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

SLIDE 1

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

M¨

bius number systems

Alexandr Kazda (with thanks to Petr K˚ urka)

Charles University, Prague

NSAC Novi Sad August 17–21, 2009

SLIDE 2

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Outline

1 M¨

bius transformations

2 Convergence 3 M¨

bius number systems

4 Examples 5 Subshifts admitting a number system 6 Conclusions

SLIDE 3

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

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

represent points on R = R ∪ {∞}.

A M¨
bius transformation (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.
These are precisely the MTs with a, b, c, d real and

ad − bc = 1.

SLIDE 4

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

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

represent points on R = R ∪ {∞}.

A M¨
bius transformation (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.
These are precisely the MTs with a, b, c, d real and

ad − bc = 1.

SLIDE 5

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

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

represent points on R = R ∪ {∞}.

A M¨
bius transformation (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.
These are precisely the MTs with a, b, c, d real and

ad − bc = 1.

SLIDE 6

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

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

represent points on R = R ∪ {∞}.

A M¨
bius transformation (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.
These are precisely the MTs with a, b, c, d real and

ad − bc = 1.

SLIDE 7

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 8

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 9

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 10

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 11

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 12

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 13

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 14

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 15

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Classifying M¨

bius transformations

M0(x) = x/2

Hyperbolic, two fixed points.

M1(x) = x + 1

Parabolic, one fixed point.

M2(x) = − 1 x + 1

Elliptic, no fixed points in R.

SLIDE 16

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Defining convergence

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

Mn(i) → 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 compact lying above the real line.

SLIDE 17

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Defining convergence

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

Mn(i) → 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 compact lying above the real line.

SLIDE 18

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Defining convergence

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

Mn(i) → 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 compact lying above the real line.

SLIDE 19

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Defining convergence

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

Mn(i) → 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 compact lying above the real line.

SLIDE 20

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Let A⋆ denote the set of all finite

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

A⋆ with the operation of concatenation is a monoid.
Let wi denote the i-th letter of the word w.
A set Σ ⊂ Aω is a subshift if Σ can be defined by a set of

forbidden (finite) factors.

For v = v1 . . . vn a word, denote by Fv the transformation

Fv1 ◦ · · · ◦ Fvn.

SLIDE 21

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Let A⋆ denote the set of all finite

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

A⋆ with the operation of concatenation is a monoid.
Let wi denote the i-th letter of the word w.
A set Σ ⊂ Aω is a subshift if Σ can be defined by a set of

forbidden (finite) factors.

For v = v1 . . . vn a word, denote by Fv the transformation

Fv1 ◦ · · · ◦ Fvn.

SLIDE 22

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Let A⋆ denote the set of all finite

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

A⋆ with the operation of concatenation is a monoid.
Let wi denote the i-th letter of the word w.
A set Σ ⊂ Aω is a subshift if Σ can be defined by a set of

forbidden (finite) factors.

For v = v1 . . . vn a word, denote by Fv the transformation

Fv1 ◦ · · · ◦ Fvn.

SLIDE 23

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Let A⋆ denote the set of all finite

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

A⋆ with the operation of concatenation is a monoid.
Let wi denote the i-th letter of the word w.
A set Σ ⊂ Aω is a subshift if Σ can be defined by a set of

forbidden (finite) factors.

For v = v1 . . . vn a word, denote by Fv the transformation

Fv1 ◦ · · · ◦ Fvn.

SLIDE 24

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Preliminaries from Symbolic dynamics

Let A be finite alphabet. Let A⋆ denote the set of all finite

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

A⋆ with the operation of concatenation is a monoid.
Let wi denote the i-th letter of the word w.
A set Σ ⊂ Aω is a subshift if Σ can be defined by a set of

forbidden (finite) factors.

For v = v1 . . . vn a word, denote by Fv the transformation

Fv1 ◦ · · · ◦ Fvn.

SLIDE 25

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system 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 {Fw1...wn}∞

n=1 represents

some point Φ(w) ∈ R.

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

SLIDE 26

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system 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 {Fw1...wn}∞

n=1 represents

some point Φ(w) ∈ R.

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

SLIDE 27

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system 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 {Fw1...wn}∞

n=1 represents

some point Φ(w) ∈ R.

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

SLIDE 28

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 29

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 30

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 31

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Getting the idea: Binary system

Take transformations F0(x) = x/2 and F1(x) = (x + 1)/2.
Take the full shift Σ = {0, 1}ω.
The function Φ maps Σ to an interval on R corresponding

to [0, 1].

