Cantor Minimal System

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NTNU Norwegian University of Science and Technology Faculty of Information Technology and Electrical Engineering Department of Mathematical Sciences

Max Lunde Hauge

Cantor Minimal Systems

Bachelor’s project in Mathematical Sciences Supervisor: Eduardo Ortega Esparza

May 2021

Bachelor ’s pr oject

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Max Lunde Hauge

Cantor Minimal Systems

May 2021

Norwegian University of Science and Technology

Faculty of Information Technology and Electrical Engineering Department of Mathematical Sciences

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c

(9)

⇥ C

C

⇥

{0, 1}^N

i

(⇥,)) (+,⇢)

(10)

(11)

⇥

C

R

C

[0, 1] ⇢ R C

⇠0 =[0, 1] ⇠= ⇢ C

= 1

⇠₀= [0, 1],

⇠1=⇠0\

✓1 3,2

3

◆

=

 0,1

3 [

2 3, 1 , ...

⇠₌ = ⇠₌ ₁

3 [

✓2

3 +⇠₌ ₁ 3

◆ .

(12)

C= Ÿ1

==1

⇠₌.

⇠= 2⁼ 1/3⁼

0, 1 2 C

1 C

3,²₃ 2 C

C

(00. . .0=) = 2N

(0₌) 0₌

0 1 (0=) =

= 0= 2 {0, 1}⁼ (0=)

(0₌) 2{0, 1}^N

0 1

|A| < 1

’1 :=0

0A^: = 0 1 A. A < 1

’= :=0

0A^: =0

✓1 A⁼⁺¹ 1 A

◆ .

G 2C

’1

==1

20₌

3⁼ 0₌ 2{0, 1}.

< 2 N < 1 ⇠< < <

(13)

⇥

⇠_< ⇠_<

<

G 2 C

’<

==1

20₌ 3⁼

| {z }

403

+

’1

==<+1

20₌ 3⁼

| {z }

) 08;

0₌ 2{0, 1}.

0₌ 0= =1 =

’1

==<+1

20=

3⁼  2

’1

==<+1

1 3⁼

!

=2

’1

==0

1 3⁼

’<

==0

1 3⁼

! ,

 2 1 1 ¹₃

1 ¹₃ ^<⁺¹ 1 ¹₃

!

=2 1+ ( 1) + ¹₃ ^<⁺¹

2 3

! ,

 3

✓1 3

◆<+1

= 1 3^<.

⇠<

G G 2⇠_< <

C=—₁

<=1⇠<

’1

==1

20₌

3⁼ 0₌ 2{0, 1}.

0= = 0 = 0= = 1

=

0, 1 2 C

’1

==1

2·0 3⁼ =0,

’1

==1

2·1 3⁼ =2·

1 3

1 ¹₃ =1.

G 2C

G C

G

(14)

= = 4

0₁ = 1 0₂ =0 0₃ =1 0₄ =0

’4

==1

20=

3⁼ = 2·1

3¹ + 2·0

3² + 2·1

3³ + 2·0 3⁴ = 20

27.

0= = =4

= =4 0= = =4

1/3⁴

C ✓ [0, 1] ✓ R R

( ✓ R ✓ = [0,1] =(0,1)

✓( ) =1 0.

C 2⁼

1/3⁼

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⇥

1

3 +2· 1

9 +4· 1

27 +8· 1

81 + · · ·=

’1

==1

2⁼ ¹ 3⁼ .

’1

==1

2⁼ ¹ 3⁼ =

’1

==1

2⁼ ¹·3 ⁼ = 1 3

1 1 ²₃

!

=1.

1 C

C

C (0=) 2{0, 1}^N

! C

! =C !

(0₌) (1=) ...

N (G=)

!

! (U₌)

!

(0=) 2! U0 2 (U=)

0₀ 2 (0₌) U0 =

( 1 0₀ =0, 0 0₀ =1.

