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IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2018

Fractal Sets: Dynamical, Dimensional and Topological Properties

NANCY WANG

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2018

Fraktalmängder: Dynamiska, Dimensionella och Topologiska Egenskaper

NANCY WANG

KTHSKOLAN FÖR TEKNIKVETENSKAP

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�M �Hi2`M�iBp2 �M/ Q7i2M mb27mH K2i?Q/ iQ �TT`QtBK�i2 i?2 }t2/ TQBMib Q7� 7mM+iBQM Bb i?2 ;`�T?B+�H K2i?Q/- r?B+? Bb /QM2 #v bim/vBM; i?2 BMi2`b2+iBQM#2ir22M i?2 ;`�T? Q7 7mM+iBQM �M/ i?2 HBM2 y = xX h?Bb K2i?Q/ /Q2b MQi ;Bp2 i?22t�+i p�Hm2- #mi Bi ;Bp2b � ;QQ/ B/2� Q7 i?2 BMi2`p�H �i r?B+? i?2 }t2/ TQBMi HB2bXh?2 �TTHB+�iBQM Q7 i?2 ;`�T?B+ K2i?Q/ Q7 G(x) Bb BHHmbi`�i2/ BM 6B;m`2 RX

N

.2}MBiBQM kX amTTQb2 x0 Bb � }t2/ TQBMi Q7 F (x)- r?2`2 F : R æ R Bb � bKQQi?7mM+iBQMX

Y__]

__[

A7 |F Õ(x0)| < 1, i?2M x0 Bb �M �ii`�+iBM; }t2/ TQBMi .

A7 |F Õ(x0)| > 1, i?2M x0 Bb � `2T2HHBM; }t2/ TQBMi .

A7 |F Õ(x0)| = 1, i?2M x0 Bb � M2mi`�H }t2/ TQBMi .

h?2Q`2K RX 6Q` � bKQQi? 7mM+iBQM F : R æ RX amTTQb2 x0 Bb �M �ii`�+iBM; }t2/TQBMi Q7 F - i?2M i?2`2 Bb �M BMi2`p�H I- bm+? i?�i B7 x œ I- i?2M F n(x) œ I ’ n �M/F n(x) æ x0 �b n æ ŒX aBKBH�`Hv- bmTTQb2 x0 Bb � `2T2HHBM; }t2/ TQBMi Q7 F - i?2Mi?2`2 Bb �M BMi2`p�H I- bm+? i?�i B7 x Bb BM i?2 BMi2`BQ` Q7 I �M/ x ”= x0- i?2M i?2`2Bb �M BMi2;2` n > 0 bm+? i?�i F n(x) /œ IXS`QQ7X �bbmK2 x0 Bb �M �ii`�+iBM; }t2/ TQBMi Q7 F - r?2`2 F : R æ R Bb � bKQQi?7mM+iBQMX h?2M |F Õ(x0)| < 1 #v /2}MBiBQMX G2i a #2 � MmK#2` bm+? i?�i |F Õ(x0)| <a < 1X q2 +�M +?QQb2 � M2B;?#Qm`?QQ/ Q7 x0- Ia = [x0 ≠ ‘, x0 + ‘]- 7Q` bQK2 ‘ > 0bm+? i?�i |F Õ(x)| < a ’ x œ IaX JQ`2Qp2`- #v i?2 J2�M o�Hm2 h?2Q`2K- r2 ?�p2

|F Õ(x)| = limxæx0

|F (x) ≠ F (x0)||x ≠ x0|

< a (’ x œ Ia �M/ x ”= x0).

"v i?2 /2}MBiBQM Q7 }t2/ TQBMi- r2 +�M r`Bi2

|F (x) ≠ F (x0)| = |F (x) ≠ x0| < a|x ≠ x0|.

aBM+2 0 < a < 1- F (x) Bb +HQb2` iQ x0 i?�M xX JQ`2Qp2`- i?Bb Bb p�HB/ 7Q` �`#Bi`�`vx œ Ia- ?2M+2 r2 FMQr F (x) œ IaX LQr bBM+2 an < a 7Q` �`#Bi`�`v BMi2;2` n > 1-r2 +�M +QMiBMm2 i?Bb �`;mK2Mi 7Q` F 2- F 3 . . . �M/ Q#i�BM

|F n(x) ≠ x0| < an|x ≠ x0|. URV

q2 b22 i?�i �b n æ Œ- an æ 0- ?2M+2 F n(x) æ x0XaBKBH�`Hv- �bbmK2 x0 Bb � `2T2HHBM; }t2/ TQBMi Q7 F - i?2M |F Õ(x0)| > 1 #v

/2}MBiBQMX G2i b #2 � MmK#2` bm+? i?�i 1 < b < |F Õ(x0)|X q2 +�M +?QQb2 �M2B;?#Qm`?QQ/ Q7 x0- Ib = [x0 ≠ ‘, x0 + ‘]- 7Q` bQK2 ‘ > 0 bm+? i?�i |F Õ(x)| >b ’ x œ IbX JQ`2Qp2`- #v i?2 J2�M o�Hm2 h?2Q`2K- r2 ?�p2

|F Õ(x)| = limxæx0

|F (x) ≠ F (x0)||x ≠ x0|

> b (’ x œ I �M/ x ”= x0).

>2M+2|F (x) ≠ x0| > b|x ≠ x0|.

aBM+2 b > 1- r2 b22 i?�i F (x) HB2b 7mì?2` 7`QK x0 i?�M t- Bi 7QHHQrb i?�i �b n æ Œ-bn æ Œ- ?2M+2 i?2`2 Bb � BMi2;2` n > 0 bm+? i?�i F n(x) /œ IbX

Ry

LQi2 i?�i i?2`2 Bb � TB2+2 Q7 �//BiBQM�H BM7Q`K�iBQM i?�i r2 +�M 2ti`�+i 7`QK i?2T`QQ7 �#Qp2X L�K2Hv i?2 `�i2 �i r?B+? � }t2/ TQBMi �ii`�+if`2T2H M2�`#v TQBMibX6`QK 2[m�iBQM R- r2 b22 i?�i i?2 `�i2 �M �ii`�+iBM; TQBMi �ii`�+i Bib M2�`#v TQBMibBb an �i n@i? Bi2`�iBQM- r?2`2 |F Õ(x0)| < a < 1X AM i?Bb +�b2- i?2 7mM+iBQM F n(x) Bbb�B/ iQ +QMp2`;2 2tTQM2MiB�HHv iQ x0 �b n æ ŒX �M�HQ;QmbHv- b ;Bp2b BM7Q`K�iBQM�#Qmi i?2 `�i2 �i r?B+? � `2T2HHBM; }t2/ TQBMi `2T2H TQBMib BM i?2 M2B;?#Qm`?QQ/Q7 x0X

RXkXk S2`BQ/B+ P`#Bi.2}MBiBQM jX � TQBMi x0 Bb T2`BQ/B+ rBi? T`BK2 T2`BQ/ n B7 n Bb i?2 bK�HH2biTQbBiBp2 BMi2;2` b�iBb}2b F n(x0) = x0X h?2 T2`BQ/B+ Q`#Bi Q7 x0 rBi? T`BK2 T2`BQ/n Bb i?2 b2[m2M+2 {x0, F (x0), F 2(x0), . . . , F n≠1(x0), x0, F (x0), F 2(x0), . . . }X

6Q` 2t�KTH2- i?2 TQBMi 0 Bb T2`BQ/B+ mM/2` i?2 K�TTBM; Q7 G(x) = 1 ≠ x2X Ai?�b � T2`BQ/B+ Q`#Bi {0, 1, 0, 1, . . . } rBi? T`BK2 T2`BQ/ 2X q2 +�M �HbQ b�v i?�i 0�M/ 1 7Q`K � 2@+v+H2X Ai b?QmH/ #2 MQi2/ i?�i B7 x0 Bb T2`BQ/B+ rBi? T`BK2 T2`BQ/n- i?2M F mn(x0) = x0 7Q` �Mv TQbBiBp2 BMi2;2` mX JQ`2Qp2`- �HH i?2 TQBMib rBi?BMi?2 T2`BQ/B+ Q`#Bi ?�p2 T`BK2 T2`BQ/ nX q2 rBHH b22 BM i?2 7Qì?+QKBM; +?�Ti2ì?�i Bi Bb Q7i2M /B{+mHi iQ }M/ i?2 T2`BQ/B+ Q`#Bi 2p2M 7Q` � 7�B`Hv bBKTH2 7mM+iBQMX

aBKBH�` iQ i?�i Q7 �M 2p2Mim�HHv }t2/ TQBMi- i?2`2 2tBbi TQBMib i?�i �`2 2p2M@im�HHv T2`BQ/B+X � TQBMi x0 Bb 2p2Mim�HHv T2`BQ/B+ rBi? T2`BQ/ n B7 F (x0) ”= x0 #mii?2`2 2tBbi � TQbBiBp2 BMi2;2` m > 0 bm+? i?�i F n+i(x0) = F i(x0) 7Q` �HH i Ø mX

"27Q`2 KQpBM; QM iQ M2ti b2+iBQM- r2 MQr BMi`Q/m+2 � i?2Q`2K r?B+? rBHH #2?2HT7mH r?2M +QKTmiBM; i?2 /2`Bp�iBp2 �i � T2`BQ/B+ TQBMiX

h?2Q`2K kX amTTQb2 i?2 b2i {x0, x1, . . . , xn≠1} Bb � T2`BQ/B+ n@Q`#Bi Q7 F (x)- r?2`2F : R æ R Bb � bKQQi? 7mM+iBQMX h?2M

(F n)Õ(x0) = F Õ(x0)F Õ(x1)·, . . . , ·F Õ(xn≠1)F Õ(xn≠1).

S`QQ7 X G2i i?2 b2i {x0, x1, . . . , xn≠1} /2MQi2 i?2 T2`BQ/ n@+v+H2 Q7 F : R æ RX6 #2BM; bKQQi? ;m�`�Mi22b i?2 2tBbi2M+2 Q7 i?2 /2`Bp�iBp2b Q7 F k(x) 7Q` k =

RR

0, 1, . . . , n ≠ 1X "v i?2 +?�BM `mH2- r2 ?�p2

(F n)Õ(x0) = ddx0

[F (F n≠1(x0))]

= F Õ(F n≠1(x0)) · ddx0

F n≠1(x0)

= F Õ(F n≠1(x0)) · F Õ(F n≠2(x0)) · ddx0

F n≠2(x0)XXX

= F Õ(F n≠1(x0)) · F Õ(F n≠2(x0)) · · · · · F Õ(F (x0)) · F Õ(x0)= F Õ(xn≠1) · F Õ(xn≠2) · · · · · F Õ(x1) · F Õ(x0)

RXj "B7m`+�iBQMq2 rBHH b22 BM i?2 M2ti +?�Ti2`b- � p2`v BMi2`2biBM; +?�`�+i2` Q7 � 7�KBHv Q7 QM2@/BK2MbBQM�H /vM�KB+ bvbi2Kb Bb i?2 /2T2M/2M+2 QM i?2 T�`�K2i2`X AM T�ìB+mH�`-� }t2/ TQBMi +�M #2 +`2�i2/ Q` /2bi`Qv2/- i?2B` bi�#BHBiv +�M #2 +?�M;2/- �b i?2T�`�K2i2` +?�M;2bX

.2}MBiBQM 9X � #B7m`+�iBQM Bb i?2 +?�M;2 Q7 i?2 [m�HBi�iBp2 bi`m+im`2 Q7 � /vM�KB+bvbi2KX h?2 T�`�K2i2` p�Hm2 �i r?B+? i?2 #B7m`+�iBQM Q++m`b Bb +�HH2/ i?2 #B7m`+�iBQMTQBMiX

GBi2`�HHv- #B7m`+�iBQM K2�Mb � /BpBbBQM BMiQ irQ #`�M+?2b Q` T�ìbX AM /vM�KB+b-#B7m`+�iBQMb +�M #2 /BpB/2/ BMiQ irQ K�BM +�i2;Q`B2b- HQ+�H #B7m`+�iBQM �M/ ;HQ#�H#B7m`+�iBQMX � ;HQ#�H #B7m`+�iBQM `272`b iQ � H�`;2` BMp�`B�Mi b2ib Q7 i?2 bvbi2Kr?BH2 HQ+�H #B7m`+�iBQM `272`b iQ HQ+�H bi�#BHBiB2b Q7 � }t2/ TQBMi +�mb2/ #v i?2p�`B�iBQM Q7 i?2 T�`�K2i2`X AM i?Bb T�T2`- r2 rBHH HBKBi mb iQ HQ+�H #B7m`+�iBQMbX>2`2 r2 BMi`Q/m+2 irQ ivT2b Q7 7mM/�K2Mi�H #B7m`+�iBQM- M�K2Hv i?2 b�//H2@MQ/2#B7m`+�iBQM Q` i�M;2MiB�H #B7m`+�iBQM �M/ i?2 T2`BQ/B+@/Qm#HBM; #B7m`+�iBQMX 6Q` �KQ`2 +QKTH2i2 `2�/BM;- TH2�b2 b22 *?�Ti2` e BM (R) Q` *?�Ti2` 3 �M/ 8 BM (8)X

*QMbB/2` � QM2@T�`�K2i2` 7�KBHv Q7 7mM+iBQMb F⁄- r?2`2 ⁄ Bb � T�`�K2i2` �M/F⁄ Bb � 7mM+iBQM Q7 xX

.2}MBiBQM 8X � 7mM+iBQM F⁄ mM/2`;Q2b � i2M;2MiB�H #B7m`+�iBQM �i i?2 T�`�K2i2`p�Hm2 ⁄0 B7 i?2`2 Bb �M QT2M BMi2`p�H I �M/ �M ‘ > 0 bm+? i?�i F⁄ ?�b,

RX MQ }t2/ TQBMib BM I 7Q` ⁄0 ± ‘ ? ⁄ ? ⁄0XkX QM2 M2mi`�H }t2/ TQBMi BM I 7Q` ⁄ = ⁄0XjX QM2 �ii`�+iBM; �M/ QM2 `2T2HHBM; }t2/ TQBMi BM I 7Q` ⁄0 ? ⁄ ? ⁄0 û ‘X

Rk

.2}MBiBQM eX � 7mM+iBQM F⁄ mM/2`;Q2b � T2`BQ/@/Qm#HBM; #B7m`+�iBQM �i i?2 T�@`�K2i2` p�Hm2 ⁄ = ⁄0 B7 i?2`2 Bb �M QT2M BMi2`p�H I �M/ �M ‘ > 0 bm+? i?�i F⁄ ?�b,

RX � mMB[m2 }t2/ TQBMi p⁄ BM I 7Q` 2�+? ⁄ BM i?2 BMi2`p�H [⁄0 ≠ ‘, ⁄0 + ‘]XkX MQ +v+H2 Q7 T2`BQ/ k BM I �M/ p⁄ Bb �ii`�+iBM; U`2bT2+iBp2 `2T2HHBM;V 7Q`

⁄0 ≠ ‘ ? ⁄ R ⁄0XjX � mMB[m2 �ii`�+iBM; U`2bT2+iBp2 `2T2HHBM;V k@+v+H2 q1

⁄- q2⁄ BM I rBi? F (q1

⁄) =q2

⁄ 7Q` ⁄0 ? ⁄ ? ⁄0 + ‘X J2�Mr?BH2- i?2 }t2/ TQBMi p⁄ Bb `2T2HHBM; U`2bT2+iBp2�ii`�+iBM;VX

9X �b ⁄ æ ⁄0- r2 ?�p2 qi⁄ æ p⁄0 7Q` i = {1, 2}X

Rj

k .vM�KB+b Q7 i?2 GQ;BbiB+ 6�KBHvkXR h?2 GQ;BbiB+ JQ/2HG2i mb MQr b?B7i Qm` �ii2MiBQM iQ i?2 bQ +�HH2/ HQ;BbiB+ KQ/2H Q7 TQTmH�iBQM ;`Qri?#v i?2 2+QHQ;BbibX �bbmK2 � +2ì�BM ivT2 Q7 QM2 +2HH #�+i2`B� /BpB/2 Bib2H7 BMiQ irQ2p2`v ?Qm`X 6Q` �M BMBiB�H TQTmH�iBQM x0- r2 rQmH/ ?�p2 2x0 �7i2` QM2 ?Qm` �M/4x0 �7i2` irQ ?Qm`bX � M�im`�H [m2biBQM iQ �bF Bb r?�i Bb i?2 TQTmH�iBQM �7i2` �HQM; iBK2\

J�i?2K�iB+�HHv- � 7mM+iBQM i?�i /2b+`B#2b i?2 TQTmH�iBQM ;`Qri? �7i2` QM2 ?Qm`Bb f(x) = 2xX h?2 TQTmH�iBQM Bb f(x0) = 2x0 �7i2` QM2 ?Qm` �M/ f 2(x0) = 22x0�7i2` irQ ?Qm`b- �M/ bQ QMX *QMb2[m2MiHv- r2 ?�p2 fn(x0) = 2nx0 �7i2` n ?Qm`bXh?2 7mM+iBQM f(x) /2b+`B#2b- BM/22/- i?2 /2p2HQTK2Mi Q7 i?2 TQTmH�iBQM Qp2` iBK2Xh?Bb ivT2 Q7 ;`Qri? Bb FMQrM �b i?2 2tTQM2MiB�H ;`Qri?- bBM+2 i?2 TQTmH�iBQM;`Qrb �b 2tTQM2MiB�H 7mM+iBQM BM i?2 iBK2 p�`B�#H2 nX

h?Bb Bb Q7 +Qm`b2 MQM@`2�HBbiB+ 7Q` KQ/2HHBM; i?2 `2�H HB72 bBim�iBQM bBM+2 i?2H�#Q`�iQ`v ?�b HBKBi2/ bmTTHB2bX AM `2�HBiv- i?2`2 Bb �Hr�vb �M 2MpB`QMK2Mi�H Q`bmTTHv HBKBi- bBKBH�` iQ i?�i i?2 HBKBi2/ `2bQm`+2b QM i?2 2�ì? +�M MQi bmTTHv �M�`#Bi`�`BHv H�`;2 ?mK�M TQTmH�iBQMX Ai Bb i?2`27Q`2 KQ`2 `2�bQM�#H2 iQ �bbmK2 i?2`2bQm`+2b iQ #2 +QMbi�MiX *QMb2[m2MiHv- i?2`2 2tBbib � K�tBKmK TQTmH�iBQM iQr?B+? i?2 HBKBi2/ `2bQm`+2b +�M T`QpB/2 7Q`X

�M BKT`Qp2/ KQ/2H +�M #2 Q#i�BM2/ #v �//BM; � 7�+iQ` (1 ≠ x) iQ f(x)X h?2BKT`Qp2/ KQ/2H- g(x) = 2x(1 ≠ x)- Bb #QmM/2/ #v R 7Q` �HH iBK2bX �bbmK2 r2+QmMi i?2 #�+i2`B� BM KBHHBQMX 6Q` � bK�HH BMBiB�H MmK#2` x0- b�v � 72r i?Qmb�M/b Q7#�+i2`B�- i?2 7�+iQ` (1≠x0) Bb �TT`QtBK�i2Hv QM2- ?2M+2 g(x0) Bb biBHH �TT`QtBK�i2Hvx0X PM i?2 +QMi`�`v- B7 i?2 BMBiB�H MmK#2` x0 Bb #B;- b�v M2�` � KBHHBQM- i?2 7�+iQ`(1 ≠ x0) Bb �TT`QtBK�i2Hv x2`Q- i?2M g(x0) K�v #2 Km+? bK�HH2` i?�M x0X h?mb- �bi?2 TQTmH�iBQM �TT`Q�+?2b QM2 KBHHBQM- Bi rBHH ;`�/m�HHv /2+�vX

kXk LmK2`B+�H �M�HvbBb Q7 i?2 GQ;BbiB+ 6mM+iBQMPM2 KB;?i �bF r?�i rBHH ?�TT2M iQ i?2 TQTmH�iBQM �7i2` � +2ì�BM HQM; T2`BQ/ Q7iBK2 7Q` /Bz2`2Mi BMBiB�H p�Hm2 x0\ .Q2b Bi +QMp2`;2 iQ �M 2[mBHB#`BmK �M/ B7 bQ-r?�i �`2 i?2 TQbbB#H2 2[mBHB#`B�\ qBi? i?2b2 [m2biBQMb BM KBM/- r2 +QKTmi2 i?2Bi2`�iBQM Q7 g(x) 7Q` /Bz2`2Mi BMBiB�H p�Hm2bX PM 6B;m`2 k- r2 b22 i?2 Q`#Bib 7Q` BMBiB�Hp�Hm2 x = 0.1 �M/ x = 0.8 #Qi? +QMp2`;2 iQ i?2 2[mBHB#`BmK 0.5X P#b2`p2 i?�ii?2 Q`#Bi Q7 i?2 BMBiB�H p�Hm2 #2HQr 0.5 +QMp2`;2b mTr�`/ iQr�`/ 0.5X J2�Mr?BH2-i?2 Q`#Bi Q7 i?2 BMBiB�H p�Hm2 ;`2�i2` i?�M 0.5 DmKTb }`bi iQ #2HQr 0.5 �M/ i?2M+QMp2`;2b mTr�`/ iQr�`/ i?2 2[mBHB#`BmK 0.5X

