[PPT] - M obius number systems Convergence M obius number systems PowerPoint Presentation

SLIDE 1

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

M¨

bius number systems

Alexandr Kazda, Petr K˚ urka

Charles University, Prague

Numeration Marseille, March 23–27, 2009

SLIDE 2

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Outline

1 M¨

bius transformations

2 Convergence 3 M¨

bius number systems

4 Examples 5 Existence theorem 6 Conclusions

SLIDE 3

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Our goal: To use sequences of M¨
bius transformations to

represent points on R = R ∪ {∞} or the unit circle T.

A M¨
bius tranformation (MT) is any nonconstant function

M : C ∪ {∞} → C ∪ {∞} of the form M(z) = az + b cz + d

We will consider MTs that preserve the upper half-plane
or the unit disc D.

SLIDE 4

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Our goal: To use sequences of M¨
bius transformations to

represent points on R = R ∪ {∞} or the unit circle T.

A M¨
bius tranformation (MT) is any nonconstant function

M : C ∪ {∞} → C ∪ {∞} of the form M(z) = az + b cz + d

We will consider MTs that preserve the upper half-plane
or the unit disc D.

SLIDE 5

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Our goal: To use sequences of M¨
bius transformations to

represent points on R = R ∪ {∞} or the unit circle T.

A M¨
bius tranformation (MT) is any nonconstant function

M : C ∪ {∞} → C ∪ {∞} of the form M(z) = az + b cz + d

We will consider MTs that preserve the upper half-plane
or the unit disc D.

SLIDE 6

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Our goal: To use sequences of M¨
bius transformations to

represent points on R = R ∪ {∞} or the unit circle T.

A M¨
bius tranformation (MT) is any nonconstant function

M : C ∪ {∞} → C ∪ {∞} of the form M(z) = az + b cz + d

We will consider MTs that preserve the upper half-plane
or the unit disc D.

SLIDE 7

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

2

2

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

Using the stereometric projection, we have a one-to-one

correspondence between the upper half-plane and unit disc.

This projection is itself an MT.
Therefore we can translate MTs that represent T to the
nes that represent R.

SLIDE 8

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

2

2

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

Using the stereometric projection, we have a one-to-one

correspondence between the upper half-plane and unit disc.

This projection is itself an MT.
Therefore we can translate MTs that represent T to the
nes that represent R.

SLIDE 9

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

2

2

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

Using the stereometric projection, we have a one-to-one

correspondence between the upper half-plane and unit disc.

This projection is itself an MT.
Therefore we can translate MTs that represent T to the
nes that represent R.

SLIDE 10

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

2

2

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

Using the stereometric projection, we have a one-to-one

correspondence between the upper half-plane and unit disc.

This projection is itself an MT.
Therefore we can translate MTs that represent T to the
nes that represent R.

SLIDE 11

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

We will be mostly talking about representing the unit

circle.

However, the example number systems represent R.
How to tell them apart: half-plane-preserving MTs have a

hat, disc-preserving MTs don’t.

SLIDE 12

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

We will be mostly talking about representing the unit

circle.

However, the example number systems represent R.
How to tell them apart: half-plane-preserving MTs have a

hat, disc-preserving MTs don’t.

SLIDE 13

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

R versus T

We will be mostly talking about representing the unit

circle.

However, the example number systems represent R.
How to tell them apart: half-plane-preserving MTs have a

hat, disc-preserving MTs don’t.

SLIDE 14

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Disc M¨

bius transformations

M : D → D

A direct calculation shows that all MTs that preserve D

must look like this:

M(z) = αz + β

βz + α,

where |β| < |α| are complex numbers.
Examples follow.

SLIDE 15

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Disc M¨

bius transformations

M : D → D

A direct calculation shows that all MTs that preserve D

must look like this:

M(z) = αz + β

βz + α,

where |β| < |α| are complex numbers.
Examples follow.

SLIDE 16

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Disc M¨

bius transformations

M : D → D

A direct calculation shows that all MTs that preserve D

must look like this:

M(z) = αz + β

βz + α,

where |β| < |α| are complex numbers.
Examples follow.

SLIDE 17

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Disc M¨

bius transformations

M : D → D

A direct calculation shows that all MTs that preserve D

must look like this:

M(z) = αz + β

βz + α,

where |β| < |α| are complex numbers.
Examples follow.