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U₀<0₀ (U₌) < (0₌) (1₌) 2!

U₁ 2 (U=) 1₀ 2 (1=)

U₁ =

( 1 1₁ =0, 0 1₁ =1.

U1 <11 (U=) < (1=) (G=) 2!

(U₌) (G₌) 2! !

(U₌)

! ! (U=) !

(V₌) 8 !

C C

C

⇥3 #

0 ! 1 3⁰

1 ! 3 3¹

2 ! 9 3²

3 ! 27 3³

C G ! 2 3^G

C

0 1

C ⇢ [0, 1]

[0, 3] C

C

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⇥

2=3^G G = ^{log 2}_{log 3} ' 0.631

C

1 2 + 1

6+ 1

18 + · · ·= 1 2+

’1

==1

2 2⇤3⁼ = 1

2 +

’1

==1

1 3⁼. 1

⇠₀ =

(18)

1 2 +

’1

==1

0₌

3⁼ 0= 2{0, 1}.

C

g

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⇥

C

⇥ R

R

- 3 3 :-⇥- ! R

" - 3 (-,3) 3

3(G,~) >0 3(G,~) =0 G =~ 3(G,~) =3(~,G) G,~ 2-

3(G,I) 63(G,~) +3(~,I) G,~,I 2- R

3(0,1) = |0 1| 3

0,1 2 - 3(0,1) = 0 0 = 1 3(0,1) > 0 0,1 2 R

0,1,2 2R

R R

(-,3) (0:)_:¹₌₀ 2-

A > 0 # 2N <,=>#

3(0=,0<) <A.

" = (-,3)

- -

R R

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(-,3) G 2- n > 0

n G n -

B(G;n) ={~ 2- |3(G,~) <n}.

(R,3) (G n;G+n)

G n

(-,3) ( ✓-

G 2( n B(G;n) ✓(

{ ^U} U 2N

* =–

U2N U 0 2 ^U ✓*

02* n B ✓ U ✓* 0 ⌫ ✓*

*

0,. . ., V V 2N

+ =—^V

:=0 : 02* 0

: n B ✓ ^: 0:  V

n

n =minn0,. . .,nV B(0,n) ✓ B(0,n:) B(0,n) ✓+ +

(-,3) ( ✓ - G 2-

( B(G;n) ( G

(-,3) ( ✓- G 2(

( (

(-,3) G 2- n > 0

n G n -

B(G;n) ={~ 2R |3(G,~)6n}.

(-,3) ( ✓ -

( G (

( " = (-,3)

(0_:) ( (0_:) - ( ✓- "

G 2- ( ( =(

G 2( (

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⇥

8 ✓ -

Ÿ

82 8

!2

⌘ÿ

82

( 8)² ÿ

82 8

!2

⌘ Ÿ

82

( 8)².

C

R C ⇢ R

(-,3) ( ✓ -

(-,3) ( ✓- !

( ( ( ( ✓ -

( =( [!.

( (0,1) ( = (0,1)[! = [0,1]

( =( (

(-,3) ( ✓ - (

(² (² ✓-

(² ={0 2- |08(}.

( (²

( ( ✓ - G 2 (² n

G (² G 2(

n B ✓$

G 2$²

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& & ✓- &

G 2&² G 8& &²

n G & B ✓&² $²

' ✓ - ('²)² = '

(-,3) ( ✓ - ( -

( =-

(-,3) (-,3) (-, ¯3)

5 :- ! - 3¯(5(G),5(~)) =3(G,~) G,~ 2- 5(-) ={5(G) :G 2-} -

R C

(C,3) G 2C

C

3 R 0,1 2 C 3 : C⇥C! R

3

3(0,1) =3

’1

==1

20=

3⁼ ,

’1

==1

21=

3⁼

!

=

’1

==1

2|0= 1=| 3⁼ 0=,1= 2 {0, 1}.