R9

6B;m`2 k, h?2 }`bi Ry Bi2`�iBQMb Q7 g(x) = 2x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X q2 b22 i?�i i?2 Q`#Bib �`2 +QMp2`;BM; iQ i?2 #H�+F HBM2 x = 0.5 7Q` #Qi? BMBiB�Hp�Hm2bX AM Qi?2` rQ`/b- x = 0.5 Bb �M �ii`�+iBM; }t2/ TQBMiX

LQr bmTTQb2 bQK2 7�+iQ` Bb �HHQr2/ iQ #2 p�`B2/- b�v i?2 i2KT2`�im`2X h?2;`Qri? `�i2 Q7 i?2 #�+i2`B� rBHH i?2M p�`v rBi? i?2 i2KT2`�im`2X �b �M 2+QHQ;Bbi-�MQi?2` M�im`�H [m2biBQM Bb ?Qr /Q2b i?2 TQTmH�iBQM +?�M;2b B7 i?2 ;`Qri? `�i2 Bbp�`B2/\ J�i?2K�iB+�HHv- r2 rQmH/ HBF2 iQ bim/v i?2 7mM+iBQM gµ(x) = µx(1 ≠ x)7Q` p�`vBM; µX 6Q` i?Bb `2�bQM- r2 Bi2`�i2 i?2 7mM+iBQM gµ(x) 7Q` µ > 2 �M/ µ < 2Xq2 +?QQb2- 7Q` BMbi�M+2 µ = 1.5 �M/ µ = 3X h?2 Bi2`�iBQMb �`2 T`2b2Mi2/ BM i?26B;m`2 j `2bT2+iBp2 6B;m`2 9X

R8

6B;m`2 j, h?2 }`bi ky Bi2`�iBQMb Q7 g(x) = 1.5x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X q2 b22 i?�i i?2 Q`#Bib �`2 +QMp2`;BM; iQ x ¥ 0.3333 7Q` #Qi? BMBiB�H p�Hm2bX AMQi?2` rQ`/b- x ¥ 0.3333 Bb �M �ii`�+iBM; }t2/ TQBMiX

6B;m`2 9, h?2 }`bi 8y Bi2`�iBQMb Q7 g(x) = 3x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X q2 b22 i?�i i?2 Q`#Bib �`2 +QMp2`;BM; iQ x ¥ 0.6337 �M/ x ¥ 0.6954 7Q` #Qi?BMBiB�H p�Hm2bX

q2 b22 BM 6B;m`2 j i?�i i?2 2[mBHB#`BmK Bb MQr +?�M;2/ 7`QK yX8 iQ yXjjjXAi `2[mB`2b N Bi2`�iBQMb iQ `2�+? i?2 2[mBHB#`BmK- r?B+? +�M #2 +QKT�`2/ iQ 8Bi2`�iBQMb T`2pBQmbHvX AM �//BiBQM- i?2 +QMp2`;2M+2 Bb bBKBH�` iQ T`2pBQmb +�b2X PMi?2 Qi?2` ?�M/- 7Q` µ = 3 �b b?QrM BM 6B;m`2 9- i?2 Q`#Bib Q7 yXR �M/ yX3 b22K iQ+QMp2`;2 iQ bQK2i?BM;- #mi BM � Km+? bHQr2` b2Mb2 +QKT�`2/ iQ µ = 1.5 �M/ µ = 2X

Re

AM Q`/2` iQ +QM}`K i?2 +QMp2`;2M+2 7Q` µ = 3- r2 Bi2`�i2/ i?2 7mM+iBQM 107 iBK2b�M/ i?2 Q#i�BM2/ p�Hm2b �`2 T`2b2Mi2/ BM h�#H2 RX JQ`2Qp2`- i?2 +QMp2`;2M+2 ?�b� M2r +?�`�+i2` 7Q` µ = 3X L�K2Hv- i?2 Q`#Bi Bb Qb+BHH�iBM; �#Qp2 �M/ #2HQr i?22[mBHB#`BmKX

LmK#2` Q7 Bi2`�iBQM x0 = 0.1 x0 = 0.8107 yXeeed9RR yXeee8NkR

h�#H2 R, Ai2`�iBQM Q7 g(x) = 3x(1 ≠ x)X �7i2` 107 Bi2`�iBQMb- i?2 Q`#Bib Q7 yXR �M/ yX3 �`2#Qi? +QMp2`;BM; iQ ¥yXeeeeX

Ai Bb 2pB/2Mi i?�i i?2 2[mBHB#`BmK +?�M;2b rBi? i?2 ;`Qri? `�i2X >Qr2p2`- r22M+QmMi2`2/ � M2r +?�`�+i2` i?�i i?2 Q`#Bi Qb+BHH�i2 �#Qp2 �M/ #2HQr i?2 2[mBHB#@`BmK 7Q` µ = 3X AM Q`/2` iQ mM/2`bi�M/ i?Bb #2ii2`- r2 Bi2`�i2 i?2 7mM+iBQM gµ(x)7Q` µ H2bb �M/ ;`2�i2` i?�M 3X *?QQbBM; µ = 2.5 �M/ µ = 3.3- r2 T`2b2Mi i?2 Q`#BibBM 6B;m`2 8 �M/ eX

6B;m`2 8, h?2 }`bi 3 Bi2`�iBQMb Q7 g(x) = 2.5x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X q2 b22 i?�i i?2 Q`#Bib �`2 +QMp2`;BM; iQ x ¥ 0.600 7Q` #Qi? BMBiB�H p�Hm2bX AMQi?2` rQ`/b- r2 ?�p2 � }t2/ TQBMi �i x ¥ 0.600

Rd

6B;m`2 e, h?2 }`bi k8 Bi2`�iBQMb Q7 g(x) = 3.3x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X q2 b22 i?�i i?2 Q`#Bib 7Q` #Qi? BMBiB�H p�Hm2b �`2 �TT`Q�+?BM; iQ 7Q`K � k@+v+H2-M�K2Hv {0.4794, 0.8236}X

6`QK 6B;m`2 j �M/ 8- r2 b22 i?�i i?2 +QMp2`;2M+2 7Q` µ = 1.5 �M/ µ = 2.5 `2b2K#H2QM2 �MQi?2`X >Qr2p2`- 7Q` µ = 3.3 Ub22 6B;m`2 eV i?2 Q`#Bib +QMp2`;2 iQ � T2`BQ/B+k@+v+H2X h?Bb K2�Mb � #B7m`+�iBQM ?�b Q++m``2/ 7Q` bQK2 µ œ {2.5 Æ µ Æ 3.3}X G2imb }`bi Tmi �bB/2 i?2 [m2biBQM, 7Q` r?�i µ p�Hm2 /B/ i?Bb brBi+? ?�TT2M �M/ mM/2`r?�i +QM/BiBQM\ AMbi2�/- r2 T`Q+22/ rBi? � 72r KQ`2 Bi2`�iBQMb 7Q` /Bz2`2Mi µ iQ�{`K Qm`b2H7 i?�i i?2 7�i2 7Q` �HH µ > 3.3 Bb � T2`BQ/B+ Q`#BiX hQ /Q i?Bb- r2 +?QQb2µ = 4 �M/ µ = 5 �M/ i?2 `2bmHib Q#i�BM2/ �`2 T`2b2Mi2/ BM 6B;m`2 d `2bT2+iBp2 3X

6B;m`2 d, h?2 }`bi Ryyy Bi2`�iBQMb Q7 g(x) = 4x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X LQ }t2/ TQBMi T�ii2`M Bb Q#b2`p2/- i?2 Q`#Bi Bb bT`2�/ 2p2`vr?2`2 rBi?BM i?2BMi2`p�H [0, 1]

R3

6B;m`2 3, h?2 }`bi 8y Bi2`�iBQMb Q7 g(x) = 5x(1 ≠ x) rBi? BMBiB�H p�Hm2 x0 = 0.1 �M/x0 = 0.8X h?2 Q`#Bi Q7 x0 = 0.1 /Bp2`;2b iQ ≠Œ �7i2` Rj Bi2`�iBQMb- i?2 b�K2 7�i2 BbQ#b2`p2/ 7Q` i?2 Q`#Bi Q7 x0 = 0.8 �7i2` 98 Bi2`�iBQMb

hQ Qm` bm`T`Bb2- i?2 Q`#Bi 7Q` µ = 4 Bb MQ HQM;2` `2;mH�`X Ai /Q2b MQi 7QHHQr�Mv T�ii2`M �M/ i?2 Q`#Bi Bb bT`2�/ 2p2`vr?2`2 BM i?2 BMi2`p�H 0 Æ gµ(x) Æ 1 �bb?QrM BM 6B;m`2 dX JQ`2Qp2`- i?2 Q`#Bi 7Q` µ = 5 /Bp2`;2b iQ ≠Œ Ub22 6B;m`23V X h?2`27Q`2- i?2 +QMp2`;2M+2 �M/ MmK#2` Q7 2[mBHB#`BmK Bb MQi +QMbi�Mi 7Q`µ > 3.3X lM/Qm#i2/Hv- i?2 +QMp2`;2M+2 �M/ MmK#2` Q7 2[mBHB#`BmK p�`B2b rBi? µXh?2 [m2biBQM Bb, ?Qr /Q i?2v p�`v �M/ B7 i?2`2 Bb � T�ii2`M 7Q` /Bz2`2Mi µ p�Hm2bXAM i?2 7QHHQrBM; b2+iBQM- r2 b?�HH �Mbr2` i?2b2 [m2biBQMb #v �M�HviB+�H �M�Hvb2 Q77mM+iBQM gµ(x)X

kXj �M�HviB+�H �M�HvbBb Q7 i?2 GQ;BbiB+ 6mM+iBQMq2 b�r 7`QK Qm` T`2pBQmb MmK2`B+�H �M�HvbBb- i?2 7mM+iBQM gµ(x) = µx(1 ≠ x) ?�bbQK2 BMi2`2biBM; /vM�KB+bX a2p2`�H [m2biBQMb �`Qb2- bm+? �b ?Qr K�Mv }t2/ TQBMib�`2 i?2`2 7Q` /Bz2`2Mi µ\ q?�i �`2 i?2 T`QT2ìB2b Q7 i?2b2 }t2/ TQBMib\ q?v Bbi?2 +QMp2`;2M+2 UQ` /Bp2`;2M+2V 7�bi2` 7Q` bQK2 µ #mi bHQr2` 7Q` i?2 Qi?2`\ qBi?i?2b2 [m2biBQMb BM i?2 KBM/- r2 T`Q+22/ rBi? �M�HviB+�H bim/v Q7 gµ(x) X

hQ bi�ì rBi?- r2 rQmH/ HBF2 iQ FMQr r?�i µ p�Hm2b T2`KBi i?2 K�TTBM; gµ :D æ D- 7Q` D = [0, 1]X h�F2 i?2 /2`Bp�iBp2 Q7 gµ- r2 Q#i�BM gµ

Õ(x) = µ(1 ≠ 2x)XaQHpBM; gµ

Õ(x) = 0- r2 Q#i�BM x = 12 X h?2 K�tBKmK Bb i?mb gµ(1

2) = µ4 X h?�i Bb iQ

b�v gµ : D æ D 7Q` 0 Æ µ Æ 4X AM T�ìB+mH�`- r2 Q#b2`p2 i?�i bQK2 TQBMib M2�ì?2 K�tBKmK rBHH #2 K�TT2/ QmibB/2 D 7Q` µ > 4X

�b 7Q` i?2 }t2/ TQBMib- Bi b?QmH/ #2 2pB/2Mi i?�i x = 0 Bb � }t2/ TQBMi 7Q` gµ(x)7Q` �HH µX AM Q`/2` iQ }M/ Qmi Qi?2` TQbbB#H2 }t2/ TQBMi- r2 bQHp2 i?2 2[m�iBQM

RN

gµ(x) ≠ x = 0X 6Q` µ ”= 0- r2 Q#i�BM,Y]

[p≠ = 0p+ = µ≠1

µ .

P#b2`p2 i?�i i?2 }t2/ TQBMi p+ /2T2M/b QMHv QM µ- r?B+? 2tTH�BMb r?v i?2 2[mB@HB#`BmK r�b b?B7i2/ r?2M r2 +?�M;2/ 7`QK µ = 1.5 iQ µ = 2.5X AM �//BiBQM- r2MQi2 i?�i x = 1 �M/ x = 1

µ �`2 2p2Mim�HHv }t2/ TQBMib Q7 gµ bBM+2 gµ(1) = 0 �M/gµ( 1

µ) = µ≠1µ X h?2 �#bQHmi2 p�Hm2 Q7 /2`Bp�iBp2b �i i?2 }t2/ TQBMib �`2

Y]

[|gµ

Õ(p≠)| = |µ||gµ

Õ(p+)| = |2 ≠ µ|.UkV

_2+�HH 7`QK *?�Ti2` R i?�i i?2 `�i2 Q7 r?B+? �M �ii`�+iBM;U`2T2HHBM;V }t2/ TQBMi�ii`�+i U`2T2HV Bb #2ir22M R �M/ |F Õ(x)|X 6`QK 2[m�iBQM UkV- r2 b22 r?v i?2+QMp2`;2M+2 �M/ /Bp2`;2M+2 ?�/ /Bz2`2Mi `�i2b 7Q` /Bz2`2Mi µ p�Hm2bX

q2 T`Q+22/ rBi? /2i2`KBM2 i?2 +?�`�+i2` Q7 2�+? }t2/ TQBMi rBi? ?2HT Q7 i?2.2}MBiBQM k �M/ i?2 `2bmHib �`2 T`2b2Mi2/ BM h�#H2 kX LQi2 BM T�ìB+mH�` i?�i i?2}t2/ TQBMib K2`;2 iQ QM2 }t2/ TQBMi 7Q` µ = 1 �M/ #Qi? }t2/ TQBMib �`2 `2T2HHBM;7Q` µ > 3X

µ p≠ p+

yIµIR �ii`�+iBM; `2T2HHBM;R p≠ = p+- M2mi`�H

RIµIj `2T2HHBM; �ii`�+iBM;j `2T2HHBM; M2mi`�H

jIµ `2T2HHBM; `2T2HHBM;

h�#H2 k, *?�`�+i2`b Q7 i?2 }t2/ TQBMib p≠ �M/ p+ 7Q` /Bz2`2Mi µ > 0X

AM Q`/2` iQ mM/2`bi�M/ i?2 �TT2�`�M+2 Q7 i?2 T2`BQ/@k +v+H2- r2 +QMiBMm2 iQbQHp2 gµ

2(x)≠x = 0- BX2X gµ2(x) = µ2x(1≠x)(1≠µx+µx2)X h?Bb Bb � TQHvMQKB�H Q7

/2;`22 9 r?B+? Bb ;2M2`�HHv ?�`/ iQ bQHp2 �M�HviB+�HHvX _2+�HH 7`QK T`2pBQmb +?�Ti2ì?�i � }t2/ TQBMi x0 Q7 7mM+iBQM f b�iBb}2b fn(x) = x 7Q` �HH n- i?mb i?2 TQBMib p≠

�M/ p+ �`2 �HbQ }t2/ TQBMib 7Q` g2µ(x)X >2M+2- 7Q` µ ”= 0- r2 ?�p2

µ2x(1 ≠ x)(1 ≠ µx + µx2) ≠ x

(x ≠ 0)(x ≠ µ≠1µ ) = µ[≠µ2x2 + µ(µ + 1)x ≠ (µ + 1)].

ky

h?2 Qi?2` irQ }t2/ TQBMib �`2 Q#i�BM2/ #v bQHpBM; i?2 TQHvMQKB�H,

µ[≠µ2x2 + µ(µ + 1)x ≠ (µ + 1)] = 0.

q2 HBbi �HH i?2 }t2/ TQBMib Q7 g2µ(x) �b 7QHHQrBM;

Y_______]

_______[

p≠ = 0p+ = µ≠1

µ

q≠ = µ+1≠

(µ+1)(µ≠3)2µ

q+ = µ+1+

(µ+1)(µ≠3)2µ .

