Mobius number systems

Petr KurkaCenter for Theoretical Study

Academy of Sciences and Charles University in Prague

Dynamics and Computation

Marseille, February 2010

Iterative systems

X compact metric space, A finite alphabet(Fa : X → X )a∈A continuous.(Fu : X → X )u∈A∗, Fuv = Fu ◦ Fv , Fλ = Id

Theorem(Barnsley) If (Fa : X → X )a∈A arecontractions, then there exists a unique attractorY ⊆ X with Y =

⋃a∈A Fa(Y ), and a continuous

surjective symbolic mapping Φ : AN → Y

{Φ(u)} =⋂

n>0

Fu[0,n)(X ), u ∈ AN

Iterative systems

X compact metric space, A finite alphabet(Fa : X → X )a∈A continuous.(Fu : X → X )u∈A∗, Fuv = Fu ◦ Fv , Fλ = Id

Theorem(Barnsley) If (Fa : X → X )a∈A arecontractions, then there exists a unique attractorY ⊆ X with Y =

⋃a∈A Fa(Y ), and a continuous

surjective symbolic mapping Φ : AN → Y

{Φ(u)} =⋂

n>0

Fu[0,n)(X ), u ∈ AN

Binary system A = {0, 1}, Φ2 : AN → [0, 1]

F0(x) =x

2, F1(x) =

x + 1

2

Φ2(u) =∑

i≥0

ui · 2−i−1, u ∈ AN

0 1

[0][1]

Binary signed system

A = {1, 0, 1}, Φ3 : AN → [−1, 1]

F1(x) =x − 1

2, F0(x) =

x

2, F1(x) =

x + 1

2

Φ3(u) =∑

i≥0

ui · 2−i−1, u ∈ AN

-1 1

[1] - [0] [1]

In the standard decadic system, the addition is notalgorithmic

0.3333333333333333333333333333...0.6666666666666666666666666666...

?

0.33333333332 0.333333333340.66666666664 0.66666666668

0.9999999999 1.0000000000

In the standard decadic system, the addition is notalgorithmic

0.3333333333333333333333333333...0.6666666666666666666666666666...

?

0.33333333332 0.333333333340.66666666664 0.66666666668

0.9999999999 1.0000000000

Redundant symbolic extensions

Theorem(Weihrauch) Any compact metric space Y

has a redundant continuous symbolic extensionΦ : AN → Y :Any continuous map G : Y → Y can be lifted to acontinuous F : AN → AN with ΦF = GΦ.

AN F//

��

AN

��

YG

// Y

The binary system Φ2 is not redundant.The binary signed system Φ3 is redundant.

Real orientation-preserving Mobius transformations

M : R → R, where R = R ∪ {∞}

Ma,b,c ,d(x) =ax + b

cx + d, ad − bc > 0

F0(x) = x/2hyperbolic

F1(x) = x + 1parabolic

F2(x) =4x+13−x

elliptic

1/0

-4

-2

-1

-1/2

-1/4

0

1/4

1/2

1

2

4

1/0

-3

-2

-1

0

1

2

3

1/0

-1

0

1

Probability densities, F0(x) =x2 , F1(x) = 1 + x

-3 3

1

-3 3

10

-3 3

101

-3 3

1010

-3 3

10101

-3 3

Complex sphere C = C ∪ {∞}

-2 2

1/0

-2

-1

-1/2

-1/4 0

1/4

1/2

1

2

d(z) =iz + 1

z + istereographic projection

d : R → ∂D = {z ∈ C : |z | = 1}

disc Mobius transformations

U = {x + iy ∈ C : y > 0}: upper half-planeD = {z ∈ C : |z | < 1}: unit discd : U → D,

M : U → U real Mobius transformationsM = dMd

−1 : D → D disc Mobius transformationspreserve hyperbolic metric.