3

G 2 C B(G;n) ~ 2 C

n G ~

n G

1/3^< < 2N n

1

3⁼ ¹ <n  ₃¹^<

n > 0 = 2N G 2C

G n = ₃¹= ~ 2 C 3(G,~) < ₃¹= G

= G ~

G 2 C = 2 N n = ₃¹= B(G;₃¹=) = ~ 2C |3(G,~) < ₃¹= G = Õ₁

==02G=

3⁼

~ =Õ₁

==0 2~₌ 3⁼

’1

==1

2|G= ~=|

3⁼ =

’<

==1

2|G= ~=| 3⁼

| {z }

403

+

’1

==<+1

2|G= ~=| 3⁼

| {z }

) 08;

.

1/3^< n

n = 1/3^< < n

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⇥

1/3^<

= G ~

1/3^<

(-,3) ( ✓ -

A > 0 3(0,1) < A 0,1 2 ( ( (-,3)

(-,3) ( ✓ - (

( (

( ✓ R (

R

C C

[0,1] C

1 < |A| [0, 1]

3(0,1) 6 1 < A C

C ⇢ [0, 1] ⇢ R

⇤

(-,3) ( ✓ - (

02(

[2,3] 2 <3

G 2⇠₁ ⇠₁

⇠₁ =

 0,1

3 [

2 3, 1 G1 2C\⇠1 G1 <G

|G G₁|  1/3¹ = 2 N

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⇠₌ G₌ 2 C\⇠₌ G₌ < G

|G G=|  1/3⁼ C

(-,3) (,& ✓ -

(,& ( \& (\&

(-,3) ' ✓ - '=([&

( & ' '

0,1 2' 0 2

1 2⌫ '= [⌫

C C

:,; 2 C

: 2 ; 2! [! ✓⇠? ⇠? ✓⇠=

:,;

⇠=

:,;

⇠₌

C

R

C

⇥

- T T ( ✓ -

-

- ; T

* T * T

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⇥

* T * T

(-,T )

- B T - ⌫ ✓-

G 2- ⌫ 2B G 2⌫

D 2 T G 2D ⌫ 2B G 2⌫ ⌫ ✓D

- B T

⌫ 2B T -

⌫ 2B T *

T G 2* ⌫ 2B G 2⌫ * =–

G2* ⌫ B

(-,3) n

C

B ={B(G;n) |G 2-,n > 0}.

- . 5 :- !.

* ⇢. 5 ¹(*) -

- . 5 :- !.

5 5

- . - .

5 5

⇥

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⇥

0 1

C - ={0, 1}^N 0 1

- {0, 1}

- = {0,. . .,? 1}^N ? 2

⇥

- T - T

- T -

T ;,- ✓ -

T -

G 2-

- -

? 2N ={0, 1, 2,. . .,? 1} =

N ={ }

< 2N ^< ={ < 2N }

⇤ =–₁

<=1 < ={ < 2N }.

G 2 ⁼ ~ 2 ^< =,< 2N

G ~ G ~ G~ =+<

G~ =G ~ 2 ⁼⁺^<.

G~ <~G

G 2 ⁼ ~ 2 ⁼ G =G₀G₁G₂. . .G₌

~ =~₀~₁~₂. . .~< G,~

G~ =G0. . .G=~0. . .~<,

~G =~0. . .~=G0. . .G<.

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⇥

I 2 ^N I=I₀I₁. . .I₈. . . G,I

GI =G0. . .G=I0. . .I8. . .,

IG =I₀. . .I8. . . .G₀. . .G=.

G 2 ⁼ ~ 2 ^< =,< 2N < >=

G ~ I 2 ^{< =} ~ =GI

G 2 ⁼ = ~ 2 ^N

G ~ 2 ^N

- =÷ -8. -

B = n÷

*₈ |*₈ ✓-₈ *₈ =-₈ 8 o

.