P#b2`p2 i?�i i?2 /Bb+`BKBM�Mi (µ + 1)(µ ≠ 3) Bb `2�H BM i?2 BMi2`p�H µ Ø 3 �M/µ Æ ≠1X AM �//BiBQM- (µ + 1)(µ ≠ 3) = 0 Bb ;Bp2M #v µ = ≠1 �M/ µ = 3X aBM+2 r2�`2 QMHv BMi2`2bi2/ BM i?2 +�b2 7Q` 0 < µ- r2 +�M /Bb`2;�`/ µ = ≠1X AM i?Bb +�b2-i?2 k@+v+H2 2tBbib 7Q` �HH µ > 3 �M/ i?2v K2`;2 BMiQ QM2 TQBMi �i µ = 3- M�K2Hvµ+12µ X >2M+2 i?2 #B7m`+�iBQM ?�TT2Mb µ = 3X

AM Q`/2` iQ T`Q+22/ rBi? i?2 bim/v Q7 i?2 +?�`�+i2`b Q7 i?2 }t2/ TQBMibX _2+�HH7`QK h?2Q`2K k i?�i r2 +�M +QKTmi2 i?2 /2`Bp�iBp2b Q7 F 2(x0) �b F Õ(x1)F Õ(x0) B7i?2 b2i {x0, x1} Bb � k@+v+H2 Q7 6X h?mb

(gµ2)Õ(q≠) = gµ

Õ(q+)gµÕ(q≠) = (gµ

2)Õ(q+).

r?B+? ;Bp2 mbY__]

__[

|(gµ2)Õ(p≠)| = |µ2|

|(gµ2)Õ(p+)| = |(2 ≠ µ)2|

|(gµ2)Õ(q≠)| = |(gµ

2)Õ(q+)| = |1 ≠ (µ + 1)(µ ≠ 3)|.

q2 THQi i?2 |(gµ2)Õ(x)| �b 7mM+iBQMb Q7 µ �b BM 6B;m`2 N- r2 b22 i?�i i?2 k@+v+H2

�TT2�`b 7Q` µ Ø 3X AM T�ìB+mH�`- i?2`2 Bb � bT2+B�H µ p�Hm2 �i r?B+? i?2 +v+H2 BbM2mi`�HX

kR

6B;m`2 N, h?2 THQi Q7 |(gµ2)Õ(x)| �b � 7mM+iBQM Q7 µX

G2i mb T`Q+22/ iQ }M/ i?2 bT2+B}+ µ p�Hm2 �i r?B+? i?2 +v+H2 Bb M2mi`�H- #vbQHpBM; |1 ≠ (µ + 1)(µ ≠ 3)| = 1X *�H+mH�iBQM vB2H/b µ = 1 ±

Ô6X q2 +�M /Bb`2;�`/

µ = 1≠Ô

6 bBM+2 r2 �`2 HQQFBM; 7Q` µ œ (3, 4) �M/ i?2 M2mi`�H +v+H2 Bb i?mb HQ+�i2/�i µ = 1 +

Ô6X h?2 `2bmHib �`2 bmKK�`Bb2/ �M/ T`2b2Mi2/ BM h�#H2 jX P#b2`p2

i?�i i?2 }t2/ TQBMib �M/ i?2 k@+v+H2 �`2 �HH `2T2HHBM; 7Q` µ > 1 +Ô

6X

µ p≠ p+ q±

yIµIR �ii`�+iBM; `2T2HHBM; L�R p≠ = p+ = 0 M2mi`�H L�

RIµIj `2T2HHBM; �ii`�+iBM; L�j `2T2HHBM; M2mi`�H M2mi`�H

jIµI 1 +Ô

6 `2T2HHBM; `2T2HHBM; �ii`�+iBM;1 +

Ô6 `2T2HHBM; `2T2HHBM; M2mi`�H

1 +Ô

6Iµ `2T2HHBM; `2T2HHBM; `2T2HHBM;

h�#H2 j, h?2 +?�`�+i2`b Q7 i?2 }t2/ TQBMib �M/ k@+v+H2 Q7 g2µ(x)X

A7 r2 +QMiBMm2 BM i?Bb 7�b?BQM- r2 rBHH }M/ i?�i i?2 #B7m`+�iBQMb ?�TT2Mb �iµ = 1 + 2

Ô2 7Q` gµ

3(x)- �i µ = 1 +Ô

6 7Q` gµ4(x) �M/ bQ QMXXX "2HQr BM 6B;m`2 Ry-

r2 b22 i?2 +QKTH2t /vM�KB+b 7Q` /Bz2`2Mi µ p�Hm2bX

kk

6B;m`2 Ry, "B7m`+�iBQM /B�;`�K Q7 gµ(x) = µx(1 ≠ x) 7Q` /Bz2`2Mi µX h?2 TB+im`2 Bb2ti`�+i2/ 7`QK (e)X

hQ bmKK�`Bb2- i?2 7mM+iBQM gµ ?�b � }t2/ TQBMi �ii`�+iQ` BM i?2 BMi2`p�H 7Q`0 < µ < 3X 6Q` H�`;2` p�Hm2 Q7 µ- gµ +�M ?�p2 T2`BQ/B+ Q` +?�QiB+ �ii`�+iQ`bX 6Q`3 < µ Æ 4- gµ K�v ?�p2 BM}MBi2Hv K�Mv T2`BQ/B+ TQBMib �M/ +v+H2bX �M/ 7Q` µ > 4-i?2`2 Bb MQ �ii`�+iQ` b2ib- �HH i?2 T2`BQ/B+ TQBMib �M/ +v+H2b �`2 `2T2HHBM;X

kX9 �TT2�`�M+2 Q7 � *�MiQ` a2i�b r2 b�r 2�`HB2` BM i?2 MmK2`B+�H �M�HvbBb i?�i 7Q` µ = 5- i?2 Q`#Bi /Bp2`;2b iQ≠ŒX AM T`2pBQmb b2+iBQM- r2 7QmM/ i?�i i?2`2 Bb MQ �ii`�+iQ` 7Q` µ > 4X h?2`27Q`2-r2 rQmH/ HBF2 iQ }M/ Qmi B7 i?2`2 �`2 TQBMib i?�i rBHH `2K�BM BM D = [0, 1] mM/2ì?2 K�TTBM; Q7 gµ 7Q` µ > 4X A7 bQ- r?�i �`2 i?2b2 TQBMib\

_2+�HH 7`QK T`2pBQmb +?�Ti2` i?�i i?2 K�tBKmK Q7 gµ(x) = µx(1 ≠ x) Bb ;Bp2M#v gµ(1

2) = µ4 X LQr B7 µ > 4- i?2M gµ(1

2) rBHH #2 ;`2�i2` i?�M QM2 �M/ x = 12 rBHH #2

K�TT2/ QmibB/2 DX AM 7�+i- r?�i Bb K�TT2/ QmibB/2 D Bb MQi Dmbi i?2 TQBMi x = 12

#mi i?2 QT2M b2i S1 = {x œ D|gµ(x) > 1}X AM Q`/2` iQ }M/ Qmi S1- r2 bQHp2 i?2

kj

2[m�iBQM gµ(x) = 1 �M/ Q#i�BM x = 12 ±

Ò14 ≠ 1

µ - i?mb S1 Bb i?2 QT2M b2i

S1 =A

12 ≠

Û14 ≠ 1

µ,12 +

Û14 ≠ 1

µ

B

.

aBM+2 MQ �ii`�+iQ` Bb T`2b2Mi2/ 7Q` µ > 4- r2 FMQr i?�i �HH i?2 TQBMib QM S1rBHH /Bp2`;2 iQ ≠ŒX hQ /2b+`B#2 i?2 TQBMib �7i2` i?2 K�TTBM; gµ : D æ D- r2 +�Mi?2`27Q`2 `2KQp2 S1 7`QK D �M/ Q#i�BM � b2i +QMbBbib Q7 i?2 mMBQM Q7 irQ +HQb2/BMi2`p�HbX .2MQi2 i?Bb b2i #v �1 r?B+? Bb

�1 =C

0,12 ≠

Û14 ≠ 1

µ

D€

C12 +

Û14 ≠ 1

µ, 1

D

.

LQr �1 Bb r?�i Bb H27i �7i2` QM2 Bi2`�iBQM Q7 gµX A7 r2 Bi2`�i2 gµ QM2 KQ`2iBK2- i?2`2 �`2 TQBMib QM �1 i?�i rBHH #2 K�TT2/ QmibB/2 D #v g2

µ- i?2b2 TQBMib�`2 S2 = {x œ D|g2

µ(x) > 1}- r2 FMQr �HH i?2 TQBMib BM S2 ?�b i?2 b�K2 7�i2 �bi?�i Q7 i?2 TQBMib BM S1X aQ r2 `2KQp2 S2 7`QK �1 �M/ Q#i�BM �2 = �1 ≠ S2X�2 Bb � mMBQM Q7 22 = 4 +HQb2/ BMi2`p�HbX A7 r2 `2T2�i i?Bb T`Q+2bb �M/ `2KQp2Sn = {x œ D|gn

µ(x) > 1} �i i?2 n@i? Bi2`�iBQM- i?2 TQBMib i?�i `2K�BM BM D �7i2`n Bi2`�iBQMb �`2 i?2 b2i , �n = {x œ D|gn

µ(x) œ D 7Q` �HH n.}X A7 r2 Bi2`�i2 gµ

BM}MBi2Hv K�Mv iBK2- r2 ;2i� ©

Œ‹

n=1�n

h?Bb b2i Bb +�HH2/ � *�MiQ` b2iX Ai ?�b K�Mv BMi2`2biBM; iQTQHQ;B+�H T`QT2ìB2b-r?B+? r2 rBHH bim/v BM /2i�BHb BM i?2 M2ti +?�Ti2`X

k9

j h?2 *�MiQ` JB//H2@h?B`/b a2i �M/ �HBF2h?2 *�MiQ` b2i � r2 2M+QmMi2`2/ BM i?2 T`2pBQmb +?�Ti2` ?�b K�Mv T`QKBM2MiiQTQHQ;B+�H T`QT2ìB2b- Bi �HbQ TH�vb � F2v `QH2 BM K�Mv #`�M+?2b Q7 K�i?2K�iB+bXAM T�ìB+mH�`- Bi +�M #2 mb2/ �b � ;QQ/ 2t�KTH2 BM /vM�KB+�H bvbi2K- 7`�+i�H i?2Q`v�M/ b2i i?2Q`vX �b b?QrM BM a2+iBQM kX9- � Bb B``2;mH�` /m2 iQ i?2 MQM@HBM2�`K�TTBM; Q7 gµ �M/ ?2M+2 /B{+mHi iQ bim/vX 6Q` i?Bb `2�bQM- r2 rBHH BMi`Q/m+2 �M/bim/v � KQ`2 `2;mH�` *�MiQ` b2i- M�K2Hv i?2 *�MiQ` JB//H2@h?B`/b b2iX h?2`2�7i2`-r2 rBHH i`2�i i?2 ;2M2`�H *�MiQ` b2i �M/ Qi?2` 2t�KTH2X 6Q` bBKTHB+Biv- r2 b?�HHmb2 i?2 i2`K *�MiQ` b2i BMbi2�/ Q7 i?2 *�MiQ` JB//H2@i?B`/b b2i BM i?2 `2bi Q7 i?BbT�T2`X A7 r2 rBb? iQ `272` iQ �MQi?2` bT2+B�H *�MiQ` b2i- r2 rBHH 2tTHB+BiHv 2tT`2bbi?2 M�K2X

jXR *QMbi`m+iBQM Q7 i?2 *�MiQ` JB//H2@h?B`/b a2ijXRXR :`�T?B+�H J2i?Q/q2 bi�ì rBi? � +HQb2/ BMi2`p�H D = [0, 1] �M/ i?2M i?`Qr �r�v i?2 QT2M KB//H2i?B`/- BX2X

113 , 2

3

27`QK DX h?2 `2K�BMBM; b2i Bb

C1 =50,

13

6fi

523 , 1

6.

_2T2�iBM; i?2 b�K2 T`Q+2bb #v i?`QrBM; �r�v i?2 QT2M KB//H2 b2ib1

19 , 2

9

2�M/

179 , 8

9

2- r?�i `2K�BMb Bb i?2 b2i

C2 =50,

19

6fi

529 ,

39

6fi

569 ,

79

6fi

589 , 1

6.

A7 r2 F22T `2T2�iBM; i?2 b�K2 T`Q+2bb BM}MBi2Hv K�Mv iBK2b- r?�i Bb H27i Bb i?2*�MiQ` KB//H2@i?B`/b b2i- r?B+? r2 /2MQi2 �b

C ©Œ‹

i=1Cn.

LQi2 i?�i r2 �Hr�vb i?`Qr �r�v i?2 QT2M KB//H2 b2i �M/ H27i i?2 2M/ TQBMib �i2�+? bi2TX �M BHHmbi`�iBQM Q7 i?2 }`bi 72r bi2T Bb b?QrM BM 6B;m`2 RRX

6B;m`2 RR, h?2 +QMbi`m+iBQM Q7 i?2 *�MiQ` KB//H2@i?B`/b b2iX h?2 TB+im`2 Bb 2ti`�+i2/7`QK (d)X

k8

jXRXk �M�HviB+�H J2i?Q/h?2 ;`�T?B+�H K2i?Q/ BM i?2 T`2pBQmb b2+iBQM Bb BMimBiBp2 �M/ 2�bv iQ BHHmbi`�i2-#mi r2 �HbQ M22/ �M �M�HviB+�H /2b+`BTiBQM Q7 i?2 *�MiQ` b2i BM Q`/2` iQ +�``v Qmi+HQb2` 2t�KBM�iBQMX AM i?Bb bm#@b2+iBQM- r2 rBHH /2b+`B#2 � K2i?Q/ iQ +QMbi`m+i i?2*�MiQ` b2i rBi? i?2 b2`B2b 2tT�MbBQMX _2+�HH � bT2+B�H T`QT2ìv Q7 i?2 ;2QK2i`B+b2`B2b- M�K2Hv

Œÿ

k=0ak = 1

1 ≠ aB7 |a| < 1.

.2}MBiBQM dX h?2 b2[m2M+2 Q7 BMi2;2`b 0.a1a2a3 . . . Bb +�HH2/ i?2 i2`M�`v 2tT�MbBQMQ7 x B7

x =Œÿ

i=1

ai

3ir?2`2 ai œ {0, 1, 2}.

1t�KTH2 jX h?2 b2[m2M+2 0.012012012 . . . Bb

= 03 + 1

32 + 233 + 0

34 + 135 + 2

36 + 037 + 1

38 + 239 + . . .

= 132

51 + 1

33 + 136 + . . .

6+ 2

33

51 + 1

33 + 136 + . . .

6

= 132

C1

(33)0 + 1(33)1 + 1

(33)2 + . . .

D

+ 233

C1

(33)0 + 1(33)1 + 1

(33)2 + . . .

D

= 132

Œÿ

i=0

1(33)i

+ 233

Œÿ

i=0

1(33)i

=31

9 + 227

4 11 ≠ 1

33

= 526 .

"27Q`2 r2 T`Q+22/ 7mì?2` iQ i?2 *�MiQ` b2i- r2 K�F2 �M Q#b2`p�iBQM i?�i i?2i2`M�`v 2tT�MbBQM Bb MQi mMB[m2- �b /2KQMbi`�i2/ BM i?2 7QHHQrBM; 2t�KTH2X

1t�KTH2 9X h?2 i2`M�`v 2tT�MbBQMb 0.21000 . . . �M/ 0.20222 . . . `2T`2b2Mi i?2b�K2 MmK#2`- M�K2Hv

0.21000 · · · = 23 + 1

32 = 79 .

0.20222 · · · = 23 +

Œÿ

i=3

23i

= 23 + 2

27Œÿ

i=0

13i

= 79 .

6Q` x œ D = [0, 1] rBi? i?2 i2`M�`v 2tT�MbBQM 0.a1a2a3 . . . - i?2 BMi2;2`b ai

UrBi? B4R-k-. . . V /2i2`KBM2 r?2`2 x HB2b QM DX A7 r2 /BpB/2 i?2 b2i D BMiQ i?`22

ke

bm#b2ib- M�K2Hv [0, 13 ], [1

3 , 23 ], [2

3 , 1]X h?2 p�Hm2 Q7 a1 i2HHb mb r?B+? bm#b2i x HB2b BMXhQ #2 KQ`2 T`2+Bb2- B7 a1 = 0- i?2M x œ [0, 1

3 ]- B7 a1 = 1- i?2M x œ [13 , 2

3 ] �M/ }M�HHvx œ [2

3 , 1] B7 a1 = 2X PM+2 r2 7QmM/ i?2 bm#b2i Q7 r?B+? x HB2b BM- r2 +�M /BpB/2 i?2bm#b2i BMiQ i?`22 bK�HH2` 2[m�H bBx2/ bm#BMi2`p�Hb- �M/ i?2 p�Hm2 Q7 a2 rBHH BM/B+�i2r?B+? bm#BMi2`p�H x HB2b QMX PM2 +�M i?BMF Q7 }M/BM; � /2+BK�H MmK#2` QM � r2HHb+�H2/ `mH2` rBi? KmHiBTH2 H2p2H Q7 K�`F2`bX h?2 /Bz2`2M+2 MQr Bb i?�i r2 ?�p2� #�b2@j bvbi2K BMbi2�/ Q7 #�b2@Ry bvbi2KX A7 r2 +QMiBMm2 BM i?Bb 7�b?BQM- Bi Bb2pB/2Mi i?�i i?2 i2`M�`v 2tT�MbBQMb +Q``2bTQM/ iQ i?2 `2�H MmK#2`b BM i?2 b2i DX

hQ b22 i?Bb Bb i`m2- `2+�HH i?�i ai œ {0, 1, 2}- ?2M+2 0 Æ ai Æ 2 U’ i = 1, 2, . . . VXh?2`27Q`2 i?2 i�BH BM i?2 i2`M�`v 2tT�MbBQM �7i2` a1 Bb

0 =Œÿ

i=2

03i

ÆŒÿ

i=2

ai

3iÆ

Œÿ

i=2

23i

= 13 .

aBKBH�`Hv- r2 b22 i?�i i?2 i�BH �7i2` a2 Bb 0 Æ qŒ

i=3ai3i Æ 1

9 X AM 7�+i- i?2 i�BH �7i2` an

U7Q` � TQbBiBp2 BMi2;2` MV Bb 13n≠1 X

hQ bmKK�`Bb2- 7Q` � i2`M�`v 2tT�MbBQM x = 0.a1a2a3 . . . - i?2 p�Hm2 Q7 ai, i =1, 2, 3, . . . ;Bp2 7QHHQrBM; BM7Q`K�iBQM,

ai =

Y__]

__[

0, B7 x HB2b BM i?2 H27i i?B`/.

1, B7 x HB2b BM i?2 KB//H2 i?B`/.

2, B7 x HB2b BM i?2 `B;?i i?B`/.

J2�M r?BH2- i?2 BM/2t i i2HH mb ?Qr K�Mv bi2Tb �`2 /QM2 BM i?2 ?B2`�`+?X "mi i?BbBb Dmbi i?2 *�MiQ` b2i B7 r2 i?`Qr �r�v �HH i?2 KB//H2 i?B`/ T�ìb- r?B+? +Q``2bTQM/iQ ai = 1X h?mb- �Mv MmK#2` +QMi�BM2/ BM i?2 *�MiQ` b2i +�M #2 2tT`2bb2/ �b

x = 0.a1a2a3 · · · =Œÿ

i=1

ai

3ir?2`2 ai œ {0, 2}.

qBi? i?Bb /2}MBiBQM- �HH i?2 MmK#2` BM i?2 *�MiQ` b2i �`2 mMB[m2Hv /2}M2/X�//BiBQM�HHv- i?2 *�MiQ` b2i K�v �HbQ #2 Q#i�BM2/ �b i?2 �ii`�+iQ` Q7 �M Bi2`�i2/

7mM+iBQM bvbi2K- 7Q` +m`BQmb `2�/2`b- i?Bb Bb /2KQMbi`�i2/ BM (R)X

jXk S`QT2ìB2b Q7 i?2 *�MiQ` JB//H2@h?B`/b a2i�b K2MiBQM2/ 2�`HB2` BM i?2 BMi`Q/m+iBQM Q7 i?Bb +?�Ti2`- � *�MiQ` b2i ?�b K�MvBMi2`2biBM; T`QT2ìB2b �M/ Bi b2`p2b �b �M 2t+2HH2Mi 2t�KTH2 iQ bim/v K�Mv iQTQ@HQ;B+�H T`QT2ìB2b Q7 bBKBH�` ivT2 Q7 b2ibX AM Q`/2` iQ mM/2` i?2b2 T`QT2ìB2b- r2rBHH bim/v i?2 *�MiQ` b2i KQ`2 +HQb2Hv BM i?Bb b2+iBQMX

kd

jXkXR � *HQb2/ a2ih?2Q`2K jX h?2 *�MiQ` b2i Bb � +HQb2/ b2iX

S`QQ7 X h?2 +QKTH2K2Mi Q7 D = [0, 1] Bb i?2 QT2M b2i (≠Œ, 0) fi (1, +Œ)X h?2*�MiQ` b2i Bb +QMbi`m+i2/ #v `2+m`bBp2Hv `2KQpBM; i?2 QT2M KB//H2 i?B`/ BMi2`p�Hb7`QK DX aQ i?2 +QKTH2K2Mi Q7 *�MiQ` b2i- Cc- Bb i?2 QT2M b2i (≠Œ, 0)fi (1, +Œ)fi(1

3 , 23) fi (1

9 , 29) fi (7

9 , 89) fi ( 1

27 , 227) fi . . .X A7 Cc Bb QT2M- i?2M C Kmbi #2 +HQb2/X

ZX1X.

jXkXk lM+QmMi�#H2.2}MBiBQM 3X G2i N /2MQi2 i?2 M�im`2 MmK#2`bX � b2i S BM R Bb

Y]

[+QmMi�#H2 B7 a Bb QM2@iQ@QM2 �M/ QMiQ NmM+QmMi�#H2 Qi?2`rBb2

:2M2`�HHv bT2�FBM;- � b2i Bb +QmMi�#H2 r?2i?2` Bi Bb }MBi2 Q` BM}MBi2- /2bTBi2i?2 +QmMiBM; K�v i�F2 7Q`2p2`- �b HQM; �b Bib 2H2K2Mib +�M #2 +QmMi2/ QM2 �i� iBK2 �M/ 2�+? 2H2K2Mi +�M #2 2MmK2`�i2/ rBi? � M�im`�H MmK#2`X PM i?2+QMi`�`v- �M mM+QmMi�#H2 b2i ?�b i`2K2M/QmbHv K�Mv 2H2K2Mib bQ i?�i i?2v +�MMQi#2 2MmK2`�i2/ #v i?2 b2i Q7 M�im`2 MmK#2`bX

h?2Q`2K 9X h?2 *�MiQ` b2i Bb mM+QmMi�#H2X

S`QQ7 X q2 rBHH T`Qp2 i?Bb i?2Q`2K rBi? i?2 bQ +�HH2/ *�MiQ`Ƕb /B�;QM�H �`;mK2MiX�bbmK2 C Bb +QmMi�#H2- i?2M i?2`2 2tBbi � 7mM+iBQM f : C æ N bm+? i?�i f BbQM2@iQ@QM2 �M/ QMiQX "mi B7 f Bb QM2@iQ@QM2 �M/ QMiQ- i?2M i?2`2 2tBbi � 7mM+iBQMf≠1 : N æ C bm+? i?�i f≠1 Bb QM2@iQ@QM2 �M/ QMiQX h?mb r2 +�M r`Bi2

f≠1(1) = 0. a11 a12a13a14 . . .

f≠1(2) = 0.a21 a22 a23a24 . . .

f≠1(3) = 0.a31a32 a33 a34 . . .

f≠1(4) = 0.a41a42a43 a44 . . .