Disc Mobius transformations M = d ◦M ◦ d−1

F0(z) =3z−iiz−3

F0(x) = x/2hyperbolic

F1(z) =(2i+1)z+1

2i−1

F1(x) = x + 1parabolic

F2(z) =(7+2i)z+i

−iz+(7−2i)

F2(x) =4x+13−x

elliptic

1/0

-4

-2

-1

-1/2

-1/4

0

1/4

1/2

1

2

4

1/0

-3

-2

-1

0

1

2

3

1/0

-1

0

1

Mean value E(Mℓ) =

∫

∂D

z d(Mℓ) = M(0)

Convergence

ℓ: the uniform measure on ∂D = {z ∈ C : |z | = 1},x ∈ R. Equivalent conditions:

limn→∞

Mnℓ = δd(x) point measure on d(x)

limn→∞

Mn(0) = d(x)

limn→∞

Mn(z) = d(x) for every z ∈ D

limn→∞

Mn(z) = x for every z ∈ U

∃c > 0, ∀I ∋ x , lim infn→∞

||M−1n (I )|| > c

Mobius number system(MNS) (F ,Σ)

(Fa : R → R)a∈A Mobius iterative systemΦ : XF → R symbolic map

XF = {u ∈ AN : limn→∞

Fu[0,n)(i) ∈ R}

Φ(u) = limn→∞

Fu[0,n)(i) ∈ R, u ∈ XF

Σ ⊆ XF is a subshift such that Φ : Σ → R iscontinuous and surjective.

Binary signed system A = {1, 0, 1, 0}

F1(x) = −1 + x

F0(x) = x/2

F1(x) = 1 + x

F0(x) = 2x

forbidden words:

11, 00, 11, 00, 10, 10

101, 101, 111, 111

u = 0n1x001x101x2 . . .

Φ(u) =∑

∞

i=0 xi · 2n−i

xi ∈ {−2,−1, 0, 1, 2}

1/0

-8

-4

-3

-2

-3/2

-1

-3/4

-1/2

-1/4 0

1/4

1/2

3/4

1

3/2

2

3

4

81-

0

1

0 -

11--

10 -

01 -

00

01

10

11

01-- 01-

00--

111 ---

110 --

101 - -

100 -

010 - 0

01

-

000

001

010

100

101

110 111

011 --- 011- 001 --- 001--

Continued fractions a0 −1

a1 −1

a2 − · · ·

= F a01 F0F

a11 F0 · · ·

F1(x) = −1 + x

F0(x) = −1/x

F1(x) = 1 + x

forbidden words:

00, 11, 11, 101, 101

Interval almost-cover:

W1 = (∞,−1)

W0 = (−1, 1)

W1 = (1,∞)

xa→ F−1

a (x) if x ∈ Wa

-2 -1 0 1 2

11

0

1-

1/0

-5

-4

-3

-2

-3/2

-1

-2/3

-1/2

-1/4 0

1/4

1/2

2/3

1

3/2

2

3

45

1-0

1 11--

10-

01 -01

10 11

111 ---

110--

101

-

011 --

010 -

010

011

101

-

110

111

Circle metric and derivation

the length of arc between d(x) and d(y):

(x , y) = 2 arcsin|x − y |√

(x2 + 1)(y 2 + 1)

circle derivation of M(x) = (ax + b)/(cx + d):

M•(x) =(ad − bc)(x2 + 1)

(ax + b)2 + (cx + d)2

= limy→x

(M(x),M(y))

(d(x),d(y))= |M ′(d(x))|

(MN)•(x) = M•(N(x)) · N•(x).

Contracting and expanding intervals

Uu = {x ∈ R : F •u (x) < 1}, Fu(Uu) = Vu

Vu = {x ∈ R : (F−1u )•(x) > 1}

F(x)=x/2UV

F(x)=x+1U

V

Theorem If {Vu : u ∈ A∗} is a cover of R, thenΦ(XF ) = R and there exists a subshift Σ ⊂ XF

such that (F ,Σ) is a MNS.

Interval almost-cover W = {Wa : a ∈ A}

Wa open intervals with⋃

a∈AWa = R

Expansion graph: xa→ F−1

a (x) if x ∈ Wa

Wu := Wu0 ∩ Fu0(Wu1) ∩ · · · ∩ Fu[0,n)(Wun)

x ∈ Wu iff u is the label of a path with source x :x ∈ Wu0, F

−1u0

(x) ∈ Wu1, F−1u0u1

(x) ∈ Wu2

Expansion subshift:

SW := {u ∈ AN : ∀n,Wu[0,n) 6= ∅}

Theorem If Wa ⊆ Va, then (F ,SW) is a MNS.