*8 = -8

={0,. . .,? 1} G 2

N

l l 2 ^⇤ /(l) ^N

/(l) = l 2 ^⇤,G 2 ^N |lG 2 ^N ,

={ l}.

/(l)

÷N

{0,. . .,? 1} = ^N ={0,. . .,? 1}^N ? 2.

? 2

0

/(l) /([) l [

= ~

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/(l) /([)

/(l)[/([) = 8>>>

>>><

>>>>

>>

:

/(l) [ =l~, /([) l =[~,

/(l₀. . .l= 1) [= <l= =,

/(l)[/([)

l =l₀. . .l₌ [ =[₀. . .[_< = < = <<

l8 <[8 0 8  min{=,<}

l [ [ =l~

~ /([) ✓ /(l) /(l) = /(l) [/([)

[ l

= =< = 1

= l =[₀. . .[= 1l₌ [=l₀. . .l= 1[₌

/([₀. . .[= 1)

/(l) /([) l [

= ~

8 /(l) /([)

8 =/(l)\/([) =8>>><

>>>

:

/([) l ✓[ [ =l~, /(l) [ ✓l l =[~,

;.

l =l₀. . .l₌ [ =[₀. . .[_< = < = <<

l8 <[8 0 8  min{=,<}

l [ [ = l~ [

l l = [~ ~

/([) ✓ /(l) /(l) ✓ /([) /(l) \/([) /(l) /([)

l 2 {0,. . .,? 1}^N

{0,. . .,? 1}^N

⇥ C n {0,. . .,? 1}^N ⇥

9: {0,. . .,? 1}^N ! ⇥

9 n

9

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⇥

{ 0, 1 }

^N

? = 2 9

÷N

{0, 1} ={0, 1}^N.

= =2 /(l) /([)

/(00)[/(01) =/(0), /(00)[/(0010) =/(00),

/(10)[/(110) =/(10)[/(110), /(01)[/(10) =/(01)[/(10).

/(l)

! 4

? = 2 /(l) /([)

/(00)\/(01) =;,

/(00)\/(0010) =/(0010), /(10)\/(110) =;,

/(01)\/(10) =;.

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l =01

/(01) =/(00)[/(1).

9: {0, 1}^N !⇥ 9 9([):/([0. . .[<)! B(Õ₁

8=02[₈

3⁸ , 1/3^<)

9 / (l) ✓ {0, 1}^N

< 2N l =l0. . .l<

9 (/(l)) = (’₁

==0

2G=

3⁼ :G8 =l8, 808 <

)

=B

’1 8=0

2l8

3⁸ ; 1 3^<

! .

G 2bZ_? < > 0 9 ¹ B

’1 8=0

2G8

3⁸ ; 1 3^<

! !

=/ (l₀. . .l_<).

9 ¹ B ✓ bZ_? / (G) ✓ {0, 1}^N ⌘

9 ¹ ¹ = 9 / (G)

B 9 ¹ 9 9 ¹

9 {0, 1}^N ⇥

G 2 {0, 1}^N

B(9(G); 1/3⁼) /(G),

{~ 2 C |3(G,~) < 1/3⁼} G 2 ^⇤,~ 2 ^N |G~ 2 ^N . G =01110

B(9(G); 1/3³) /(011), B(9(G); 1/3⁵) /(01110).

B(9(G); 1/3⁵) ✓ B(9(G); 1/3³), /(01110) ✓ /(011).

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{0,. . .,? 1}^N ? 2

? bZ_?

⌧ ⌧ ⇤

⌧ h⌧,⇤i

(0⇤1)⇤2 =0⇤(1⇤2) 0,1,2 2⌧

4 2⌧ 4⇤G =G⇤4 G 2⌧ G 2⌧ 0⁰2⌧ 0⇤0⁰=0⁰⇤0=4

⌧

? 2N

N[?] = G₀?⁰· · · +G8?⁸ |8 2N G8 2{0,. . .,? 1} ,

= (’_<

8=0

G₈%⁸ |G₈ 2{0,. . .,? 1} )

.