XXXf≠1(n) = 0.an1an2an3an4 . . . ann . . .

XXX

k3

q2 FMQr i?�i aij œ {0, 2} 7`QK i?2 i2`M�`v `2T`2b2Mi�iBQM Q7 i?2 *�MiQ` b2iX.2}M2 � M2r MmK#2` b = 0.b1b2b3b4 . . . bm+? i?�i

bj =Y]

[0 B7 aij = 22 B7 aij = 0

aBM+2 �HH i?2 2H2K2Mib Q7 i?2 i2`M�`v 2tT�MbBQM Q7 b Bb 2Bi?2` 0 Q` 2- r2 FMQr i?�ibj Bb BM i?2 +�MiQ` b2iX 6mì?2`KQ`2- bBM+2 �HH i?2 MmK#2`b BMbB/2 i?2 +�MiQ` b2i BbmMB[m2Hv /2}M2- r2 FMQr i?�i b ”= f≠1(k)- 7Q` k = 1, 2, . . . , n ≠ 1X LQr bmTTQb2b = f≠1(n)- #mi i?2 bn /Bz2`b 7`QK ann 7Q` �HH nX h?mb # Bb MQi BM i?2 HBbi �#Qp2 �M/f≠1 Bb MQi QM2@iQ@QM2 �M/ QMiQX "v +QMi`�/B+iBQM- i?2 *�MiQ` b2i Bb mM+QmMi�#H2X

jXkXj w2`Q G2M;i?h?2Q`2K 8X h?2 *�MiQ` b2i ?�b x2`Q H2M;i?X

S`QQ7 X hQ +QMbi`m+i i?2 *�MiQ` b2i- r2 bi�ì2/ rBi? i?2 b2i . r?B+? ?�b H2M;i?RX h?2M r2 `2+m`bBp2Hv `2KQp2/ i?2 QT2M KB//H2 i?B`/ T�ì Q7 2�+? BMi2`p�HX h?mb�i i?2 }`bi bi2T- r2 `2KQp2/ 1

3 H2M;i? �M/ Q#i�BM2/ k bm#BMi2`p�HbX �i i?2 b2+QM/bi2T- r2 `2KQp2/ k TB2+2b Q7 1

33 H2M;i? �M/ Q#i�BM2/ 9 bm#BMi2`p�Hb- �b r2 +QMiBMm2BM i?Bb 7�b?BQM- i?2 iQi�H `2KQp2/ KB//H2@i?B`/b ?�b H2M;i?,

13 + 2 1

32 + 22 133 + · · · =

Œÿ

i=02i 1

3i+1 = 13

Œÿ

i=0

323

4i

= 1.

.2/m+iBM; i?2 iQi�H `2KQp2/ H2M;i? 7`QK .- r2 b22 i?�i i?2 *�MiQ` b2i ?�b H2M;i?x2`QX

jXkX9 hQi�HHv .Bb+QMM2+i2/.2}MBiBQM NX � bT�+2 Bb iQi�HHv /Bb+QMM2+i2/ B7 Bib QMHv +QMM2+i2/ bm#b2ib �`2bBM;H2 TQBMibX

h?2Q`2K eX h?2 *�MiQ` b2i Bb iQi�HHv /Bb+QMM2+i2/X

S`QQ7X amTTQb2 C Bb MQi iQi�HHv /Bb+QMM2+i2/- i?2M i?2`2 Bb � MQM@2KTiv BMi2`p�HI œ CX A7 I Bb �M BMi2`p�H- H2i L(I) /2MQi2b i?2 H2M;i? Q7 IX "v /2}MBiBQM Q7BMi2`p�H- I Kmbi ?�p2 TQbBiBp2 H2M;i?- BX2X L(I) > 0X AM �//BiBQM- B7 I œ C- i?2ML(I) Æ L(C)X aBM+2 L(I) > 0 �M/ L(I) Æ L(C)- i?2M L(C) > 0- #mi i?Bb +QMi`�/B+ibrBi? h?2Q`2K 8X h?2 *�MiQ` b2i Bb ?2M+2 iQi�HHv /Bb+QMM2+i2/ #v +QMi`�/B+iBQMX

kN

jXkX8 S2`72+i a2i.2}MBiBQM RyX � b2i ?�b MQ BbQH�i2/ TQBMib B7 2p2`v TQBMi Q7 i?2 b2i Bb �M �++m@KmH�iBQM TQBMi Q7 Qi?2` TQBMib BM i?2 b2iX

h?2Q`2K dX h?2 *�MiQ` b2i ?�b MQ BbQH�i2/ TQBMibX

S`QQ7 X hQ T`Qp2 i?Bb i?2Q`2K- r2 M22/ iQ b?Qr i?�i 2p2`v TQBMi BM i?2 *�MiQ`b2i Bb �M �++mKmH�iBQM TQBMi Q7 i?2 TQBMib BM i?2 b2iX hQ /Q i?Bb- r2 b?Qr i?�i 7Q`�`#Bi`�`BHv bK�HH ‘ > 0 �M/ 7Q` �`#Bi`�`v x œ C- i?2`2 Bb � y œ C bm+? i?�i y ”= x�M/ |x ≠ y| < ‘ X

G2i x �M/ y #2 irQ /BbiBM+i TQBMib BM CX 6Q` bBKTHB+Biv- H2i mb �bbmK2 x > yX_2+�HH 7`QK a2+iBQM jXRXk i?�i �Mv MmK#2` +QMi�BM2/ BM i?2 *�MiQ` b2i +�M #22tT`2bb2/ �b a = 0.a1a2a3 · · · = q

Œ

i=1ai3i - r?2`2 ai œ {0, 2}X 1tTHB+BiHv- r2 +�M

r`Bi2 x, y �bx =

Œÿ

i=1

xi

3i- r?2`2 xi œ {0, 2}

y =Œÿ

i=1

yi

3i- r?2`2 yi œ {0, 2}

?2M+2|x ≠ y| =

Œÿ

i=1

|xi ≠ yi|3i

- r?2`2 |xi ≠ yi| œ {0, 2}.

LQr 7Q` �Mv x œ C- r2 +�M +?QQb2 y œ C, bm+? i?�i y ”= x �M/ yi = xi 7Q` bQK2�`#Bi`�`BHv H�`;2 BMi2;2` k- r?2`2 k œ {1, 2, 3, . . . }X h?mb

|x ≠ y| =Œÿ

i=k

|xi ≠ yi|3i

= 13k

Œÿ

i=0

|xi ≠ yi|3i

Æ 13k

Œÿ

i=0

23i

= 13k≠1 .

aBM+2 k Bb �`#Bi`�`BHv H�`;2- r2 +�M +?QQb2 k bm+? i?�i 13k≠1 < ‘- ?2M+2 i?2 T`QQ7 Bb

+QKTH2i2X

.2}MBiBQM RRX � +HQb2/ b2i Bb T2`72+i B7 Bi /Q2b MQi ?�p2 �Mv BbQH�i2/ TQBMibX

h?2Q`2K 3X h?2 *�MiQ` b2i Bb T2`72+iX

S`QQ7 X h?2 `2bmHi 7QHHQrb 7`QK h?2Q`2K 8 �M/ h?2Q`2K dX

jy

jXkXe a2H7@bBKBH�`Biv.2}MBiBQM RkX G2i S #2 � +HQb2/ bm#b2i Q7 RnX � K�TTBM; F : S æ S Bb +�HH2/� +QMi`�+iBQM QM S B7 i?2`2 2tBbi � `2�H MmK#2` c- rBi? 0 < c < 1- bm+? i?�i|F (x) ≠ F (y)| Æ c|x ≠ y|, ’ x, y œ SX q?2`2 i?2 2[m�HBiv ?QH/b B7 F i`�Mb7Q`Kb b2ibBMiQ ;2QK2i`B+�HHv bBKBH�` QM2b- F Bb i?2M +�HH2/ � bBKBH�`Biv �M/ c Bb +�HH2/ i?2`�iBQ Q7 F X

_2K�`F, h?2 +QMbi�Mi M = 1c Bb +�HH2/ i?2 K�;MB}+�iBQM 7�+iQ`X

Ai b?QmH/ #2 Q#pBQmb 7`QK i?2 ;`�T?B+�H +QMbi`m+iBQM Q7 i?2 *�MiQ` b2i i?�i r2�`2 ;2iiBM; irB+2 �b K�Mv BMi2`p�Hb �i 2�+? bi2TX 1p2`v BMi2`p�H ?�b � H2M;i? Q713 Q7 T`2pBQmb bi2TX h?�i Bb iQ b�v- i?2`2 �`2 2n bBKBH�`BiB2b �i bi2T n- 2�+? rBi?`�iBQ 1

3n X *QMp2`b2Hv- B7 r2 K�;MB7v �Mv Q7 i?2b2 bK�HH2` BMi2`p�Hb #v � K�;MB}+�@iBQM 7�+iQ` 3n- r2 b?QmH/ ;2i � b2i i?�i Bb ;2QK2i`B+�HHv bBKBH�` iQ Qm` Q`B;BM�H b2iD = [0, 1]X

6B;m`2 Rk, �i bi2T n- 2p2`v BMi2`p�H +�M #2 K�TT2/ #�+F iQ i?2 BMBiB�H b2i (y-R)X

.2}MBiBQM RjX G2i S #2 � +HQb2/ bm#b2i Q7 Rn �M/ {F1, . . . , Fm} #2 � +QHH2+iBQM Q7+QMi`�+iBQMbX q2 +�HH � bm#b2i U Q7 S BMp�`B�Mi 7Q` i?2 i`�Mb7Q`K�iBQMb Fi B7

U =m€

i=1Fi(U).

.2}MBiBQM R9X G2i {F1, . . . , Fm}, Fi : Rn æ Rn #2 bBKBH�`BiB2b- � b2i i?�i BbBMp�`B�Mi mM/2` bm+? � +QHH2+iBQM Q7 bBKBH�`BiB2b Bb +�HH2/ � bi`B+iHv b2H7@bBKBH�` b2iX

AM HQb2 rQ`/b- bi`B+i b2H7@bBKBH�`Biv K2�Mb i?�i r2 +�M HQQF �i �M �`#Bi`�`BHvbK�HH b+�H2 Q7 i?2 b2i �M/ i?2 bi`m+im`2 r2 b22 �7i2` i?2 K�;MB}+�iBQM Bb 2t�+iHv �bi?2 Q`B;BM�H bi`m+im`2 �b � r?QH2X aQK2iBK2- i?2 K�;MB}2/ b2i Bb MQi 2t�+iHv b�K2�b i?2 Q`B;BM�H b2i #mi �TT`QtBK�i2Hv bBKBH�`- i?2M r2 /`QT i?2 rQ`/ bi`B+iHv �M/+�HH i?2 b2i b2H7@bBKBH�`X

jR

G2i mb +QMbB/2` i?2 *�MiQ` b2i CX hrQ Q7 i?2 7mM/�K2Mi�H b2H7@bBKBH�`BiB2b �`2F1(x) = 1

3x �M/ F2(x) = 13x + 2

3 X AM T�ìB+mH�`- F1(C) Bb i?2 H27i BMi2`p�H Q7 C1 QM6B;m`2 Rk �M/ F2(C) Bb i?2 `B;?i BMi2`p�H Q7 C1 QM 6B;m`2 Rk- ?2M+2 C = F1(C)fiF2(C)X*QMb2[m2MiHv- i?2`2 �`2 2n bBKBH�`BiB2b 7Q` CnX h?2 *�MiQ` b2i Bb i?2`27Q`2 � bi`B+iHvb2H7@bBKBH�` b2iX

�++Q`/BM;Hv- i?2`2 Bb � +QHH2+iBQM Q7 BMp2`b2 K�TTBM; {F ≠11 , . . . , F ≠1

2n } i?�i +�MK�T i?2 BMi2`p�Hb QM Cn #�+F iQ D 7Q` �HH nX 6Q` BMbi�M+2- F ≠1

1 (x) = 3nx K�T i?2KQbi H27i BMi2`p�H QM 2�+? Cn #�+F iQ Dc i?2 7mM+iBQM F ≠1

2n (x) = 3nx ≠ (3n ≠ 1)K�T i?2 KQbi `B;?i BMi2`p�H QM 2�+? Cn #�+F iQ DX 6mì?2`KQ`2- ;`�T?B+�HHv- i?2BMi2;2` n +�M #2 b22K �b i?2 xQQKBM; H2p2H- r?2`2 i?2 MmK#2` 3n +�M #2 b22K �b �K�;MB}+�iBQM +QMbi�MiX >2M+2- r2 +�M xQQK BM n iBK2b QM i?2 ?�H7 Q7 i?2 *�MiQ`b2i �M/ i?2M K�;MB7v i?2 BMi2`p�H #v 3n iQ Q#i�BM i?2 Q`B;BM�H bi`m+im`2 Q7 i?2 b2iX

jXj � *�MiQ` a2ih?2 *�MiQ` b2i i?�i r2 2M+QmMi2`2/ BM i?2 T`2pBQmb b2+iBQMb Bb MQi mMB[m2X hQ+QMbi`m+i Qi?2` *�MiQ` b2ib- r2 +�M bBKTHv `2KQp2 �MQi?2` 7`�+iBQM 7`QK i?2 b2iD = [0, 1] �i 2�+? bi2TX

.2}MBiBQM R8X � *�MiQ` b2i Bb � +HQb2/- iQi�HHv /Bb+QMM2+i2/ �M/ T2`72+i bm#b2iQ7 DX

1t�KTH2 8X h?2 bm#b2i � r?B+? �`Qb2 BM a2+iBQM kX9 Bb � *�MiQ` b2iXh?2 +QKTH2K2Mi Q7 � Bb � b2i +QMbBbi Q7 � mMBQM Q7

(≠Œ, 0)€

A12 ≠

Û14 ≠ 1

µ,12 +

Û14 ≠ 1

µ

B€

· · ·€

(1, Œ),

?2M+2 � Bb � +HQb2/ bm#b2i Q7 DX"v �M�HQ;Qmb �`;mK2Mi �b i?�i 7Q` i?2 *�MiQ` JB//H2@h?B`/ b2i- Bi +�M #2

T`Qp2/ i?�i � Bb iQi�HHv /Bb+QMM2+i2/ �M/ T2`72+iX

jXjXR h?2 :2M2`�HBb2/ *�MiQ` a2i6Q` �M �`#Bi`�`v BMi2;2` m rBi? 2 Æ m < Œ- r2 +�M /BpB/2 D BMiQ 2m ≠ 1bm#BMi2`p�Hb �M/ `2KQp2 i?2 QT2M KB//H2 bm#BMi2`p�Hb r?B+? Bb i?2 b2i

3 12m ≠ 1 ,

22m ≠ 1

4 € 3 32m ≠ 1 ,

42m ≠ 1

4 €· · ·

€ 32m ≠ 32m ≠ 1 ,

22m ≠ 1

4.

h?2 `2K�BMBM; b2i Bb i?2 +HQb2/ b2i

�1 =50,

12m ≠ 1

6 € 5 22m ≠ 1 ,

32m ≠ 1

6 €· · ·

€ 52m ≠ 12m ≠ 1 , 1

6.

jk

�7i2` QM2 Bi2`�iBQM- i?2`2 �`2 m `2K�BMBM; bm#BMi2`p�Hb �M/ 2�+? Q7 i?2b2 bm#BM@i2`p�Hb #2+QK2 m bm#@bm#BMi2`p�Hb �7i2` i?2 b2+QM/ Bi2`�iBQMX >2M+2- i?2`2 �`2 mn

+HQb2/ bm#BMi2`p�Hb �7i2` n bi2T- 2�+? rBi? � H2M;i? 1(2m≠1)n X _2T2�i i?Bb T`Q+2bb

Qp2` �M/ Qp2`- i?2 `2K�BMBM; b2i �7i2` n Bi2`�iBQM Bb

�n =C

0,1

(2m ≠ 1)n

D€

C2

(2m ≠ 1)n,

3(2m ≠ 1)n

D€

· · ·€

C(2m ≠ 1)n ≠ 1

(2m ≠ 1)n, 1

D

.

h?2 ;2M2`�HBb2/ *�MiQ` b2i Bb /2}M2/ �b

� =Œ‹

n=1�n.

h?2Q`2K NX h?2 ;2M2`�HBb2/ *�MiQ` b2i Bb +HQb2/X

S`QQ7X h?2 +QKTH2K2Mi Q7 D Bb (≠Œ, 0) fi (1, Œ)X >2M+2 i?2 +QKTH2K2Mi Q7 � Bb

(≠Œ, 0)€

(�HH i?2 `2KQp2/ QT2M KB//H2 BMi2`p�Hb )€

(1, Œ).

aBM+2 i?2 +QKTH2K2Mi Q7 � Bb � mMBQM Q7 BM}MBi2 QT2M BMi2`p�Hb- i?2 b2i � Bb +HQb2/X

h?2Q`2K RyX h?2 ;2M2`�HBb2/ *�MiQ` b2i ?�b x2`Q H2M;i?X

S`QQ7X �i i?2 }`bi bi2T- r2 `2KQp2 m ≠ 1 BMi2`p�Hb- 2�+? Q7 H2M;i? 12m≠1 X h?2 b2i

�7i2` i?2 }`bi bi2T- �1- ?�b m BMi2`p�HX �i i?2 b2+QM/ bi2T- r2 `2KQp2 (m ≠ 1)BMi2`p�Hb 7`QK 2�+? Q7 i?2 m BMi2`p�Hb BM �1- 2�+? rBi? H2M;i? 1

(2m≠1)2 X >2M+2 r2�`2 `2KQpBM; iQi�HHv m(m ≠ 1) BMi2`p�Hb �i bi2T irQX *QMiBMm2 BM i?Bb 7�b?BQM- r2�`2 `2KQpBM; mn≠1(m ≠ 1) BMi2`p�Hb �i bi2T n- 2�+? rBi? � H2M;i? 1

(2m≠1)n X h?mbi?2 iQi�H `2KQp2/ H2M;i? Bb

Œÿ

n=1mn≠1(m ≠ 1) 1

(2m ≠ 1)n= m ≠ 1

2m ≠ 1Œÿ

n=0

3m

2m ≠ 1

4n

= 1.

h?mb- i?2 `2K�BMBM; b2i � ?�b x2`Q H2M;i? bBM+2 i?2 BMBiB�H b2i D ?�b H2M;i? R �M/i?2 iQi�H `2KQp2/ H2M;i? Bb �HbQ QM2X

h?2Q`2K RRX h?2 ;2M2`�HBb2/ *�MiQ` b2i Bb iQi�HHv /Bb+QMM2+i2/

h?2 T`QQ7 Q7 i?Bb i?2Q`2K Bb �M�HQ;Qmb iQ i?�i 7Q` h?2Q`2K eX

h?2Q`2K RkX h?2 ;2M2`�HBb2/ *�MiQ` b2i ?�b MQ BbQH�i2/ TQBMibX

jj

S`QQ7 X q2 T`Qp2 i?Bb i?2Q`2K BM � bBKBH�` 7�b?BQM �b i?�i 7Q` h?2Q`2K dX q2M22/ iQ b?Qr i?�i 2p2`v TQBMi BM i?2 ;2M2`�HBb2/ *�MiQ` b2i Bb �M �++mKmH�iBQMTQBMi Q7 i?2 TQBMib BM i?2 b2iX hQ /Q i?Bb- r2 b?Qr i?�i 7Q` �`#Bi`�`BHv bK�HH ‘ > 0�M/ 7Q` �`#Bi`�`v x œ �- i?2`2 Bb � y œ � bm+? i?�i y ”= x �M/ |x ≠ y| < ‘X

G2i x �M/ y #2 irQ /BbiBM+i TQBMib BM �X 6Q` bBKTHB+Biv- H2i mb �bbmK2 x > yXLQi2 i?�i r2 +�M r`Bi2 �Mv TQBMi BM i?2 ;2M2`�HBb2/ *�MiQ` b2i BM � bBKBH�` 7�b?BQM�b i?�i 7Q` i2`M�`v 2tT�MbBQM 7Q` i?2 *�MiQ` JB//H2@h?B`/b b2iX A7 r2 2tT`2bb �TQBMi BM i?2 ;2M2`�HBb2/ *�MiQ` b2i �b a = 0.a1a2a3 · · · = q

Œ

i=1ai

(2m≠1)i - r?2`2 ai œ{0, 2, 4, . . . , 2m}X q2 +�M i?2M r`Bi2 x, y �b

x =Œÿ

i=1

xi

(2m ≠ 1)i, r?2`2 xi œ {0, 2, . . . , 2m ≠ 2, 2m}.

y =Œÿ

i=1

yi

(2m ≠ 1)i, r?2`2 yi œ {0, 2, . . . , 2m ≠ 2, 2m}.