Expansion quotient Q(W) of F and W = {Wa : a ∈ A}

q(u) = inf{(F−1u )•(x) : x ∈ Wu}

Qn(W) = min{q(u) : u ∈ An, Wu 6= ∅}

Q(W) = limn→∞

n

√Qn(W)

q(uv) ≥ q(u) · q(v),Qn+m(W) ≥ Qn(W) ·Qm(W)

Theorem If Q(W) > 1, then (F ,SW) is a MNS andΦ([u]) = Wu for each u ∈ L(SW).If W is a cover, then (F ,SW) is redundant.

Theorem If (F ,SW) is a MNS and Φ([u]) = Wu foreach u ∈ L(SW), then Q(W) ≥ 1.

Arithmetical algorithms: expansion graph

M1 = {M(a,b,c ,d) : a, b, c , d ∈ Z, ad − bc > 0}Q = Q ∪ {∞} = {x0

x1: x0, x1 ∈ Z, |x0|+ |x1| > 0}

W is a cover of R with rational endpoints(F ,SW) is redundant.

xa

−→ F−1a (x) if x ∈ Wa

Proposition For x ∈ Q there exists an infinite pathwith source x . If u is its label, then u ∈ SW andΦ(u) = x .

Linear graph

vertices: (M , u) ∈ M1 × SW ,

(M , u)a

−→ (F−1a M , u) if M(Wu0) ⊆ Wa

(M , u)λ

−→ (MFu0, σ(u))

Proposition There exists a path with source (M , u)whose label w = f (u) ∈ SW and Φ(w) = M(Φ(u)).

The map f : SW → SW is continuous andΦf = MΦ.

Fractional bilinear functions M(1,1)

P(x , y) =axy + bx + cy + d

exy + fx + gy + h, M(x) =

ax + b

cx + d.

Mx =

a 0 b 00 a 0 b

c 0 d 00 c 0 d

, My =

a b 0 0c d 0 00 0 a b

0 0 c d

P(Mx , y) = PMx(x , y), P(x ,My) = PMy(x , y),MP(x , y) are fractional bilinear functions.

Singular and zero MT M(z) = (az + b)/(cz + d)

orientation reversing: ad − bc < 0singular: ad − bc = 0, |a|+ |b|+ |c |+ |d | > 0,zero MT : M(0,0,0,0) = 0.

M = {(x0x1, y0y1) ∈ R

2: (ax0+bx1)y1 = (cx0+dx1)y0}.

stable point: sM ∈ {ac, bd} ∩ R,

unstable point: uM ∈ {−ba,−d

c} ∩ R.

singular MT: M = (R× {sM}) ∪ ({uM} × R).

zero MT: M = R2.

The value M(I ) = {y ∈ R : ∃x ∈ I , (x , y) ∈ M}on I = [I0, I1] is

M(I ) =

[M(I0),M(I1)] if ad − bc > 0[M(I1),M(I0)] if ad − bc < 0{sM} if M 6= 0 & uM 6∈ I

R if M 6= 0 & uM ∈ I

R if M = 0

Fractional bilinear function

P

(x0

x1,y0

y1

)=

ax0y0 + bx0y1 + cx1y0 + dx1y1

ex0y0 + fx0y1 + gx1y0 + hx1y1.

P = {(x0x1, y0y1, z0z1) ∈ R

3:

(ax0y0 + bx0y1 + cx1y0 + dx1y1)z1 =

(ex0y0 + fx0y1 + gx1y0 + hx1y1)z0}

P(I , J) = {z ∈ R : ∃x ∈ I , ∃y ∈ J , (x , y , z) ∈ P}

= P(I0, J) ∪ P(I , J1) ∪ P(I1, J) ∪ P(I , J0).

The bilinear graph

vertices: (P , u, v), P ∈ M(1,1), u, v ∈ ΣW .