3? N[?]

3?(G,~) =3?(0₀+ · · · +0=?⁼,1₀+ · · · +1<?^<) = |0: 1:|

?^: ,

: 0: < 1: N[?] 3?

? b Z_? =

(’₁

8=0

G8%⁸ |G8 2 {0,. . .,? 1} )

.

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0 = 00. . .0< {0,. . .,? 1} <

⌘(0)=Õ₁

8=00₈?⁸

⌘ / (l) ✓ {0,. . .,? 1}^N

< 2N l =l0. . .l<

⌘(/(l)) = (’₁

==0

G=%⁼ :G8 =l8, 808 <

)

=B

’1 8=0

l8?⁸; 1

?^<

! .

G 2bZ_? < > 0

⌘ ¹ B

’1 8=0

G8%⁸; 1

?^<

! !

=/ (l₀. . .l<).

⌘ ¹ B ✓ bZ_? / (G) ✓ {0,. . .,? 1}^N

⌘ ⌘ ¹ ¹ / (G)

B ⌘ ¹ ⌘ ⌘ ¹

⌘ {0,. . .,? 1}^N bZ?

i

? 2N @ 2N i_@: {0,. . .,? 1}^N ! {0,. . .,? 1}^N

4 2bZ_?

0,1, 0, 1 2bZ? 0+1 =2

= 2N 0 1

2₀ =(G+~)0 =G₀+~₀+Y₀ mod ? Y₀=0, ...

2₌ =(G+~)= =G₌+~₌+Y₌ mod? Y₌ =

( 0 G= 1+~= 1+Y= 1 < ?, 1 G= 1+~= 1+Y= 1 =?.

G 2bZ_? G G G 2bZ_?

G+ ( G) =0.

@ 2 N ? 2 N

q@: bZ? ! bZ? q@(G) =G +@ q_@¹(G) =G @ G 2bZ_?

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q = 2N q

q_@⁼ =q_@ . . . q_@

| {z }

=

=q_=? =? =?+ · · · +?

| {z }

=

.

q bZ_?

q bZ_? q

@ 2N ? 2N i@: {0,. . .,? 1}^N !

{0,. . .,? 1}^N G 2 {0,. . .,? 1}^N i

i@ (G) =⌘ ¹(q@(⌘(G))).

⌘ :{0,. . .,? 1}^N !bZ_? q_@:bZ_? !bZ_? @

-,.,/ 5 :- !. 6:. !/

5 6 6 5: - !/

* ✓ / / (6 5) ¹(*) = {8G 2- |6(5(G)) 2*} = 8G 2- |G 2 5 ¹(6 ¹(*)) = 5 ¹(6 ¹(*)) 6 5 6 ¹(*)

. 5 ¹(6 ¹(*)) - (6 5)

5 6

(6 5) (6 5)

(6 5) ¹

i@ {0,. . .,? 1}^N

⇥ {0,. . .,? 1}^N {0,. . .,? 1}^N

bZ? bZ?

i_@

⌘ ⌘ ¹

q_@

' '

i = ⌘ ¹q⌘ ⌘

i

(34)

@=1 ? =2

q1:bZ₂!bZ₂ q1(G) =G+1 q₁¹(G) =G 1 G 2bZ₂ 1, 12bZ₂ : 2N

1=1+0·2¹+0·2²+ · · ·=1+

’1

==1

0·?⁼ 1=1+1·2¹+1·2²+ · · · =

’1

==0

1·?⁼. q⁼(G)

@< 1

G G 2 bZ₂

{0, 1}^N

G =1+2¹+2²+2³+ · · ·= (111111111111. . .), G =1+0+0+0+0+ · · ·= (100000000000. . .), G+ ( G) =0+0+0+0+0+ · · ·= (000000000000. . .).