SLIDE 18

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Examples of M¨

bius

transformations

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

M0(z) = 3z−i

iz−3

ˆ M0(x) = x/2 hyperbolic M1(z) = (2i+1)z+1

2i−1

ˆ M1(x) = x + 1 parabolic M2(z) =

(7+2i)z+i −iz+(7−2i)

ˆ M2(x) = 4x+1

3−x

elliptic

SLIDE 19

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Examples of M¨

bius

transformations

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

M0(z) = 3z−i

iz−3

ˆ M0(x) = x/2 hyperbolic M1(z) = (2i+1)z+1

2i−1

ˆ M1(x) = x + 1 parabolic M2(z) =

(7+2i)z+i −iz+(7−2i)

ˆ M2(x) = 4x+1

3−x

elliptic

SLIDE 20

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Examples of M¨

bius

transformations

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 8

M0(z) = 3z−i

iz−3

ˆ M0(x) = x/2 hyperbolic M1(z) = (2i+1)z+1

2i−1

ˆ M1(x) = x + 1 parabolic M2(z) =

(7+2i)z+i −iz+(7−2i)

ˆ M2(x) = 4x+1

3−x

elliptic

SLIDE 21

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Defining convergence

A sequence M1, M2, . . . represents the number x ∈ T if

Mn(0) → x for n → ∞.

Isn’t it a bit arbitrary?
No. This definition is quite natural.
For example, if M1, M2, . . . represents x then

Mn(K) → {x} for any K ⊂ Do compact.

SLIDE 22

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Defining convergence

A sequence M1, M2, . . . represents the number x ∈ T if

Mn(0) → x for n → ∞.

Isn’t it a bit arbitrary?
No. This definition is quite natural.
For example, if M1, M2, . . . represents x then

Mn(K) → {x} for any K ⊂ Do compact.

SLIDE 23

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Defining convergence

A sequence M1, M2, . . . represents the number x ∈ T if

Mn(0) → x for n → ∞.

Isn’t it a bit arbitrary?
No. This definition is quite natural.
For example, if M1, M2, . . . represents x then

Mn(K) → {x} for any K ⊂ Do compact.

SLIDE 24

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Defining convergence

A sequence M1, M2, . . . represents the number x ∈ T if

Mn(0) → x for n → ∞.

Isn’t it a bit arbitrary?
No. This definition is quite natural.
For example, if M1, M2, . . . represents x then

Mn(K) → {x} for any K ⊂ Do compact.

SLIDE 25

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Then A+ is the set of all finite

nonempty words over A, Aω the set of all one-sided infinite words.

Recall that Σ ⊂ Aω is a subshift if Σ can be defined by a

set of forbidden (finite) words.

For v = v0v1 . . . vn a word, denote by Fv the

transformation Fv0 ◦ Fv1 ◦ · · · ◦ Fvn.

SLIDE 26

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Then A+ is the set of all finite

nonempty words over A, Aω the set of all one-sided infinite words.

Recall that Σ ⊂ Aω is a subshift if Σ can be defined by a

set of forbidden (finite) words.

For v = v0v1 . . . vn a word, denote by Fv the

transformation Fv0 ◦ Fv1 ◦ · · · ◦ Fvn.

SLIDE 27

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Then A+ is the set of all finite

nonempty words over A, Aω the set of all one-sided infinite words.

Recall that Σ ⊂ Aω is a subshift if Σ can be defined by a

set of forbidden (finite) words.

For v = v0v1 . . . vn a word, denote by Fv the

transformation Fv0 ◦ Fv1 ◦ · · · ◦ Fvn.

SLIDE 28

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

What is a M¨

bius number system?

Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω is a M¨

bius number system if:
For every w ∈ Σ, the sequence {Fw0w1...wn}∞

n=0 represents

some point Φ(w) ∈ T.

The function Φ : Σ → T is continuous and surjective.

SLIDE 29

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

What is a M¨

bius number system?

Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω is a M¨

bius number system if:
For every w ∈ Σ, the sequence {Fw0w1...wn}∞

n=0 represents

some point Φ(w) ∈ T.

The function Φ : Σ → T is continuous and surjective.

SLIDE 30

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

What is a M¨

bius number system?

Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω is a M¨

bius number system if:
For every w ∈ Σ, the sequence {Fw0w1...wn}∞

n=0 represents

some point Φ(w) ∈ T.

The function Φ : Σ → T is continuous and surjective.

SLIDE 31

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 32

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 33

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 34

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 35

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 36

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Getting the idea: Binary system

Take transformations ˆ

F0(x) = x/2 and ˆ F1(x) = (x + 1)/2.

Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on T corresponding

to [0, 1].

Essentialy, it is the ordinary binary system; Φ(w)

corresponds to 0.w.

Note that this is not a M¨
bius number system yet, as it is

not surjective. . .

. . . we will fix that soon.

SLIDE 37

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Binary signed system A = {1, 0, 1, 2}

6
5
4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 5 6 3/2

3/2

2/3

2/3

8

1

1

2

11

10
01
00

01 1 11 21

21

22 1 1

100
1

1

1

1

1
1
000

1 1 1 1 1 1

100

1 1 211

210
210

211 221

221

222

ˆ F1(x) = (x − 1)/2 ˆ F0(x) = x/2 ˆ F1(x) = (x + 1)/2 ˆ F2(x) = 2x Forbidden words: 20, 02, 12, 12, 11, 11

SLIDE 38

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Regular continued fractions A = {1, 0, 1}

6
5
4
1/4
3
1/3
2
1/2
1

1 2 1/2 3 1/3 4 1/4 5 6 3/2

3/2

2/3

2/3

8

1

1

11

10
01
01

10 11

111

110
101
011
1
1

011 101

110

111

ˆ F1(x) = −1 + x ˆ F0(x) = −1/x ˆ F1(x) = 1 + x Forbidden words: 00, 11, 11, 101, 101

SLIDE 39

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Existence problem

The question: Given a system of MTs {Fa : a ∈ A}, does

there exist a M¨

bius number system?
The answer: It depends on whether {Va : a ∈ A} cover T

in a certain way.

SLIDE 40

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Existence problem

The question: Given a system of MTs {Fa : a ∈ A}, does

there exist a M¨

bius number system?
The answer: It depends on whether {Va : a ∈ A} cover T

in a certain way.

SLIDE 41

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Intervals of contraction and expansion

U V U V Uu = {z ∈ T : |F ′

u(z)| < 1},

Vu = {z ∈ T : |(F −1

u )′(z)| > 1}

Fu(Uu) = Vu, u ∈ A+

SLIDE 42

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Theorem

Let {Fa : a ∈ A} be MTs.

1 If {Vu : u ∈ A+} = T, then there does not exist any

M¨

bius number system.

2 If there exists a finite B ⊂ A+ such that {V u : u ∈ B}

cover T, then there exists a M¨

bius number system.

Note that there can still be some situation in between (1) and (2).

SLIDE 43

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Theorem

Let {Fa : a ∈ A} be MTs.

1 If {Vu : u ∈ A+} = T, then there does not exist any

M¨

bius number system.

2 If there exists a finite B ⊂ A+ such that {V u : u ∈ B}

cover T, then there exists a M¨

bius number system.

Note that there can still be some situation in between (1) and (2).

SLIDE 44

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Theorem

Let {Fa : a ∈ A} be MTs.

1 If {Vu : u ∈ A+} = T, then there does not exist any

M¨

bius number system.

2 If there exists a finite B ⊂ A+ such that {V u : u ∈ B}

cover T, then there exists a M¨

bius number system.

Note that there can still be some situation in between (1) and (2).

SLIDE 45

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Conclusions

Sequences of MTs can represent numbers.
We have some sufficient and some necessary conditions for

a M¨

bius number system to exist.
Continued fractions are a special case of a M¨
bius number

system.

SLIDE 46

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Conclusions

Sequences of MTs can represent numbers.
We have some sufficient and some necessary conditions for

a M¨

bius number system to exist.
Continued fractions are a special case of a M¨
bius number

system.

SLIDE 47

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions

Conclusions

Sequences of MTs can represent numbers.
We have some sufficient and some necessary conditions for

a M¨

bius number system to exist.
Continued fractions are a special case of a M¨
bius number

system.

SLIDE 48

M¨

bius

number systems Alexandr Kazda, Petr K˚ urka M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Existence theorem Conclusions