M¨ obius number systems Petr K˚ urka Center for Theoretical Study Academy of Sciences and Charles University in Prague Dynamics and Computation Marseille, February 2010
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--