?2M+2

|x ≠ y| =Œÿ

i=1

|xi ≠ yi|(2m ≠ 1)i

- r?2`2 |xi ≠ yi| œ {0, 2, . . . , 2m ≠ 2, 2m}.

LQr 7Q` �Mv x œ �- r2 +�M +?QQb2 y œ �, bm+? i?�i y ”= x �M/ yi = xi 7Q`bQK2 �`#Bi`�`BHv H�`;2 BMi2;2` k- r?2`2 k œ {1, 2, 3, . . . }X h?mb

|x ≠ y| =Œÿ

i=k

|xi ≠ yi|(2m ≠ 1)i

= 1(2m ≠ 1)k

Œÿ

i=0

|xi ≠ yi|(2m ≠ 1)i

aBM+2 xi, yi œ {0, 2, . . . , 2m ≠ 2, 2m}- i?2 `B;?i ?�M/ bB/2 Q7 �#Qp2 2[m�iBQM Bb

1(2m ≠ 1)k

Œÿ

i=0

|xi ≠ yi|(2m ≠ 1)i

Æ 1(2m ≠ 1)k

Œÿ

i=0

2m

(2m ≠ 1)i= m

(m ≠ 1)(2m ≠ 1)k≠1 .

aBM+2 k Bb �`#Bi`�`BHv H�`;2- r2 +�M +?QQb2 k bm+? i?�i m(m≠1)(2m≠1)k≠1 < ‘- ?2M+2

i?2 T`QQ7 Bb +QKTH2i2X

h?2Q`2K RjX h?2 ;2M2`�HBb2/ *�MiQ` b2i Bb T2`72+iX

S`QQ7X h?2Q`2K Rk �M/ h?2Q`2K N T`Qp2 iQ;2i?2` i?�i ?2 ;2M2`�HBb2/ *�MiQ` b2iBb T2`72+iX

j9

jXjXk h?2 6�i *�MiQ` a2iPM2 KB;?i rQM/2` B7 �HH i?2 *�MiQ` b2ib ?�p2 x2`Q H2M;i?\ h?2 �Mbr2` Bb MQX q2b?�HH /2KQMbi`�i2 i?Bb #v +?QQbBM; iQ `2KQp2 � 7`�+iBQM Q7 1

2n �i i?2 n@i? bi2TXai�ì rBi? i?2 mMBi b2i D = [0, 1] �M/ /2H2i2 i?2 QT2M KB//H2 1

2 T�ì- BX2X (14 , 3

4)Xh?2 `2K�BMBM; b2i Bb

�1 =50,

14

6 € 534 , 1

6.

�i i?2 i?2 b2+QM/ bi2T- r2 /2H2i2 i?2 QT2M KB//H2 14 Q7 2�+? BMi2`p�H- i?2 `2K�BMBM;

b2i #2+QK2b�2 =

50,

116

6 € 5 316 ,

416

6 € 51216 ,

1316

6 € 51516 , 1

6.

aBKBH�`Hv- r2 `2KQp2 i?2 QT2M KB//H2 18 Q7 2�+? BMi2`p�H �i i?2 i?B`/ bi2T- i?2

`2K�BMBM; b2i #2+QK2b

�3 =50,

164

6 € 5 364 ,

464

6 € 51264 ,

1364

6 € 51564 ,

1664

6 € 54864 ,

4964

6 € 55164 ,

5264

6 € 56064 ,

6164

6 € 56364 , 1

6.

A7 r2 `2T2�i i?Bb T`Q+2bb BM}MBi2Hv K�Mv iBK2b �M/ /2MQi2 i?2 b2i �b

� =Œ‹

i=1�i.

A7 r2 i`v iQ +�H+mH�i2 i?2 iQi�H H2M;i? Q7 �X �i i?2 }`bi bi2T- r2 `2KQp2/ 12 X �i

i?2 b2+QM/ bi2T- r2 `2KQp2/ k TB2+2b Q7 14 ú 1

4 X �i i?2 i?B`/ bi2T- r2 `2KQp2/ 9TB2+2b Q7 1

8 ú 164 �M/ bQ QMX >2M+2 i?2 iQi�H `2KQp2/ H2M;i? Bb

Œÿ

i=12i≠1 1

4i≠112i

=Œÿ

i=1

2i

244i

12i

= 2Œÿ

i=1

14i

= 12

Œÿ

i=0

14i

= 23

h?�i Bb iQ b�v- i?2 b2i � ?�b � H2M;i? Q7 13 - r?B+? Bb MQi x2`QX h?Bb ivT2 Q7 *�MiQ`

b2i rBi? TQbBiBp2 H2M;i? Bb +�HH2/ i?2 7�i *�MiQ` a2iX

j8

q2 ?�p2 T`2pBQmbHv 2M+QmMi2`2/ i?2 *�MiQ` JB//H2@h?B`/b b2i �M/ i?2 ;2M2`�HBb2/*�MiQ` b2i- i?2v �HH ?�p2 x2`Q H2M;i?X PM2 KB;?i rQM/2` r?�i Bb i?2 /Bz2`2M+2#2ir22M i?2 ;2M2`�HBb2/ *�MiQ` b2ib- �HH Q7 r?B+? ?�p2 H2M;i? x2`Q\ hQ #2 KQ`2bT2+B}+- QM2 KB;?i rQM/2` ?Qr i?2 +QKT�`BbQM Q7 /Bz2`2Mi ;2M2`�HBb2/ *�MiQ` b2ibrBi? `2bT2+i iQ bBx2 +�M #2 /QM2\ h?2 �Mbr2`b iQ i?2b2 [m2biBQMb Bb +`m+B�H 7Q` Qm`mM/2`bi�M/BM; Q7 i?2 *�MiQ` b2i � r?B+? �`Qb2 BM b2+iBQM kX9X AM i?2 M2ti T�ì Q7i?Bb T�T2`- r2 b?�HH i`v iQ �Mbr2` i?Bb [m2biBQM #v /BK2MbBQM bim/B2bX 6mì?2`KQ`2-r2 b?�HH i`v iQ }M/ Qmi i?2 bBx2@/Bz2`2M+2 #2ir22M i?2 *�MiQ` JB//H2@h?B`/b b2i�M/ i?2 *�MiQ` b2i �X

je

S�ì AA

jd

9 6`�+i�H a2ib BM >B;?2` .BK2MbBQMbAM i?Bb +?�Ti2`- r2 b?�HH T`2b2Mi bQK2 Qi?2` b2ib i?�i b?�`2 bQK2 +QKKQM T`QT2`@iB2b rBi? i?2 *�MiQ` b2iX h?2b2 b2ib rBHH b2`p2 �b ;QQ/ 2t�KTH2b BM i?2 7Qì?+QKBM;+?�Ti2`bX

9XR h?2 pQM EQ+? aMQr~�F2 R

*QMbB/2` �M 2[mBH�i2`�H i`B�M;H2 +QMbBbib Q7 i?`22 HBM2@b2;K2Mib rBi? H2M;i? RX 6Q`2�+? HBM2@b2;K2Mi- mb2 i?2 KB//H2 i?B`/ Q7 i?2 HBM2 b2;K2Mi �b � #�b2- #mBH/ �MQi?2`2[mBH�i2`�H i`B�M;H2 rBi? bB/2 H2M;i? 1

3 X h?2 +Q`M2` QTTQbBi2 iQ i?2 #�b2 Bb �``�M;2/iQ #2 TQBMiBM; Qmir�`/bX PM+2 i?Bb Bb /QM2- `2KQp2 i?2 KB//H2 i?B`/b mTQM r?B+?r2 #mBHi i?2 M2r i`B�M;H2b bQ i?�i i?2 }M�H bi`m+im`2 Bb � bi�` b?�T2/ +HQb2/ +m`p2Xh?2 pQM EQ+? bMQr~�F2 Bb Q#i�BM2/ #v `2+m`bBp2Hv `2T2�iBM; i?Bb T`Q+2bb BM}MBi2iBK2bX h?2 }`bi 72r bi2Tb �`2 BHHmbi`�i2/ BM 6B;m`2 RjX

6B;m`2 Rj, h?2 }`bi 72r bi2T Q7 i?2 pQM EQ+? bMQr~�F2 +QMbi`m+iBQMX h?2 TB+im`2 Bb2ti`�+i2/ 7`QK (3)X

LQr BMbi2�/ Q7 TH�+BM; i?2 i`B�M;H2b TQBMiBM; Qmir�`/b- r2 +�M +?QQb2 iQ ?�p2i?2 +Q`M2` QTTQbBi2 i?2 #�b2 TQBMiBM; BMr�`/b iQ Q#i�BM �M �MiB@bMQr~�F2 �b b?QrMBM 6B;m` R9

Rh?2 EQ+? bMQr~�F2 Bb QM2 Q7 i?2 2�`HB2bi 7`�+i�H +m`p2b iQ #2 /2b+`B#2/- M�K2/ �7i2` i?2ar2/Bb? K�i?2K�iB+B�M >2H;2 pQM EQ+? Uk8 C�Mm�`v R3dy @ RR J�`+? RNk9VX >2H;2 pQM EQ+?`2+2Bp2/ ?Bb S?X.X BM lTTb�H� BM R3NkX >2 r�b �TTQBMi2/ T`Q72bbQ` Q7 K�i?2K�iB+b �i i?2 _Qv�HAMbiBimi2 Q7 h2+?MQHQ;v BM aiQ+F?QHK BM RNy8 #27Q`2 ?2 #2+�K2 T`Q72bbQ` Q7 Tm`2 K�i?2K�iB+b�i aiQ+F?QHK lMBp2`bBiv *QHH2;2 BM RNRRX

j3

6B;m`2 R9, h?2 }`bi 72r bi2T Q7 i?2 pQM EQ+? �MiB@bMQr~�F2 +QMbi`m+iBQMX h?2 TB+im`2Bb 2ti`�+i2/ 7`QK (Ry)X

9Xk h?2 aB2`TBMbFB �``Qr?2�/ *m`p2*QMbB/2` � HBM2 b2;K2Mi rBi? H2M;i? RX h?2M /BpB/2 i?2 b2;K2Mi BM i?2 KB//H2-F22TBM; i?2 irQ 2M/ TQBMi bi�iB+- �M/ HB7i i?2 KB//H2 2M/b ey /2;`22 7`QK i?2Q`B;BM�H TQbBiBQMX PM+2 i?Bb Bb /QM2- }HH BM i?2 ;�T bQ i?�i i?2 }M�H bi`m+im`2+QMbBbib Q7 j HBM2@b2;K2Mib +QMM2+i2/ �b � +?�BM rBi? Rky /2;`22 �i +QMM2+iBM;TQBMib- 2�+? HBM2@b2;K2Mi rBi? � H2M;i? 1

2 Q7 i?2 Q`B;BM�H b2;K2MiX _2T2�iBM; i?BbT`Q+2bb `2+m`bBp2Hv BM}MBi2 iBK2b- r2 ;2i i?2 aB2`TBMbFB �``Qr?2�/ +m`p2 r?B+?HQQFb p2`v bBKBH�` iQ i?2 r2HH FMQrM aB2`TBMbFB i`B�M;H2X

h?2 H2M;i? Q7 i?2 aB2`TBMbFB �``Qr?2�/ +m`p2 Bb 32 �7i2` QM2 bi2T �M/ 32

22 �7i2ìrQ bi2TX �M�HQ;QmbHv- �7i2` n bi2T- i?2 iQi�H H2M;i? Bb (3

2)nX >2M+2 i?2 +m`p2;`Qrb iQ Œ �b n æ ŒX

6B;m`2 R8, h?2 }`bi 72r bi2Tb Q7 i?2 +QMbi`m+iBQM Q7 aB2`TBMbFB �``Qr?2�/ +m`p2X P#b2`p2i?�i i?2 }M�H bi`m+im`2 `2b2K#H2b i?2 r2HH@FMQrM aB2`TBMbFB i`B�M;H2X

jN

9Xj h?2 aB2`TBMbFB *�`T2ih?2 aB2`TBMbFB +�`T2i Bb i?2 ǴirQ /BK2MbBQM�HǴ +Q``2bTQM/2M+2 iQ i?2 *�MiQ` a2iXai�ì rBi? � [0, 1] ◊ [0, 1] b[m�`2- b�K2 �b i?�i BM *�MiQ` b2i +QMbi`m+iBQM- ;`B/2�+? bB/2b iQ [0, 1

3 ]- (13 , 2

3)- [23 , 1] �M/ i?2M `2KQp2 i?2 QT2M KB//H2 i?B`/ �`2� �b

b?QrM BM 6B;m`2 ReX h?2 aB2`TBMbFB *�`T2i Bb Q#i�BM2/ B7 r2 `2T2�i i?Bb T`Q+2bbBM}MBi2 iBK2bX

A7 r2 �ii2KTi iQ +�H+mH�i2 i?2 �`2� Q7 i?2 aB2`TBMbFB +�`T2iX h?2 BMBiB�H �`2� Q7i?2 b[m�`2 Bb RX �7i2` i?2 }`bi bi2T- i?2 `2K�BMBM; �`2� Bb Q7 8

9 X �7i2` i?2 b2+QM/bi2T- i?2 `2K�BMBM; �`2� Bb (8

9)2X *QMiBMmBM; BM i?Bb 7�b?BQM- i?2 �`2� Bb (89)n �i i?2

n@i? bi2TX LQi2 i?�i i?2 �`2� ;Q2b iQ y �b n æ ŒX

6B;m`2 Re, h?2 }`bi 72r bi2Tb Q7 i?2 +QMbi`m+iBQM Q7 aB2`TBMbFB +�`T2iX h?2 TB+im`2 Bb2ti`�+i2/ 7`QK (Rk)X

9X9 J2M;2` aTQM;2h?2 J2M;2` aTQM;2- M- Bb i?2 Ǵi?`22 /BK2MbBQM�HǴ +Q``2bTQM/2M+2 Q7 i?2 *�MiQ`b2i �M/ i?2 aB2`TBMbFB +�`T2iX *QMbB/2` � [0, 1] ◊ [0, 1] ◊ [0, 1] +m#2- r2 /BpB/2i?2 +m#2 BMiQ kd bK�HH2` 2[m�H bBx2/ +m#2b- 2�+? rBi? bB/2 H2M;i? 1

3 X h?2`2�7i2`-`2KQp2 �HH i?2 QT2M KB//H2 +m#2bX h?2 J2M;2` aTQM;2 Bb Q#i�BM2/ B7 r2 `2T2�ii?Bb T`Q+2bb BM}MBi2 iBK2bX q2 b?Qr i?2 BHHmbi`�iBQM Q7 i?2 }`bi 72r bi2Tb �b BM6B;m`2 RdX

9y

6B;m`2 Rd, h?2 }`bi 72r bi2Tb Q7 i?2 +QMbi`m+iBQM Q7 J2M;2` bTQM;2X h?2 TB+im`2 Bb2ti`�+i2/ 7`QK (N)X

�b r2 +�M b22 7`QK 6B;m`2 Rd- �7i2` QM2 bi2T- i?2`2 �`2 ky +m#2b rBi? pQHmK2(1

3)3- bQ i?2 iQi�H pQHmK2 Bb 2027 X �7i2` irQ bi2T- i?2`2 �`2 202 bK�HH2` +m#2b rBi?

pQHmK2 (19)3- ?2M+2 i?2 iQi�H pQHmK2 Bb (20

27)2X *QMiBMm2 BM i?Bb 7�b?BQM- r2 b22 i?�ii?2 pQHmK2 �7i2` n bi2T Bb (20

27)nX h?2 iQi�H bm`7�+2 �`2� Bb 2(209 )n + 4(8

9)n �7i2` nbi2T (RR)X

*QMb2[m2MiHv- �b n æ Œ- i?2 iQi�H pQHmK2 ;Q2b iQ y �M/ i?2 bm`7�+2 ;Q2b iQBM}MBivX JQ`2Qp2`- �Mv bm`7�+2 BM i?2 J2M;2` aTQM;2 rBHH #2 i?Q`Qm;?Hv TmM+im`2/�b i?2 pQHmK2 ;2i ?QHHQr2` �M/ ?QHHQr2`X PM2 KB;?i rQM/2` B7 r2 +�M +QMbB/2ì?2 J2M;2` bTQM;2 bi`m+im`2 �b � bQHB/ Q` � bm`7�+2\ A7 bQ- +�M � bQHB/ ?�p2 x2`QpQHmK2 �M/ BM}MBi2 bm`7�+2\

9R

8 hQTQHQ;B+�H .BK2MbBQM.2}MBiBQM ReX � b2i S ?�b iQTQHQ;B+�H /BK2MbBQM x2`Q B7 2p2`v TQBMi BM S ?�b�`#Bi`�`BHv bK�HH M2B;?#Qm`?QQ/b r?Qb2 #QmM/�`B2b /Q MQi BMi2`b2+i i?2 b2iX � b2i S?�b iQTQHQ;B+�H /BK2MbBQM k B7 2p2`v TQBMi BM S ?�b �`#Bi`�`BHv bK�HH M2B;?#Qm`?QQ/br?Qb2 #QmM/�`B2b K22i S BM � b2i Q7 /BK2MbBQM k≠1- r?2`2 k Bb i?2 H2�bi MQMM2;�iBp2BMi2;2` 7Q` r?B+? i?Bb ?QH/bX

1t�KTH2 eX � HBM2 b2;K2Mi ?�b iQTQHQ;B+�H /BK2MbBQM RXG2i L /2MQi2b i?2 HBM2 b2;K2Mi- 7Q` 2p2`v TQBMi p QM i?2 b2;K2Mi- r2 +�M /`�r

� ‘ M2B;?#Qm`?QQ/ S‘ Q7 pX A7 p Bb � 2M/TQBMi- i?2 #QmM/�`v Q7 S‘ K22i L �i QM2TQBMiX Pi?2`rBb2- i?2 #QmM/�`v Q7 S‘ K22i L �i irQ b2T�`�i2/ TQBMibX "mi �Mvb2i +QMbBbib Q7 QM2 Q` irQ TQBMib ?�b /BK2MbBQM x2`Q- ?2M+2 L ?�b /BK2MbBQM QM2�++Q`/BM; iQ .2}MBiBQM ReX

1t�KTH2 dX � TH�M2 ?�b iQTQHQ;B+�H /BK2MbBQM kXG2i P /2MQi2b i?2 TH�M2- 7Q` 2p2`v TQBMi p BM i?2 TH�M2- i?2 b2i Q+? r?B+? �M

