWf—‰Œ‹ÌS_a’!nˆFkãÈWf•ˆFz“⇐⇒B“⇐⇒KÕÕG ⇐⇒I SŒoXI SnˆFjv

UŒ_phS psb
0, 1, 2, 3, . . .
xn+1 = F (xn )LnâÇëgB‹

D•’ ¢p F kˆ‹ CTomoki Kawahira (Nagoya University)

Cosmos & Chaosn›fûhˆ

NHK Culture Center 20070324

20 / 1Õ¡È¦h¸åê¢

1918t°éó×êI Ñt³óÔåü¿ünzUkˆŠ›fûn ÖoWWOz

UI WKW]n
I 1915t
Io³óÔåü¿üåMkÉKŒfD_Õéó¹Ñf¢«Çßüo!nøÞOL’zhÍ©

I Œºnpfkˆ‹›fûkdDfn'ß„ÖnŒÕ¡È¦h¸åê¢n‰Dhjc_

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

21 / 1Õ¡È¦h¸åê¢)‡f g‚Bc_ûÕ¡È¦Ô¨üëûÕ¡È¦(Pierre Fatou, 1878–1929)
I Õéó¹qèn:íê¢ók~Œ‹I 1901tK‰¡t~gÑê)‡ðkäÙI ]n•‰pfn

ã•IMv’šQ_Ö)‡fkJDfUWjm>

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

22 / 1Õ¡È¦h¸åê¢'&nñÄû¸åê¢¬¹Èóû¸åê¢(Gaston Julia, 1893–1978)
I Õéó¹q»
I ,!

n¢ë¸§ê¢ ~Œ

L'&gÍºhWf;•I SnhM

TbkñD·’D ;’1c_I

Tomoki Kawahira (Nagoya University)

ã•
Îkm>

Cosmos & Chaos

pÖ¢pã•ji EƒDNHK Culture Center 20070324

23 / 1Õ¡È¦h¸åê¢°éó×ênPœoI 1917t¸åê¢oÖ‡’¢«Çßük ØI ]nŒÕ¡È¦L¸åê¢h^<nPœ’Comptes RendusŒkzhI ¸åê¢oComptes RendusŒJˆs¢«ÇßükþW—pI P@&II°éó×êo¸åê¢ ÞÑ3000ÕéóÕ¡È¦nm>‚ðH‰Œ ÞÑ2000ÕéóL

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos‰Œ_NHK Culture Center 20070324

24 / 1Õ¡È¦h¸åê¢Õ¡È¦h¸åê¢n ÖgoÕ¡È¦h¸åê¢nÖ’'ÑŠk‹f•~W‡FI ›fûoÕ¡È¦Æ‰šjè‰šjèé•=³¹â
÷Œ=«ª¹ k2 UŒ‹I Õ¡È¦Æ‰šèQŒpÌSo{h“iI ‹$LÖìf‚ ÖìLUf (x) = x3 − 1 nËåüÈóÕn D(1), D(ω), D(ω 2 ).

I ¸åê¢ÆÌSLÀI ‹hoWjDèh¸åê¢Æ‰šèWfW~FèhogD(1), D(ω), D(ω 2 ) å

Tomoki Kawahira (Nagoya University)$LWg‚Öì‹hnè

Cosmos & Chaos

NHK Culture Center 20070324

25 / 1Õ¡È¦h¸åê¢Õ¡È¦h¸åê¢nšÕ¡È¦h¸åê¢oìËk!nš’:W_Õ¡È¦h¸åê¢nšÕ¡È¦Æ‰šè2‚WOo!PrUŒ‹ShojDo¸åê¢Æ‰šèkˆcf1nè k rUŒ‹ yk a‡Fi3dkSŒoOKg‚SŒoÀágに収束する部分

SFjcfW~Fに収束する部分

どちらにも収束しない

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

26 / 1

x2 + c kˆ‹›fûhÞóÇëÖíüÆF (x) = x2 + c K‰UŒ‹¸åê¢Æ _ac n$’ H‹h¸åê¢Æ nb‚ •‹Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

27 / 1

x2 + c kˆ‹›fûhÞóÇëÖíüÆÞóÇëÖíüÆ¸åê¢Æn«¿í° ÞóÇëÖíüÆ kdDf

Tomoki Kawahira (Nagoya University)

Cosmos & Chaos

NHK Culture Center 20070324

28 / 1

x2 + c kˆ‹›fûhÞóÇëÖíüÆ

BŠLhFTVD~W_Â ‡.IIIÕé¯¿ënŽ›fûn¤áü¸ H.O.Ñ¤È²óÖó¬üûÕ§¢éü¯q¬Õé¯¿ënLe€û›fû ‡wÍƒ å,UÖ>

I Án¦§ÖÚü¸

http://www.math.nagoya-u.ac.jp/ e kawahira(W_×í°éà„uLJDfBŠ~YTomoki Kawahira (Nagoya University)·å×êCosmos & Chaos

Êå•NHK Culture Center 20070324

29 / 1