(P , u, v)a

−→ (F−1a P , u, v) if P(Wu0,Wv0) ⊆ Wa

(P , u, v)λ

−→ (PF xu0, σ(u), v)

(P , u, v)λ

−→ (PF yv0, u, σ(v))

Proposition If u, v ∈ ΣW and w ∈ AN is a label of apath with source (P , u, v), then w ∈ ΣW andΦ(w) = P(Φ(u),Φ(v)).

Rational functions Mn

P(x) =a0 + a1x + · · ·+ anx

n

b0 + b1x + · · ·+ bnxn

If M ∈ M1, then PM ,MP ∈ Mn.

t(x) := arg d(x) = 2 arctan x , t : R → (−π, π).

P•(x) := (tPt−1)′(t(x)) =

P ′(x)(1 + x2)

1 + P2(x)

monotone element: (P , I ), ∀x ∈ I ,P•(x) 6= 0sign-changing element: P(I0)P(I1) < 0

Expansions of rational numbers

M•(x0x1) =

(ad − bc)(x20 + x21 )

(ax0 + bx1)2 + (cx0 + dx1)2

M•(x0x1) =

det(M) · ||x ||2

||M(x))||2

R(M) = {x ∈ R : (M−1)•(x) > det(M)}

If x ∈ R(Fa) and y = F−1a (x), then ||y || < ||x ||.

If Wa ⊆ R(Fa) then rational numbers have periodicexpansions.

Circle derivations

1

2

3

3- 2- 1- 0 1 2 3

1 10

101

1012

0

1 11

110

1102

1

2 20

201

2011

2

2 21

210

2101

3

2 21

-210

-

2101

-

3-

2 20

201

-

2011

-

2-

1 11

-110

-

1102

-

1-

1 10

101

-

1012

-

0-F3(x) = 2x − 1 F2(x) =

2x−x+1 , F1(x) =

x−12 , F0(x) =

x

−x+2 ,

F0(x) =x

x+2 , F1(x) =x+12 , F2(x) =

2xx+1 , F3(x) = 2x + 1.

R(Fa) : (∞,−2), (−3,−1), (−1,−13), (−1

2 , 0),(0, 12), (13 , 1), (1, 2), (2,∞),

V(Fa) : (3,−1), (∞,−12), (−2, 0), (−1, 13),

(−13 , 1), (0, 2) (12 ,∞), (1,−3).

The bimodular octanic system A = {3, 2, 1, 0, 0, 1, 2, 3}

1/0

-5

-3

-2

-3/2

-1

-2/3

-1/2

-1/3 -1/5

0

1/5

1/31/2

2/3

1

3/2

2

3

5

0 1

2

3 3-

2-1-

0-

F3(x) = 2x − 1

F2(x) = 2x/(1− x)

F1(x) = (x − 1)/2

F0(x) = x/(2− x)

F0(x) = x/(x + 2)

F1(x) = (x + 1)/2

F2(x) = 2x/(x + 1)

F3(x) = 2x + 1

Wa = R(Fa)

Modular group

G1 = {M ∈ M1 : det(M) = 1}

Theorem If (F ,Σ) is a modular system, thenrational numbers have periodic expansions but(F ,Σ) is not redundant.

R(M) = V(M) for M ∈ G1

R(M) ⊆ (0,∞) or R(M) ⊆ (∞, 0) for M ∈ M1

Biternary system A = {1, 0, 1, 0}

F1(x) = (2x − 1)/(2− x)

F0(x) = x/2

F1(x) = (2x + 1)/(x + 2)

F0(x) = 2x

intervals:

W1 = (−2,−12)

W0 = (−12 ,

12)

W1 = (12 , 2)

W0 = (2,−2)

SW = {1, 0}N ∪ {0, 1}N ∪ {1, 0}N ∪ {0, 1}N

1/0

-8

-4

-5/2

-2

-3/2

-5/4

-1

-4/5

-2/3

-1/2 -2/5

-1/4 -1/8

0 1/8

1/4

2/5

1/2

2/3

4/5

1

5/4

3/2

2

5/2

4

8

1 -

0

1

0 -

11--

10 -

10

--

01

-

00

01

10

11

10 -

01 -- 01

-

00--