�`#Bi`�`BHv bK�HH M2B;?#Qm`?QQ/ Q7 T K22i i?2 TH�M2 P Bb 2Bi?2` � +B`+H2 Q` �M �`+/2T2M/b B7 i?2 TQBMi Bb QM i?2 #QmM/�`v Q7 P Q` BMbB/2 P X r?B+? ?�b /BK2MbBQMQM2X >2M+2 P ?�b /BK2MbBQM irQ �++Q`/BM; iQ .2}MBiBQM ReX

6`QK 1t�KTH2 e �M/ d- r2 b22 i?�i i?2 iQTQHQ;B+�H /BK2MbBQM +QBM+B/2 rBi? Qm`BMimBiBQM 7Q` BMi2;2` /BK2MbBQMbX "v BM/m+iBQM- r2 +�M T`Qp2 i?�i i?2 iQTQHQ;B+�H/BK2MbBQM 7Q` � +m#2 Bb j �M/ bQ QMX q2 +QKTmi2 i?2 iQTQHQ;B+�H /BK2MbBQM 7Qì?2 *�MiQ` JB//H2@h?B`/b b2i �M/ i?2 J2M;2` bTQM;2 iQ b22 B7 i?2B` iQTQHQ;B+�H/BK2MbBQM +Q``2bTQM/ iQ Qm` BMimBiBQMX

1t�KTH2 3X h?2 *�MiQ` JB//H2@h?B`/b b2i ?�b iQTQHQ;B+�H /BK2MbBQM yX"v h?2Q`2K e- i?2 *�MiQ` JB//H2@h?B`/b b2i Bb iQi�HHv /Bb+QMM2+i2/X >2M+2

i?2 b2i ?�b iQTQHQ;B+�H /BK2MbBQM y #v .2}MBiBQM ReX

1t�KTH2 NX h?2 iQTQHQ;B+�H /BK2MbBQM Q7 i?2 J2M;2` bTQM;2- M- Bb RX M Bb�++Q`/BM;Hv B/2MiB}2/ �b � +m`p2X

h?2 J2M;2` bTQM;2 ?�b x2`Q pQHmK2- i?mb i?2 bi`m+im`2 Bb ?QHHQr 2p2`vr?2`2Xh?2`27Q`2- 7Q` 2p2`v TQBMi p QM i?2 bi`m+im`2- r2 +�M TH�+2 � �`#Bi`�`BHv bK�HH ‘M2B;?#Qm`?QQ/ S‘ Q7 pX h?2 #QmM/�`v rBHH K22i M BM � MmHH b2iX >2M+2- M ?�biQTQHQ;B+�H /BK2MbBQM RX

� `2�/2` KB;?i }M/ Bi �++2Ti�#H2 iQ �bbB;M x2`Q /BK2MbBQM iQ i?2 *�MiQ`JB//H2@h?B`/b b2iX >Qr2p2`- Bi Bb +QmMi2`@BMimBiBp2 iQ /2b+`B#2 i?2 J2M;2` bTQM;2�b � QM2@/BK2MbBQM�H +m`p2X 6Q` i?Bb `2�bQM- r2 rBHH T`Q;`2bb iQ i?2 M2ti +?�Ti2`BM b2�`+? 7Q` �M BKT`Qp2/ /BK2MbBQM /2}MBiBQMX

9k

e "Qt .BK2MbBQMAM i?Bb +?�Ti2`- r2 rBHH BMi`Q/m+2 i?2 JBMFQrbFB /BK2MbBQM- r?B+? Bb +QKKQMHvFMQrM i?2 "Qt /BK2MbBQM Q` i?2 "Qt@+QmMiBM; /BK2MbBQMX 6Q` +QMbBbi2M+2 �M/bBKTHB+Biv- r2 b?�HH `2bi`B+i mb iQ mb2 i?2 #Qt /BK2MbBQMX q2 bi�ì #v BMi`Q/m+BM;i?2 +QM+2Ti Q7 i?2 +m#2 �b 7QHHQrbX

G2i R /2MQi2 � +HQb2/ `2+i�M;H2 BM RnX R Bb i?2M i?2 T`Q/m+i Q7 d TB2+2bQM2@/BK2MbBQM�H +HQb2/ �M/ #QmM/2/ BMi2`p�Hb

R = [a1, b1] ◊ [a2, b2] ◊ · · · ◊ [an, bn].

r?2`2 ai Æ bi, i = {1, 2, , . . . , n} �`2 `2�H MmK#2`bX �Hi2`M�iBp2Hv- r2 +QmH/ r`Bi2R = {(x1, x2, . . . , xn) œ Rn : ai Æ xi Æ bi, ’ i = 1, 2, . . . , n}. aBM+2 i?2 H2M;i? Q7i?2 bB/2b �`2 b1 ≠ a1, b2 ≠ a2, . . . , bn ≠ an- r2 +�M r`Bi2 i?2 pQHmK2 Q7 i?2 `2+i�M;H2-|R| - Bb

|R| = (b1 ≠ a1)(b2 ≠ a2) . . . (bn ≠ an).AM ;2M2`�H- B7 n = 1- i?2M i?2 `2+i�M;H2 Bb � HBM2- B7 n = 2- i?2 `2+i�M;H2 Bb �M �`2��M/ bQ QMX *QMb2[m2MiHv- i?2 BMi2`BQ` Q7 i?2 `2+i�M;H2 +�M #2 r`Bii2M �b

(a1, b1) ◊ (a2, b2) ◊ · · · ◊ (an, bn).

.2}MBiBQM RdX � +m#2 U�HbQ +�HH2/ #Qt V- Q- Bb � `2+i�M;H2 i?�i b�iBb}2b ,

b1 ≠ a1 = b2 ≠ a2 = · · · = bn ≠ an = l.

h?2 pQHmK2 Q7 i?2 +m#2- /2MQi2/ �b |Q|- Bb lnX

eXR "Qt .BK2MbBQM*QMbB/2` � bK�HH QM2@/BK2MbBQM�H #Qt Q7 H2M;i? ‘- r?2`2 ‘ π 1X A7 r2 i`v iQ +Qp2`� +m`p2 Q7 mMBi H2M;i? rBi? Qm` #Qt- i?2 MmK#2` Q7 #Qt2b M22/2/ iQ 7mHHv +Qp2` i?2+m`p2 Bb 1

‘ X aBKBH�`Hv- r2 +�M +Qp2` � mMBi �`2� rBi? bK�HH irQ@/BK2MbBQM�H #Qt Q7bB/2 H2M;i? ‘- i?2M i?2 MmK#2` Q7 #Qt2b M22/2/ iQ 7mHHv +Qp2` i?2 bm`7�+2 Bb 1

‘2 XA7 r2 2ti2M/ i?Bb B/2� 7Q` �M �`#Bi`�`v /BK2MbBQM- r2 +�M +Qp2` � n@/BK2MbBQM�H

b2i Q7 mMBi pQHmK2 rBi? 1‘n TB2+2b Q7 /@/BK2MbBQM�H #Qt2b Q7 bB/2@H2M;i? ‘X h?2

MmK#2` Q7 #Qt2b M22/2/ iQ +Qp2` �M mMBi b2i Bb i?2 7mM+iBQM

N(‘) = 1‘n

=31

‘

4n

,

r?2`2 n bi�M/b 7Q` i?2 /BK2MbBQM Q7 i?2 b2iX >Qr2p2`- B7 i?2 b2i ?�b �MQi?2` n@/BK2MbBQM�H pQHmK2 i?�M i?2 mMBi n@/BK2MbBQM�H pQHmK2- b�v Vn- i?2M i?2 MmK#2`Q7 #Qt2b M22/2/ iQ +Qp2` � b2i #2+QK2b

N(‘) = Vn

31‘

4n

,

9j

r?2`2 Vn Bb � +QMbi�Mi /2T2M/b QMHv QM i?2 pQHmK2 Q7 i?2 b2i BM [m2biBQMX aQHpBM;i?Bb 2[m�iBQM 7Q` n- r2 ;2i

n = lnN(‘) ≠ lnVn

ln1

1‘

2 .

6mì?2`KQ`2- B7 r2 /2bB`2 iQ +Qp2` �M B``2;mH�` b2i rBi? ;QQ/ �TT`QtBK�iBQM- r2+�M H2i i?2 #Qt bBx2 iQ #2 �`#Bi`�`BHv bK�HHX AM Qi?2` rQ`/b-

n = lim‘æ0

lnN(‘) ≠ lnVn

ln1

1‘

2 = {Vn Bb � +QMbi�Mi } = lim‘æ0

lnN(‘)ln

11‘

2 . UjV

AM �//BiBQM- i?2`2 �`2 BM ;2M2`�H irQ b2H2+iBQM `mH2b i?�i +�M #2 �TTHB2/ r?2M+Qp2`BM; � b2i S BM RnX q2 +�M +?QQb2 iQ +QmMi QMHv i?Qb2 #Qt2b i?�i �`2 7mHHv+QMi�BM2/ BMbB/2 S Q` r2 +�M +?QQb2 iQ +QmMi �HH i?2 Q#D2+i i?�i Bb 7mHHv Q` T�ìB�HHv+QMi�BM2/ BM SX h?Bb +�M #2 bmKK�`Bb2/ �b i?2 /2}MBiBQM 7QHHQrbX.2}MBiBQM R3X 6Q` � #QmM/2/ b2i S µ RnX h?2 mTT2` #Qt /BK2MbBQM Bb /2}M2/ �b

dimB(S) = lim‘æ0

sup lnN(‘)ln

11‘

2.

h?2 HQr2` #Qt /BK2MbBQM Bb /2}M2/ �b

dimB(S) = lim‘æ0

inf lnN(‘)ln

11‘

2.

A7 dimB(S) = dimB(S)- i?2M i?2 +QKKQM p�Hm2 Bb /2MQi2/ �b dimB(S) �M/ bBKTHv+�HH2/ i?2 #Qt /BK2MbBQM Q7 SX_2K�`F,

RX h?2 HBKBib BM i?2 /2}MBiBQM Q7 i?2 mTT2` �M/ HQr2` #Qt /BK2MbBQM K�v MQi2tBbi bBKmHi�M2QmbHv- i?2M i?2 #Qt /BK2MbBQM 7�BHb iQ /2b+`B#2 i?2 /BK2MbBQM Q7 i?2b2i BM [m2biBQMX

kX A7 i?2 mTT2` #Qt /BK2MbBQM 2tBbib #mi MQi i?2 HQr2` #Qt /BK2MbBQM Q` pB+2p2`b�- r2 +�M �bbB;M i?2 #Qt /BK2MbBQM �b i?2 HBKBi i?�i /Q2b 2tBbiX

jX AM i?2 +�b2 r?2M #Qi? HBKBi 2tBbi- r2 +�M r`Bi2 dimB(S) = lim‘æ0lnN(‘)ln

! 1‘

" X

P#b2`p2 i?�i i?2 JBMFQrbFB /BK2MbBQM M22/ MQi #2 � BMi2;2`- BM T�ìB+mH�`- Bi+�M i�F2 �Mv p�Hm2 BM R+X h?2`27Q`2- r2 K�F2 i?2 7QHHQrBM; /2}MBiBQMX.2}MBiBQM RNX � 7`�+i�H /BK2MbBQM Bb � /BK2MbBQM i?�i K�v i�F2 �Mv MQM@M2;�iBp2p�Hm2 BM RX � b2H7@bBKBH�` bm#b2i Q7 Rn r?Qb2 7`�+i�H /BK2MbBQM 2t+22/b Bib iQTQ@HQ;B+�H /BK2MbBQM Bb +�HH2/ � 7`�+i�HX

q2 T`Q+22/ iQ i?2 M2ti b2+iBQM iQ +QKTmi2 i?2 /BK2MbBQM Q7 bQK2 7`�+i�H b2ibX

99

eXk "Qt .BK2MbBQM Q7 6`�+i�H a2ib1t�KTH2 RyX h?2 #Qt /BK2MbBQM Q7 i?2 *�MiQ` KB//H2@i?B`/b b2i Bb ln2

ln3 .6Q` i?2 *�MiQ` KB//H2@i?B`/b b2i- r2 ?�p2 N = 2n TB2+2b �7i2` n bi2T Q7 `2@

KQpBM; i?2 KB//H2 T�ìb- 2�+? rBi? � H2M;i? Q7 bBx2 ‘ = (1/3)nX h?mb i?2 #Qt/BK2MbBQM Bb

dimB(C) = ln2n

ln 1(1/3)n

= ln2n

ln3n= nln2

nln3 = ln2ln3 ¥ 0.6309 . . .

6B;m`2 R3, PM2 bB/2 Q7 i?2 }`bi 72r bi2T Q7 i?2 pQM EQ+? bMQr~�F2 +QMbi`m+iBQMX h?2TB+im`2 Bb 2ti`�+i2/ 7`QK (Rj)X

1t�KTH2 RRX h?2 #Qt /BK2MbBQM Q7 i?2 aB2`TBMbFB �``Qr?2�/ Bb ln3ln2 X

�7i2` i?2 n@i? bi2T- i?2`2 �`2 3n HBM2@b2;K2Mib BM i?2 b2i- 2p2`v HBM2@b2;K2Mi?�b H2M;i? 1

2n X >2M+2 i?2 #Qt /BK2MbBQM +�M #2 +�H+mH�i2/ �b

dimB = ln3n

ln 1(1/2)n

= ln3n

ln2n= nln3

nln2 = ln3ln2 ¥ 1.5849 . . .

1t�KTH2 RkX h?2 #Qt /BK2MbBQM Q7 i?2 pQM EQ+? bMQr~�F2 Bb ln4ln3 .

6Q` i?2 pQM EQ+? bMQr~�F2- QM2 +�MMQi K�;MB7v QM2 TB2+2 Q7 i?2 b2i iQ Q#i�BMi?2 Q`B;BM�H i`B�M;H2X >Qr2p2` i?2 bB/2b �`2 b2H7@bBKBH�`X �7i2` i?2 n@i? bi2T- i?2`2�`2 4n TB2+2b Q7 iBMv HBM2@b2;K2Mib- 2�+? Q7 H2M;i? (1/3)nX h?mb

dimB = ln4n

ln 1(1/3)n

= ln4n

ln3n= nln4

nln3 = ln4ln3 ¥ 1.261 . . .

1t�KTH2 RjX h?2 #Qt /BK2MbBQM Q7 i?2 aB2`TBMbFB +�`T2i Bb ln8ln3 X

h?2`2 �`2 8n b[m�`2b �7i2` i?2 n@i? bi2T- 2�+? b[m�`2 ?�b bB/2@H2M;i? 13n X >2M+2

i?2 #Qt /BK2MbBQM +�M #2 +�H+mH�i2/ �b

dimB = ln8n

ln 1(1/3)n

= ln8n

ln3n= nln8

nln3 = ln8ln3 ¥ 1.8927 . . .

98

1t�KTH2 R9X h?2 #Qt /BK2MbBQM Q7 i?2 J2M;2` bTQM;2 Bb ln20ln3 X

�i i?2 n@i? bi�;2- i?2 J2M;2` aTQM;2 +QMbBbib Q7 20n bK�HH +m#2bX �HH Q7 i?2b2bK�HH +m#2b ?�p2 � bB/2@H2M;i? Q7 (1/3)nX *QMb2[m2MiHv-

dimB = ln20n

ln 1(1/3)n

= ln20n

ln3n= nln20

nln3 = ln20ln3 ¥ 2.7268 . . .

1t�KTH2 R8X h?2 ;2M2`�HBb2/ *�MiQ` b2i ?�b #Qt /BK2MbBQM ln(m)ln(2m≠1) .

h?2`2 �`2 iQi�HHv mn TB2+2b Q7 HBM2@b2;K2Mib �7i2` i?2 n@i? bi2T- 2�+? Q7 H2M;i?1

(2m≠1)n X h?2 #Qt /BK2MbBQM Q7 i?2 ;2M2`�HBb2/ *�MiQ` b2i Bb i?2M

dimB(�) = ln(mn)ln 1

(1/(2m≠1))n

= ln(mn)ln(2m ≠ 1)n

= n ln(m)n ln(2m ≠ 1) = ln(m)

ln(2m ≠ 1).

P#b2`p2 i?�i �b m æ Œ- i?2 /BK2MbBQM dimB(�) æ 1X

q2 b22 7`QK Qm` +�H+mH�iBQMb �#Qp2 i?�i i?2 #Qt /BK2MbBQM +Q``2bTQM/b iQQm` BMimBiBQM #2ii2` i?�M i?2 iQTQHQ;B+�H /BK2MbBQMX AM T�ìB+mH�`- i?2 +�H+mH�iBQMBb bBKTH2 �M/ bi`�B;?i 7Q`r�`/X AM ;2M2`�H- i?2 #Qt /BK2MbBQM +�M #2 �TTHB2/ iQ7`�+i�H b2ib i?�i �`2 bi`B+iHv b2H7@bBKBH�` Q` b2H7@bBKBH�` �b HQM; �b QM2 Q7 i?2 mTT2` Q`HQr2` #Qt /BK2MbBQM HBKBi 2tBbibX >Qr2p2`- i?2 b2i BM [m2biBQM ?�b iQ #2 #QmM/2/#v /2}MBiBQM- Qi?2`rBb2- r2 +�MMQi B;MQ`2 i?2 HMVn i2`K BM 2[m�iBQM UjVX �MQi?2`Bbbm2 Bb i?�i i?2 #Qt /BK2MbBQM K�v MQi 2tBbib 7Q` � b2i r?2M i?2 mTT2` �M/ HQr2`#Qt /BK2MbBQM /Q MQi 2tBbi bBKmHi�M2QmbHvX JQ`2Qp2`- Bi Bb /B{+mHi iQ +QKTmi2 i?2#Qt /BK2MbBQM 7Q` i?2 *�MiQ` b2i � r?B+? �`Qb2 BM b2+iBQM kX9X h?2 K�TTBM; Q7i?2 HQ;BbiB+ 7mM+iBQM gµ(x) = µx(1 ≠ x) Bb MQi HBM2�`- �7i2` n Bi2`�iBQMb Q7 gµ- i?2`2�`2 2n BMi2`p�Hb BM i?2 b2i- #mi rBi? /Bz2`2Mi H2M;i?X >2M+2 r2 ?�p2 /B{+mHiv iQ+QKTmi2 i?2 /BK2MbBQM Q7 � �M�HviB+�HHv rBi? i?2 #Qt /BK2MbBQM /2}MBiBQMX qBi?i?Bb #�+F;`QmM/- r2 +QMiBMm2 iQ b2�`+? 7Q` � #2ii2` /BK2MbBQM /2}MBiBQMX

9e

d >�mb/Q`z .BK2MbBQMAM i?2 T`2pBQmb +?�Ti2`- r2 bim/B2/ i?2 #Qt /BK2MbBQM r?B+? Bb +QMp2MB2Mi BM+QKTmi�iBQM �M/ r2HH@/2}M2/ 7Q` 7`�+i�H b2ibX >Qr2p2`- i?2`2 Bb biBHH bQK2 `QQK7Q` BKT`Qp2K2Mi bBM+2 i?2 /2}MBiBQM �TTHB2b QMHv iQ #QmM/2/ b2ib �M/ i?2 mTT2`�M/ HQr2` #Qt /BK2MbBQM K�v MQi 2tBbi bBKmHi�M2QmbHvX JQ`2Qp2`- r2 ?�p2 }M/Bi /B{+mHi iQ +QKTmi2 i?2 *�MiQ` b2i � r?B+? �`Qb2 BM b2+iBQM kX9 /m2 iQ i?2MQM@HBM2�` K�TTBM; Q7 i?2 HQ;BbiB+ 7mM+iBQM gµX h?2`27Q`2- r2 b?�HH BMi`Q/m+2 i?2>�mb/Q`z /BK2MbBQM BM i?Bb +?�Ti2`X h?2 >�mb/Q`z /BK2MbBQM BMpQHp2b i?2 i?2Q`vQ7 2ti2`BQ` K2�bm`2X Ai Bb QM2 Q7 i?2 KQbi /2HB+�i2 /2}MBiBQM Q7 /BK2MbBQM- r2b?�HH b22 BM i?Bb +?�Ti2` i?�i i?2 >�mb/Q`z /BK2MbBQM ?�b rB/2` b+QT2 �M/ #2ii2`~2tB#BHBiv i?�M i?2 #Qt /BK2MbBQMX

dXR 1ti2`BQ` J2�bm`2 �M/ "Q`2H b2ib.2}MBiBQM kyX � mMBQM Q7 +m#2b /2}M2/ �b BM .2}MBiBQM Rd Bb b�B/ iQ #2 �HKQbi/BbDQBMi B7 i?2 BMi2`BQ`b Q7 i?2 +m#2b �`2 /BbDQBMiX

G2KK� RX A7 � +m#2 Bb i?2 �HKQbi /BbDQBMi mMBQM Q7 }MBi2Hv K�Mv Qi?2` +m#2b- BX2Q = fiN

i=1QiX h?2M

|Q| =Nÿ

i=1|Qi|.