02bZ₂ 0=0+0·2¹+0·2²+. . .

i² q_@²

{0, 1}^N {0, 1}^N {0, 1}^N

b

Z₂ bZ₂ bZ₂

i

i²

i

⌘

q₁

q₁²

⌘

q₁

⌘

'

0+0=0 0+1=1 1+1=0 1

G G

(35)

? =2 @ =1 / (l) ✓ {0, 1}^N i

l =l8l⁰=l0. . .l< l8

l⁰ l₈

i(l8l⁰) =

( 1l⁰ l₈ ₁ =0,

0i(l⁰) l₈ ₁ =1 08 =.

i

? = 2 @ = 1 / (G) ✓ {0, 1}^N

<

l =l0. . .l< =l0l1l⁰.

l =l₀+l₁·2¹+l₂·2²+ · · · +l<·2^<, 1=1+0·2¹+0·2²+ · · · +0·2^<.

l₀+1 l₁+0+n

n l₀ = 1

0 = 2 mod 2 1 =2¹

l₁+0+2¹ 1 mod 2 1 2bZ₂

0 02bZ₂

i i(l₀l₁l⁰) = 0l₁l⁰ l₀ = 1

i(l₁l⁰) 1 = 2¹

i(l0l1l⁰) = 1l1l⁰ l0 = 0

l G,~ 2 {0, 1}^N

G =G₀G⁰=G₀G₁G⁰⁰ ~ =~₀~⁰=~₀G₁~⁰⁰ G⁰,~⁰ G⁰⁰,~⁰⁰

i ¹ l =01 i ¹

12bZ₂ 1=1000. . . {0, 1}^N

i ¹(/(l)) = G 2{0, 1}^N |i(G)=l~ .

i ¹ G =1G⁰ i(1G⁰) =0i(G⁰) =0~⁰ i ¹ G =10G⁰⁰ i(10G⁰⁰) =0i (0G⁰⁰) =01i(G⁰⁰) =01~⁰⁰

i(/(01)) = G 2{0, 1}^N |i(G)=01~ , i(/(01)) =/(10).

(36)

(37)

( ⇥, ) )

i = 2 Z

⇥

G 2⇥ i

i

i: ⇥ !⇥

G 2⇥ G

G

i = 2Z

G ⇥

⇥

i

G

(⇥,)) ⇥

) :⇥ !⇥

(⇥,)) G 2 ⇥

⇥

{)⁼(G) |= 2Z}.

) G

G

(38)

(⇥,)) G 2 ⇥

= <0 )⁼(G) =G

(⇥,)) G 2⇥

~ 2⇥ n > 0 = 2Z

3()⁼(G),~) <n.

G

.

n )⁼(G)

Tⁿx

(⇥,)) )

n

G ~

G ⇥

(⇥,)) ) : ⇥ ! ⇥

i ?

@

?,@ 2 N G,~ 2 ⇥ i: ⇥ ! ⇥

? @ gcd(?,@) =1

i = 2N 3(i(G),~) < 1/3⁼

= G ~

⇧: bZ? ! Z/?⁼Z Õ₁

<=00<?^< ! Õ= 1

<=00<?^<

mod (2⁼) ⇧

?⁼

bZ? Z/?⁼Z

G 2 Z_? iG =G +@ ⇧(G +@)

(39)

(⇥,))

G 2bZ? Z/?⁼Z

bZ? Z/?⁼Z

⇧

i i

⇧

⇧(G) i(⇧(G)) = ⇧(G) +@ i b

Z_? i

i

i bZ? gcd(?,@) = 1

gcd(?,@) =1

? =2 @ =3 gcd(?,@) =1

G 2 {0, 1}^N G < =3 G =100G⁰

G i⁼: {0, 1}^N ⁺!³ {0, 1}^N = 2Z

: 100 ⁺!³ 001 ⁺!³ 111 ⁺!³ 010 ⁺!³ 101 ⁺!³ 000 ⁺!³ 110 ⁺!³ 011 ⁺!³ 100.