S`QQ7X G2i Q /2MQi2 i?2 +m#2- �M/ H2i Qi, i = 1, 2, . . . , N /2MQi2 i?2 +m#2b BMbB/2Q- bQ i?�i Q = fiN

i=1QiX aBM+2 i?2 +m#2b Qi �`2 /BbDQBMi- i?2 BMi2`BQ` Q7 i?2 +m#2b/Q MQi BMi2`b2+iX >2M+2 i?2 pQHmK2 Q7 Q Bb Dmbi i?2 bmK Q7 i?2 pQHmK2 Q7 2�+?+m#2 Qi- BX2X |Q| = qN

i=1 |Qi|.

.2}MBiBQM kRX 6Q` S � bm#b2i Q7 Rn- i?2 2ti2`BQ` K2�bm`2 Q7 S Bb /2}M2/ �b

mú(S) = infŒÿ

i=1|Qi|.

q?2`2 i?2 BM}KmK Bb i�F2M Qp2` �HH +QmMi�#H2 +Qp2`b S = fiŒ

i=1Qi #v +HQb2/ +m#2bQi

AM HQb2 rQ`/b- i?2 2ti2`BQ` K2�bm`2 Q7 � b2i Bb #�b2/ QM i?2 B/2� i?�i r2 i`v iQ+Qp2` � b2i 7`QK QmibB/2 #v +m#2b- H2iiBM; i?2 +m#2b iQ #2 bK�HH2` �M/ bK�HH2` bQi?�i i?2 +Qp2`BM; ;2i }M2` �M/ }M2` bm+? i?�i i?2 pQHmK2 Q7 i?2 +Qp2`BM; +m#2b Bb�TT`Q�+?BM; i?2 pQHmK2 Q7 i?2 b2i SX

9d

h?2Q`2K R9X h?2 2ti2`BQ` K2�bm`2 ?�b 7QHHQrBM; T`QT2ìB2b ,BVX UJQMQiQMB+BivV A7 S1 µ S2- i?2M mú(S1) Æ mú(S2)XBBVX U*QmMi�#H2 bm#@�//BiBpBivV A7 S = fiŒ

i=1Si- i?2M mú(S) Æ qŒ

i=1 mú(Si)XBBBVX A7 S µ Rn- i?2M mú(S) = inf mú(O)- r?2`2 i?2 BM}KmK Bb i�F2M Qp2` �HH

QT2M b2ib O +QMi�BMBM; SXBpVX A7 d(S1, S2) > 0- i?2M mú(S1 fi S2) = mú(S1) + mú(S2).pVX A7 � b2i S Bb i?2 +QmMi�#H2 mMBQM Q7 �HKQbi /BbDQBMi +m#2b S = fiŒ

i=1Qi- i?2Mmú(S) = q

Œ

i=1 |Qi|X

h?2 QmiHBM2 Q7 i?Bb T`QQ7 7QHHQrb i?2 QM2 BM (9)- ?Qr2p2` /Bz2`2Mi MQi�iBQMb �M/KQ`2 2tTHB+Bi 7Q`KmH�iBQM �`2 mb2/ BM Q`/2` iQ �/�Ti iQ i?Bb T�T2`X

S`QQ7X BVX h?Bb T`QT2ìv 7QHHQrb /B`2+iHv 7`QK i?2 /2}MBiBQM- B7 S1 µ S2- S1 +�M #2+Qp2` #v �Mv +QHH2+iBQM Q7 +Qp2`b i?�i +�M +Qp2`b S2X

BBVX amTTQb2 mú(S) Bb BM}MBi2Hv H�`;2- Bi Bb Q#pBQmbHv i`m2 i?�i i?2 BM2[m�HBiv?QH/bX 6Q` mú(S) < Œ- i?2 /2}MBiBQM Q7 i?2 2ti2`BQ` K2�bm`2 vB2H/b i?�i 7Q` ‘ > 0-i?2`2 2tBbi +Qp2`b Q7 +m#2b Qij 7Q` 2�+? Si bm+? i?�i

Si ™Œ€

j=1Qij �M/

Œÿ

j=1|Qij| Æ mú(Si) + ‘

2j

>2M+2 i?2 +Qp2`b tŒ

i,j=1 Qi,j �HbQ +Qp2`b tŒ

i=1 Si = SX h?2`27Q`2- i?2 7QHHQrBM;`2H�iBQMb?BT ?QH/b 7Q` 2p2`v ‘ > 0-

mú(S) = mú(Œ€

i=1Si)

ÆŒÿ

i,j=1|Qi,j|

=Œÿ

i=1

Œÿ

j=1|Qi,j|

ÆŒÿ

i=1

3mú(Si) + ‘

2j

4

=Œÿ

i=1mú(Si) + ‘.

BBBVX q2 rBHH T`Qp2 mú(S) Æ inf mú(O) Æ mú(S) r?B+? BKTHB2b mú(S) =inf mú(O)X AM T�ìB+mH�`- r2 rBb? iQ T`Qp2 i?2 `2H�iBQMb?BTb ,

Y]

[�VX mú(S) Æ inf mú(O)#VX inf mú(O) Æ mú(S)

.

93

�VX aBM+2 O Bb � +QHH2+iBQM Q7 QT2M b2i i?�i +QMi�BMBM; S- BX2X S µ OX Ai 7QHHQrb#v i?2 T`QT2ìv BV i?�i mú(S) Æ mú(O)X

#VX G2i Qi #2 � +QHH2+iBQM Q7 +m#2b i?�i +Qp2` SX h?2M #v i?2 /2}MBiBQM Q7 i?22ti2`BQ` K2�bm`2- i?2 7QHHQrBM; `2H�iBQMb?BT ?QH/b 7Q` �Mv ‘ > 0-

Œÿ

i=1|Qi| Æ mú(S) + ‘

2 .

amTTQb2 2�+? +m#2 Qi Bb +QMi�BM2/ BM �M QT2M b2i QÕ

i bQ i?�i i?2 pQHmK2 Q7 i?2+m#2b ?�p2 7QHHQrBM; `2H�iBQMb?BT

|Qi| = |QÕ

i| + ‘

2i+1 .

�M/ O = tŒ

i=1 QÕ

i Bb �M QT2M b2i- i?mb

mú(O) Æ(1)

Œÿ

i=1mú(QÕ

i)

=Œÿ

i=1|QÕ

i|

ÆŒÿ

i=1

3|QÕ

i| + ‘

2i+1

4

Æÿ

i=1Œ|Qi| + ‘

2Æ mú(S) + ‘.

q?2`2 bi2T URV 7QHHQrb 7`QK T`QT2ìv BBVX aBM+2 ‘ > 0 Bb �`#Bi`�`v- Bi 7QHHQrb i?�iinf mú(O) Æ mú(S)X

BpVX aBKBH�` �b i?�i 7Q` T`QT2ìv BBBV- r2 rBHH T`Qp2 7Q` d(S1, S2) > 0- i?27QHHQrBM; `2H�iBQMb?BTb ?QH/,

Y]

[�VX mú(S1 fi S2) Æ mú(S1) + mú(S2)#VX mú(S1) + mú(S2) Æ mú(S1 fi S2).

U9V

h?2 `2H�iBQMb?BT �V 7QHHQrb /B`2+iHv 7`QK T`QT2ìv BBVX AM Q`/2` iQ T`Qp2 #V- H2iQi #2 � +QHH2+iBQM Q7 +m#2b i?�i +Qp2`b S1 fi S2X "v i?2 /2}MBiBQM Q7 i?2 2ti2`BQ`K2�bm`2- i?2 7QHHQrBM; `2H�iBQMb?BT ?QH/b 7Q` �Mv ‘ > 0-

Œÿ

i=1|Qi| Æ mú(S1 fi S2) + ‘.

9N

AM �//BiBQM- r2 K�v +?QQb2 � ” bm+? i?�i d(S1, S2) > ” > 0 �M/ BKTQb2 � +QM/BiBQM7Q` i?2 +m#2b Qi bm+? i?�i i?2 /B�K2i2` Q7 Qi Bb H2bb i?�M ” r?2`2 diam Qi =sup{|x ≠ y| : x, y œ Qi}X *QMb2[m2MiHv-

S1 =Œ€

iœE1

Qi, S2 =Œ€

iœE2

Qi

r?2`2 i?2 b2i E1 +QMi�BMb �HH i?2 BM/B+2b Q7 i?2 +m#2b i?�i +Qp2` S1X aBKBH�`Hv- i?2b2i E2 +QMi�BMb �HH i?2 BM/B+2b Q7 i?2 +m#2b i?�i +Qp2` S2X P#b2`p2 i?�i E1flE2 = ?bBM+2 2p2`v +m#2 Qi +�MMQi BMi2`b2+i #Qi? S1 �M/ S2 bBKmHi�M2QmbHv /m2 iQ i?2BKTQb2/ +QM/BiBQMX Ai 7QHHQrb i?2M

mú(S1) + mú(S2) Æÿ

iœE1

|Qi| +ÿ

iœE2

|Qi|

ÆŒÿ

i=1|Qi|

Æ mú(S1 fi S2) + ‘.

aBM+2 ‘ > 0 Bb �`#Bi`�`v- Bi 7QHHQrb i?�i mú(S1) + mú(S2) Æ mú(S1 fi S2)X 6m`@i?2`KQ`2- bBM+2 #Qi? �V �M/ #V ?QH/ BM U9V- r2 ?�p2 i?2M T`Qp2/ mú(S1 fi S2) =mú(S1) + mú(S2)X

pVX AM � bBKBH�` 7�b?BQM �b BM T`2pBQmb +�b2b- r2 rBHH T`Qp2 i?2 7QHHQrBM; `2H�@iBQMb?BTb ?QH/ 7Q` S � +QmMi�#H2 mMBQM Q7 �HKQbi /BbDQBMi +m#2b,

Y]

[�VX mú(S) Æ q

Œ

i=1 |Qi|#VX q

Œ

i=1 |Qi| Æ mú(S).U8V

G2i S = fiŒ

i=1QiX _2H�iBQM �V BM U8V ?QH/b #v T`QT2ìv BBVX AM Q`/2` iQ T`Qp2#V- Q#b2`p2 i?�i 7Q` 2�+? +m#2 Qi- r2 +�M }M/ �MQi?2` +m#2 QÕ

i i?�i Bb bi`B+iHv+QMi�BM2/ BM Qi- BX2X QÕ

i µ Qi, ’ i- bm+? i?�i i?2 pQHmK2 Q7 i?2 +m#2b ?�p2 7QHHQrBM;`2H�iBQMb?BT

|Qi| Æ |QÕ

i| + ‘

2i,

r?2`2 ‘ > 0 Bb �`#Bi`�`v #mi }t2/ 7Q` �HH iX .2MQi2 S Õ = tŒ

i=1 QÕ

i- i?2M S Õ µ SXJQ`2Qp2`- i?2 +m#2b {QÕ

i} �`2 /BbDQBMi- BX2X d(QÕ

i, QÕ

j) > 0 7Q` 2p2`v i ”= jX >2M+2r2 +�M `2T2�i2/Hv �TTHv T`QT2ìv BpV iQ i?2 +m#2b {QÕ

i} iQ Q#i�BM

mú

AŒ€

i=1QÕ

i

B

=Œÿ

i=1mú(QÕ

i) =Œÿ

i=1|QÕ

i| ØŒÿ

i=1(|Qi| ≠ ‘

2i) =

Œÿ

i=1|Qi| ≠ ‘.

8y

aBM+2 S Õ µ S- mú(S) Ø mú(S Õ) ?QH/b #v T`QT2ìv BBVX 6BM�HHv- r2 +�M H2i ‘ æ 0 iQQ#i�BM

Œÿ

i=1|Qi| Æ mú(S).

ZX1X.

dXk >�mb/Q`z J2�bm`2 �M/ .BK2MbBQM:2M2`�HHv bT2�FBM;- i?2 >�mb/Q`z K2�bm`2 Bb � ivT2 Q7 2ti2`BQ` K2�bm`2 i?�i �b@bB;Mb � MmK#2` BM R+ fi {Œ} iQ 2�+? b2i BM RnX AM Q`/2` iQ ;�BM bQK2 BMimBiBQM-Q#b2`p2 i?�i � k@/BK2MbBQM�H +m#2 ?�b BM}MBi2Hv H�`;2 R@/BK2MbBQM K�bb B7 Bib +QK@T�`2/ rBi? � R@/BK2MbBQM�H +m#2X AM i?2 K2�MiBK2- � k@/BK2MbBQM�H +m#2 ?�bM2;HB;B#H2 j@/BK2MbBQM�H K�bb BM +QKT�`BbQM rBi? � j@/BK2MbBQM�H +m#2X

*QMbB/2` i?2 [m�MiBiv m–(S) 7Q` 2�+? �TT`QT`B�i2 b2i S �M/ 2�+? – > 0- i?2[m�MiBiv m–(S) +�M #2 mM/2`biQQ/ �b i?2 –@/BK2MbBQM�H K�bb Q7 SX h?2M r2+QKT�`2 S rBi? b2ib Q7 /BK2MbBQM –X A7 – Bb ;`2�i2` i?�M i?2 /BK2MbBQM Q7 S- i?2Mi?2 –@/BK2MbBQM�H K�bb Q7 S Bb M2;HB;B#H2- BX2X m–(S) = 0X PM i?2 +QMi`�`v- B7 –Bb bK�HH2` i?�M i?2 /BK2MbBQM Q7 S- i?2M i?2 –@/BK2MbBQM�H K�bb Q7 S Bb BM}MBi2HvH�`;2 #v +QKT�`BbQM- ?2M+2 m–(S) = ŒX AM T�ìB+mH�`- B7 i?2 /BK2MbBQM Q7 i?2 b2iS +QBM+B/2b rBi? –- i?2M – Bb i?2 �+im�H /BK2MbBQM Q7 SX

q2 MQr T`Q+22/ iQ BMi`Q/m+2 i?2 ”@+Qp2` mb2/ #v 6�H+QM2` BM (k)X G2i U #2�Mv MQM@2KTiv bm#b2i Q7 Rn- i?2 /B�K2i2` Q7 U Bb /2}M2/ �b i?2 H�`;2bi /Bbi�M+2#2ir22M �Mv T�B` Q7 TQBMib BM i?2 b2iX JQ`2 2tTHB+BiHv- r2 K2�M diam U © sup{|x≠y| : x, y œ U}X LQr H2i S #2 � bm#b2i Q7 RnX amTTQb2 {Ui} Bb � +QHH2+iBQM Q7+QmMi�#H2 b2ib Q7 /B�K2i2` �i KQbi ” i?�i +Qp2` S- BX2X S µ fiŒ

i Ui rBi? 0 <diam Ui Æ ” 7Q` �HH i œ Z+- r2 b�v i?�i {Ui} Bb � ”@+Qp2` Q7 SX.2}MBiBQM kkX 6Q` �Mv S ™ Rn, – Ø 0 �M/ ” > 0- i?2 2ti2`BQ` –@/BK2MbBQM�H>�mb/Q`z K2�bm`2 Q7 S Bb /2}M2/ �b

H–(S) © mú

–(S) = lim”æ0

H–” (S),

r?2`2H–

” (S) © infI

ÿ

i

(diam Ui)– : {Ui} Bb � ”@+Qp2` Q7 S

J

Smi /Bz2`2MiHv- 7Q` 2�+? ” > 0- r2 +QMbB/2` i?2 +Qp2`b Q7 S #v +QmMi�#H2 +QH@H2+iBQMb Q7 b2ib- Ui, i = 1, 2, . . . - rBi? /B�K2i2` H2bb i?�M ”X h�F2 i?2 BM}KmK Q7i?2 bmK q

k(diam Ui)- i?2 2ti2`BQ` –@/BK2MbBQM�H >�mb/Q`z K2�bm`2 Bb /2}M2/ �bi?2 HBKBi Q7 i?2 BM}K� Q7 i?2 bmK �b ” æ 0X P#b2`p2 i?�i i?2 [m�MiBiv H–

” (S)BM+`2�b2b �b ” /2+`2�b2bX AM T�ìB+mH�`- i?2 BM2[m�HBiv H–

” (S) Æ H–(S) ?QH/b 7Q` �HH” > 0X

8R

h?2Q`2K R8X h?2 >�mb/Q`z 2ti2`BQ` K2�bm`2 TQbb2bb2b i?2 7QHHQrBM; T`QT2ìB2b,BVX UJQMQiQMB+BivV A7 S1 µ S2- i?2M H–(S1) Æ H–(S2)XBBVX Uam#@�//BiBpBivV H–(fiŒ

i=1Si) Æ qŒ

i=1 H–(Si) 7Q` �Mv +QmMi�#H2 7�KBHv {Si}Q7 b2ib BM RdX

BBBVX A7 d(S1, S2) Ø 0- i?2M H–(S1 fi S2) = H–(S1) + H–(S2)XBpVX A7 H–(S) < Œ �M/ — > –- i?2M H—(S) = 0X aBKBH�`Hv- B7 H–(S) > 0 �M/

— < –- i?2M H—(S) = Œ

S`QQ7X h?2 T`QQ7 Q7 T`QT2ìB2b BV- BBV- �M/ BBBV 7QHHQrb /B`2+iHv 7`QK h?2Q`2K R9X6Q` T`QT2ìv BpV- `2+�HH i?�i

H–(S) = lim”æ0

H–” (S) = lim

”æ0inf

Iÿ

i

(diam Ui)– : {Ui} Bb � ”@+Qp2` Q7 S

J

.

amTTQb2 – < —- bBM+2 0 Æ diam Ui Æ ”- r2 ?�p2

(diam Ui)— = (diam Ui)—≠–(diam Ui)– Æ ”—≠–(diam Ui)–.

h?mbH—

” (S) Æ ”—≠–H–” (S).

JQ`2Qp2`- H–” (S) Æ H–(S)- — ≠ – > 0 �M/ H–(S) < Œ- ?2M+2

H—(S) Æ ”—≠–H–” (S) =

”æ00

�M�HQ;QmbHv- i?2 +QMi`�TQbBiBp2 ;Bp2b H—(S) = Œ 7Q` H–(S) > 0 �M/ — < –.

.2}MBiBQM kjX � ‡@�H;2#`�- B- Bb � 7�KBHv Q7 bm#b2ib Q7 W bm+? i?�i ,BVX B œ W ,BBVX A7 S œ W- i?2M B\S = Sc œ W ,BBBVX A7 Si œ W , i = 1, 2, . . . - i?2M t

i=1 Si œ W .