3 0 1

8 i⁸(100) = 100

G

0 1 i⁼(G) <G = 2 N

==0 G

0 1 i n

i⁼(G) ~ 2{0, 1}^N :

=

? =2 @ =2 gcd(?,@) =2 : 100 ⁺!² 110 ⁺!² 101 ⁺!² 111 ⁺!² 100,

: 010 ⁺!² 001 ⁺!² 011 ⁺!² 000 ⁺!² 010.

2 0 1

8

G

n G

: i⁼(G) ~

=

⇥,)

(40)

( + , ⇢ )

(⇠,))

G 2C

+₁

+2

+₃

+4

⇢₁

⇢₂

⇢3

= 1 (+,⇢)

+ +₀. . .+₌. . .

+₀ = {E₀} +₁ = {E₁,. . .,}

⇢ ⇢0. . .⇢=. . .

⇢₀ = {4₀,. . .,} ⇢₁ = {4₁,. . .,}

B: ⇢= ! += 1 B ¹({E}) E [⁼ 0+= A: ⇢= ! +=

A ¹({E}) E [⁼ ¹+=

E₀

(+,⇢) 0  < < =

+< += %<,=

%_<,= ={(?_<₊₁,?_<₊₂,. . .,?₌) |?₈ 2⇢₈,< <8 =, A (?₈) =B (?₈₊₁),< < 8 <=}.

(41)

(⇥,))

+₀ ={E₀} ?

%⇢ ={(?1,?2,. . .) |?: 2⇢:, A (?:) =B(?:+1),: 1}.

?_:

?_:₊₁

= 2N (⇢,+) = |E₌|=:

= "= 2"_:_⇥_: (N)

"=(8,9)= ⇢= 8 2+= 1 9 2+=

"₌ ="_< =,< 2N

"= ="⇢

"_⇢ "_⇢ = 2 N

"_⇢⁼ ="_⇢· · · · ·"_⇢

| {z }

=

"_⇢

"_⇢ = (2),

| {z } "_⇢ = 1 1 1 1

! ,

| {z }

"_⇢ = 1 0 0 1

! .

| {z }

"_⇢ (1)

%⇢

40. . .4= E0

l =4₀. . .4₌

/ (l) ={lG 2%⇢ |l =40. . .4=,G 2%⇢},

={ l}.

%⇢

(+,⇢, ) (+,⇢)

⇢ 4,4⁰ 2 ⇢ A(4) = A(4⁰)

(42)

W = 4₁. . .4₌ d = 5₁. . .5₌ A(W) = A(d) (+,⇢, )

-_<0G =U₀U₁. . . U 2⇢_<0G -_<8= =V₀V₁. . . V 2⇢_<8=

(+,⇢, )

%_⇢ ): %_⇢ !%_⇢

)(-<0G) =)(-<8=) )(G) =~(G=+1)G=+1 G= 2⇢8

G: 8 ⇢<0G : )(G) = (~1~2. . .~: 1(4: +

1)4_:₊₁. . .) 4_: +1 5

4_: +1

G 2 {0, 1}^N G = 1110011G⁰ G⁰

) G

)(G)=)(1110011G⁰) =000(0+1) 011G⁰=0001011G⁰.

: = 4 4: 8 ⇢<0G @ = 1 ? =2

{0, 1}^N i₁ G

i1(1110011) =0i1(110011) =00i1(10011) =· · ·=0001011.

{0, 1}^N

(43)

(⇥,))

(44)

E_max E_min

0 1

0

0 1

0

0 0

1

(45)

(46)

NTNU Norwegian University of Science and Technology Faculty of Information Technology and Electrical Engineering Department of Mathematical Sciences

Cantor Minimal Systems

May 2021