AM Qi?2` rQ`/b- i?2 ‡@�H;2#`� B Bb Bb +HQb2/ mM/2` +QKTH2K2Mib �M/ +QmMi�#H2mMBQMb #v +QM/BiBQM ii) `2bT2+iBp2 +QM/BiBQM iii)X JQ`2Qp2`- B7 B Bb +HQb2/ mM/2`+QmMi�#H2 mMBQMb- i?2M B Bb �HbQ +HQb2/ mM/2` +QmMi�#H2 BMi2`b2+iBQM #v i?2 /2JQ`;�M H�r (A fi B)c = Ac fl BcX

.2}MBiBQM k9X h?2 "Q`2H ‡@�H;2#`� Bb i?2 bK�HH2bi ‡@�H;2#`� i?�i +QMi�BMb �HHQT2M b2ibX 1H2K2Mib Q7 i?2 "Q`2H ‡@�H;2#`� Bb +�HH2/ i?2 "Q`2H b2ibX

AM i?Bb T�T2`- r2 b?�HH HBKBi mb iQ i?2 "Q`2H b2ib Q7 "Q`2H ‡@�H;2#`� BM RnX h?2`2bi`B+iBQM Q7 H– iQ i?2 "Q`2H b2ib BM Rn Bb +�HH2/ i?2 –@/BK2MbBQM�H >�mb/Q`zK2�bm`2X q2 b?�HH mb2 i?2 b�K2 MQi�iBQM H–X

8k

h?2Q`2K ReX A7 {Si} Bb � +QmMi�#H2 +QHH2+iBQM Q7 /BbDQBMi "Q`2H b2ib �M/ S =fiŒ

i=1Si- i?2MH–(

Œ€

i=1Si) =

Œÿ

i=1H–(Si).

h?2 T`QQ7 Q7 i?Bb i?2Q`2K Bb QKBii2/ /m2 iQ i?2 `2[mB`2K2Mi Q7 FMQrH2/;2 BM"Q`2H K2�bm`2 �M/ *�`�i?ûQ/Q`vǶb +`Bi2`BQMX � T`QQ7 Q7 i?Bb i?2Q`2K +�M #2 b22MBM (j)X

h?2Q`2K RdX >�mb/Q`z K2�bm`2 Bb BMp�`B�Mi mM/2` i`�MbH�iBQMb �M/ `Qi�iBQMbXJQ`2 Qp2`- Bi b+�H2b �b H–(⁄S) = ⁄–H–(S), ’ ⁄ > 0.

S`QQ7X h?2 /B�K2i2` Q7 � b2i S Bb /2}M2/ �b diam S = sup{|x≠y| : x, y œ S}- bBM+2|x ≠ y| Bb BMp�`B�Mi mM/2` i`�MbH�iBQMb �M/ `Qi�iBQMb 7Q` �HH x, y œ S- Bi 7QHHQrb i?�ii?2 >�mb/Q`z K2�bm`2 Bb BMp�`B�Mi mM/2` i`�MbH�iBQMb �M/ `Qi�iBQMbX 6mì?2`KQ`2-i?2 /B�K2i2` b�iBb}2b diam ⁄S = ⁄diam S, ’⁄ > 0- ?2M+2 i?2 b+�HBM; `mH2 Bb p�HB/X

6B;m`2 RN, �M BHHmbi`�iBQM Q7 i?2 >�mb/Q`z /BK2MbBQMX

*QMbB/2` � "Q`2H bm#b2i Q7 Rn- BRn - 7`QK T`QT2ìv iv) BM h?2Q`2K R8- r2 +�M+QM+Hm/2 i?�i i?2`2 Bb � mMB[m2 – bm+? i?�i

H—(S) =Y]

[Œ if — < –,

0 if – < —.

�M BHHmbi`�iBQM Bb b?QrM BM 6B;m`2 RNX h?�i Bb iQ b�v- – Bb i?2 +`BiB+�H 2tTQM2Mib�iBb}2b 0 Æ H–(S) Æ Œ- r?2`2 i?2 BM2[m�HBiB2b �`2 bi`B+i B7 S Bb #QmM/2/X

8j

.2}MBiBQM k8X h?2 >�mb/Q`z /BK2MbBQM Q7 � "Q`2H bm#b2i Q7 Rn- S- Bb /2}M2/ �b

– = sup{— Ø 0 : H—(S) = Œ} = inf{— Ø 0 : H—(S) = 0}

q2 b?�HH /2MQi2 – = dimH(S) �M/ – Bb +�HH2/ i?2 >�mb/Q`z /BK2MbBQM Q7 SXhQ 2KT?�bBx2- Bi b?QmH/ #2 MQi2/ i?�i i?2 >�mb/Q`z /BK2MbBQM M22/ MQi #2 �MBMi2;2`X

1t�KTH2 ReX h?2 >�mb/Q`z /BK2MbBQM Q7 i?2 *�MiQ` JB//H2@h?B`/b b2i Bb ln2ln3 X

q2 b?�HH mb2 i?2 >2m`BbiB+ +�H+mH�iBQM (k)X q?2M +QMbi`m+i i?2 *�MiQ` KB//H2@i?B`/b b2i- i?2 b2i D bTHBi BMiQ � H27i T�ì CL

1 = D fl [0, 13 ] �M/ � `B;?i T�ì CR

1 =D fl [2

3 , 1]X �b r2 FMQr 7`QK b2+iBQM jXkXe- i?2 *�MiQ` KB//H2@i?B`/b b2i Bb b2H7@bBKBH�` #v � b+�HBM; `�iBQ 1

3 X 6mì?2`KQ`2- C1 = CL1 fi CR

1 Bb � /BbDQBMi mMBQMX h?mb#v iii) BM h?2Q`2K R8 �M/ i?2 b+�HBM; T`QT2ìv BM h?2Q`2K Rd- r2 ?�p2

Hd(C1) = Hd(CL1 ) + Hd(CR

1 ) =31

3

4d

Hd(C1) +31

3

4d

Hd(C1) = 231

3

4d

Hd(C1)

7Q` �Mv MQM M2;�iBp2 `2�H MmK#2` dX aBM+2 i?2 *�MiQ` KB//H2@i?B`/b b2i Bb #QmM/2/-r2 +�M �bbmK2 i?2 bi`B+i BM2[m�HBiv 0 < Hd(C1) < Œ �i i?2 +`BiB+�H p�Hm2 d =dimH(C)X Ai 7QHHQrb i?2M d = ln2

ln3 X

_2K�`F,RX � KQ`2 /2i�BH2/ T`QQ7 Q7 1t�KTH2 Re +�M #2 b22M BM (k)XkX Ai +�M #2 b?QrM i?�i i?2 ln2

ln3 @/BK2MbBQM�H >�mb/Q`z K2�bm`2 Bb 2t�+iHv R (k)X

.2bTBi2 i?2 +mK#2`bQK2 +�H+mH�iBQM- Bi b22Kb i?�i i?2 >�mb/Q`z /BK2MbBQM+QBM+B/2b rBi? i?2 #Qt /BK2MbBQM BM Qm` +�H+mH�iBQMX PM2 KB;?i rQM/2` B7 i?2>�mb/Q`z /BK2MbBQM Bb `2�HHv M2+2bb�`vX h?2 �Mbr2` Bb v2b- i?2 /Bz2`2M+2 #2ir22Mi?2 >�mb/Q`z /BK2MbBQM �M/ i?2 #Qt /BK2MbBQM +�M #2 +`m+B�H BM bQK2 +�b2bX AMT�ìB+mH�`- 6�H+QM2` b?Qr2/ BM (k) i?�i i?2 7QHHQrBM; `2H�iBQMb?BT ?QH/b 7Q` �MvS µ Rn,

dimH(S) Æ dimB(S) Æ dimB(S).q2 /2KQMbi`�i2 i?2 MQM@+QBM+B/2M+2 Q7 i?2 >�mb/Q`z �M/ i?2 #Qt /BK2MbBQM

rBi? �M 2t�KTH2X

1t�KTH2 RdX G2i � /2MQi2 Q fl [0, 1]- i?2M dimB(A) = 1 �M/ dimH(A) = 0._2+�HH i?�i i?2 +HQbm`2 Q7 � b2i a- cl(S) Bb i?2 bK�HH2bi +HQb2/ b2i +QMi�BMBM;

SX h?2 #Qt /BK2MbBQM Q7 � +�M #2 Q#i�BM2/ #v `2�HBbBM; i?�i B7 � }MBi2 +QHH2+iBQMQ7 +HQb2/ +m#2b rBi? bB/2 H2M;i? ” +QMi�BMb �- i?2M Bi �HbQ +QMi�BMb �Ƕb +HQbm`2-

89

M�K2Hv cl(A) = [0, 1]X >2M+2- i?2 KBMBKmK MmK#2` Q7 +m#2b M22/2/ iQ +Qp2` �+�M �HbQ +Qp2` cl(A)X Ai 7QHHQrb i?�i

dimB(A) = dimB(cl(A)) = 1.

JQ`2Qp2`- � bBM;H2iQM Ai ?�b x2`Q@/BK2MbBQM�H >�mb/Q`z K2�bm`2 H0(Ai) = 1-?2M+2 dimH(Ai) = 0. A7 r2 /2MQi2 � �b fiŒ

i=1Ai- i?2M dimH(A) = 0 #v h?2Q`2K ReX.2}MBiBQM keX h?2 QT2M b2i +QM/BiBQM, G2i {F1, . . . , Fm}, Fi : Rn æ Rn #2 bBK@BH�`BiB2b- r2 b�v i?�i Fi b�iBb7v i?2 QT2M b2i +QM/BiBQM B7 i?2`2 Bb � MQM@2KTiv#QmM/2/ QT2M b2i V bm+? i?�i

m€

i=1Fi(V ) µ V

rBi? i?2 mMBQM /BbDQBMiXq2 bi�i2 i?2 7QHHQrBM; i?2Q`2K rBi?Qmi T`QQ7 BM Q`/2` iQ /2KQMbi`�i2 bBim�iBQMb

r?2M >�mb/Q`z /BK2MbBQM �M/ #Qt /BK2MbBQM +QBM+B/2X h?2 T`QQ7 +�M #2 7QmM/BM (k)Xh?2Q`2K R3X amTTQb2 i?�i i?2 QT2M b2i +QM/BiBQM ?QH/b 7Q` i?2 bBKBH�`BiB2b {F1, . . . , Fm}, Fi :Rn æ Rn rBi? `�iBQ ci œ (0, 1) r?2`2 (1 Æ i Æ m)X A7 S Bb i?2 BMp�`B�Mi b2i b�iBb@7vBM; S = fim

i=1Fi(S)- i?2M

dimH(S) = dimB(S) = d, r?2`2 d Bb ;Bp2M #v,mÿ

i=1cd

i = 1.

JQ`2Qp2`- i?2 d@/BK2MbBQM�H >�mb/Q`z K2�bm`2 b�iBb}2b 0 < Hd(S) < Œ 7Q` i?BbbT2+B}+ p�Hm2 Q7 dX

h?2 7QHHQrBM; irQ `2bmHib 7`QK (k) r?B+? r2 �HbQ bi�i2 rBi?Qmi T`QQ7- rBHH #2mb2/ BM i?2 M2ti b2+iBQMX h?2v rBHH �HHQr mb iQ 2biBK�i2 i?2 >�mb/Q`z /BK2MbBQM7Q` i?2 *�MiQ` b2i r?B+? �TT2�`2/ 7`QK /vM�KB+b Q7 HQ;BbiB+ K�Tb BM a2+iBQM kX9�M/ r?Qb2 #Qt /BK2MbBQM r2 7QmM/ 2t+22/BM;Hv /B{+mHi iQ +QKTmi2Xh?2Q`2K RNX G2i {F1, . . . , Fm}, Fi : Rn æ Rn #2 +QMi`�+iBQMb QM � +HQb2/ bm#b2iS Q7 Rn bm+? i?�i |Fi(x) ≠ Fi(y)| Æ ci|x ≠ y|, rBi? ci œ (0, 1), ’x, y œ S, i?2M

dimH(S) Æ d, r?2`2 d Bb ;Bp2M #v,mÿ

i=1cd

i = 1.

h?2Q`2K kyX G2i {F1, . . . , Fm}, Fi : Rn æ Rn #2 +QMi`�+iBQMb QM � +HQb2/ bm#b2iS Q7 Rn bm+? i?�i ci|x ≠ y| Æ |Fi(x) ≠ Fi(y)|, rBi? ci œ (0, 1), ’x, y œ S. amTTQb2U Bb �M BMp�`B�Mi b2i b�iBb7vBM; U = fim

i=1Fi(U) rBi? mMBQMb /BbDQBMiX h?2M

d Æ dimH(S), r?2`2 d Bb ;Bp2M #v,mÿ

i=1cd

i = 1.

88

dXj *QM+HmbBQM �M/ 6mì?2` aim/B2bG2i mb MQr `2im`M iQ b2+iBQM kX9X h?2 *�MiQ` b2i � © flŒ

i=1�n r�b Q#i�BM2/ #v`2KQpBM; i?2 b2ib {Si} 7`QK D = [0, 1]- r?2`2 Sn = {x œ D|gn

µ(x) > 1, 7Q` µ > 4}X6Q` 2t�KTH2- S1 +�M #2 Q#i�BM2/ #v bQHpBM; i?2 2[m�iBQM gµ(x) = 1- BX2X

S1 =A

12 ≠

Û14 ≠ 1

µ,12 +

Û14 ≠ 1

µ

B

= (›, 1 ≠ ›).

A7 r2 /2MQi2 › = 12 ≠

Ò14 ≠ 1

µ �M/ 1 ≠ › = 12 +

Ò14 ≠ 1

µ X h?2`2 �`2 irQ K�TTBM; F1�M/ F2 i?�i K�T D #BD2+iBp2Hv QMiQ [0, ›] `2bT2+iBp2 [1 ≠ ›, 1]X h?2b2 K�TTBM;b �`2

Y]

[F1(x) = 1

2 ≠Ò

14 ≠ x

µ

F2(x) = 12 +

Ò14 ≠ x

µ .

"v i?2 J2�M o�Hm2 i?2Q`2K- i?2 7QHHQrBM; `2H�iBQMb?BT ?QH/b 7Q` x, y œ D �M/x ”= y,

FziœD

Õ

i(zi) = |Fi(x) ≠ Fi(y)|

|x ≠ y| rBi? i = 1, 2.

h?mb-infxœD

|F Õ

i (x)| Æ |Fi(x) ≠ Fi(y)||x ≠ y| Æ sup

xœD|F Õ

i (x)|.

JQ`2Qp2`- bBM+2

|F Õ

i (x)| = 12µ

A14 ≠ x

µ

B≠

12

, i = 1, 2,

q2 ?�p2

1µ

|x ≠ y| Æ |Fi(x) ≠ Fi(y)| Æ 12µ

A14 ≠ 1

µ

B≠

12

|x ≠ y|, rBi? i = 1, 2.

AM Q`/2` 7Q` F1 �M/ F2 iQ #2 +QMi`�+iBQMb- r2 `2[mB`2 i?2 7QHHQrBM; +QM/BiBQM iQ #2b�iBb}2/

0 <1

2µ

A14 ≠ 1

µ

B≠

12

< 1, BX2X µ > 2 +Ô

5.

LQr i?�i F1, F2 �`2 +QMi`�+iQ`b rBi? `�iBQ 12µ(1

4 ≠ 1µ)≠

12 < 1- r2 ?�p2 dimH(�) Æ d

#v h?2Q`2K RN- r?2`2 d Bb ;Bp2M #v 2( 12µ(1

4 ≠ 1µ)≠

12 )d = 1X >2M+2 r2 +�M bQHp2 7Q`

d �M/ Q#i�BMdimH(�) Æ ln2

ln3

µ11 ≠ 4

µ

2 124

.UeV

8e

AM �//BiBQM- bBM+2 � Bb BMp�`B�Mi 7Q` F1, F2X q2 ?�p2 s Æ dimH(�) #v h?2Q`2K ky-r?2`2 s Bb ;Bp2M #v 2( 1

2µ)s = 1X h?mb r2 ?�p2

ln2lnµ

Æ dimH(�). UdV

*QK#BMBM; 2[m�iBQM UeV �M/ 2[m�iBQM UdV- r2 ?�p2 ,

ln2lnµ

Æ dimH(�) Æ ln2

ln3

µ11 ≠ 4

µ

2 124

.U3V

P#b2`p2 i?�i dimH(�) ¥ ln2lnµ r?2M µ Bb H�`;2X AM T�ìB+mH�`- dimH(�) æ 0 �b

µ æ Œ.AM i?Bb T�T2`- r2 bi�ì2/ rBi? i?2 HQ;BbiB+ 7mM+iBQM �M/ bim/B2/ i?2 /BK2MbBQM

Q7 i?2 *�MiQ` b2i � ;2M2`�i2/ #v i?2 K�TTBM; gµ r?B+? Bb �M B``2;mH�` *�MiQ` b2i;2M2`�i2/ #v `2KQpBM; QT2M BMi2`p�Hb Q7 /Bz2`2Mi H2M;i? 7`QK (y-R)X .m`BM; i?2+Qm`b2 Q7 i?Bb bim/v- r2 �HbQ 2tTHQ`2/ /Bz2`2Mi /2}MBiBQMb Q7 /BK2MbBQM r?2`2 r2H2�`M2/ i?�i i?2 >�mb/Q`z /BK2MbBQM Bb � KQ`2 /2HB+�i2 /2}MBiBQM Q7 /BK2MbBQMrBi? KQ`2 ~2tB#BHBiv �M/ rB/2` b+QT2X

6mì?2` /B`2+iBQM Q7 bim/v Bb iQ +QKTmi2 i?2 2t�+i /BK2MbBQM Q7 i?2 *�MiQ`b2i � �M/ iQ bim/v /vM�KB+b Q7 MQM@HBM2�` K�TTBM;b bm+? �b i?2 7mM+iBQM f(x) =⁄xsin(“x) r?2`2 ⁄ Q` “ Bb � T�`�K2i2`X

8d

_272`2M+2b(R) _XGX .2p�M2v- � 6B`bi *Qm`b2 BM *?�QiB+ .vM�KB+�H avbi2Kb- S2`b2mb "QQFb

Sm#HBb?BM;- GXGX*X- RNNk

(k) EXCX 6�H+QM2`- 6`�+i�H ;2QK2i`v, K�i?2K�iB+�H 7QmM/�iBQMb �M/ �TTHB+�iBQMb-CQ?M rBH2v � aQMb Gi/X- RNNy

(j) AX J�MQH2b+m- h?ûQ`B2 /2 H� K2bm`2 2i BMiû;`�iBQM- lMBp2`bBiû /2 6`B#Qm`;-kyRd

(9) 1XJX ai2BM- _X a?�F�`+?B- _2�H �M�HvbBb- S`BM+2iQM lMBp2`bBiv S`2bb- kyy8

(8) aX>X ai`Q;�ix- LQMHBM2�` .vM�KB+b �M/ *?�Qb- S2`b2mb "QQFb Sm#HBb?BM;-GXGX*X- RNN9

(e) ?iiTb,ff2MXrBFBT2/B�XQ`;frBFBf"B7m`+�iBQMn/B�;`�KOfK2/B�f6BH2,GQ;BbiB+J�Tn"B7m`+�iBQM.B�;`�KXTM;U�++2bb2/ d J�`+? kyR3V

(d) ?iiTb,ff2MXrBFBT2/B�XQ`;frBFBf*�MiQ`nb2i U�++2bb2/ N J�`+? kyR3V

(3) ?iiT,ffrrrX+K�i?XBM7Qf?iKHfFQ+?S2`BK2i2`X?iKH U�++2bb2/ Ry �T`BHkyR3V

(N) ?iiT,ff7`�+i�H7QmM/�iBQMXQ`;fP6*fP6*@Ry@jX?iKH U�++2bb2/ ky �T`BHkyR3V

(Ry) ?iiT,ffBM7Q?QbiXMKiX2/mf�#H2rBbf?iKHfFQ+?�MiB@bMQr7H�F2@bim/2MiX?iKH U�++2bb2/ k8 �T`BH kyR3V

(RR) ?iiT,ffb+B2M+2`2b@2/+T@2/m+XbBi2bXQHiXm#+X+�f7BH2bfkyR8fyRfb2+nK�i?n;2QK2i`vnK2M;2`XT/7 U�++2bb2/ ke �T`BH kyR3V

(Rk) ?iiT,ffK�i?rQ`H/XrQH7`�KX+QKfaB2`TBMbFB*�`T2iX?iKH U�++2bb2/ k3�T`BH kyR3V

(Rj) ?iiTb,ff7`�+i�H7QmM/�iBQMXQ`;f`2bQm`+2bf7`�+iBpBiB2bfFQ+?@+m`p2f U�++2bb2/ k3 �T`BH kyR3V

83

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