Essentially, 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 (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system 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

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

SLIDE 35

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Why forbid words?

We forbid words to get rid of troublesome combinations.
In the binary signed system F0 and F2 are inverse to each
ther, so F02 = F20 = id.
Forbidding 12 and 12 keeps twos at the beginning of every

word.

Finally, 11 and 11 are forbidden because Φ((11)∞) and

Φ((11)∞) are not defined.

We shall see that unregulated concatenation can break

any M¨

bius number system.

SLIDE 36

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Why forbid words?

We forbid words to get rid of troublesome combinations.
In the binary signed system F0 and F2 are inverse to each
ther, so F02 = F20 = id.
Forbidding 12 and 12 keeps twos at the beginning of every

word.

Finally, 11 and 11 are forbidden because Φ((11)∞) and

Φ((11)∞) are not defined.

We shall see that unregulated concatenation can break

any M¨

bius number system.

SLIDE 37

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Why forbid words?

We forbid words to get rid of troublesome combinations.
In the binary signed system F0 and F2 are inverse to each
ther, so F02 = F20 = id.
Forbidding 12 and 12 keeps twos at the beginning of every

word.

Finally, 11 and 11 are forbidden because Φ((11)∞) and

Φ((11)∞) are not defined.

We shall see that unregulated concatenation can break

any M¨

bius number system.

SLIDE 38

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Why forbid words?

We forbid words to get rid of troublesome combinations.
In the binary signed system F0 and F2 are inverse to each
ther, so F02 = F20 = id.
Forbidding 12 and 12 keeps twos at the beginning of every

word.

Finally, 11 and 11 are forbidden because Φ((11)∞) and

Φ((11)∞) are not defined.

We shall see that unregulated concatenation can break

any M¨

bius number system.

SLIDE 39

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Why forbid words?

We forbid words to get rid of troublesome combinations.
In the binary signed system F0 and F2 are inverse to each
ther, so F02 = F20 = id.
Forbidding 12 and 12 keeps twos at the beginning of every

word.

Finally, 11 and 11 are forbidden because Φ((11)∞) and

Φ((11)∞) are not defined.

We shall see that unregulated concatenation can break

any M¨

bius number system.

SLIDE 40

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Forbidding words is necessary

A non-erasing substitution is monoid homomorphism ρ : A∗ → B∗ such that ρ(v) is the empty word only for v empty.

Theorem

If Σ is a M¨

bius number system then Σ = ρ(Aω) for all

alphabets A and all non-erasing substitutions ρ. In particular, for ρ identity we obtain that Σ is never the full shift Aω.

SLIDE 41

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Forbidding words is necessary

A non-erasing substitution is monoid homomorphism ρ : A∗ → B∗ such that ρ(v) is the empty word only for v empty.

Theorem

If Σ is a M¨

bius number system then Σ = ρ(Aω) for all

alphabets A and all non-erasing substitutions ρ. In particular, for ρ identity we obtain that Σ is never the full shift Aω.

SLIDE 42

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Forbidding words is necessary

A non-erasing substitution is monoid homomorphism ρ : A∗ → B∗ such that ρ(v) is the empty word only for v empty.

Theorem

If Σ is a M¨

bius number system then Σ = ρ(Aω) for all

alphabets A and all non-erasing substitutions ρ. In particular, for ρ identity we obtain that Σ is never the full shift Aω.

SLIDE 43

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Sketch of a proof

To simplify notation, we consider only the case ρ(v) = v.
We first prove that for every w ∈ Aω and every x ∈ R it is

true that limn→∞ Fw1w2...wn(x) = Φ(w).

This is highly suspicious. . .
The only way we can obtain such pointwise convergence is

when Fv is parabolic (like x → x + 1) for every v nonempty finite word.

But a simple case consideration shows that then all the Fv

have the same fixed point and Φ is a constant map.

SLIDE 44

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Sketch of a proof

To simplify notation, we consider only the case ρ(v) = v.
We first prove that for every w ∈ Aω and every x ∈ R it is

true that limn→∞ Fw1w2...wn(x) = Φ(w).

This is highly suspicious. . .
The only way we can obtain such pointwise convergence is

when Fv is parabolic (like x → x + 1) for every v nonempty finite word.

But a simple case consideration shows that then all the Fv

have the same fixed point and Φ is a constant map.

SLIDE 45

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Sketch of a proof

To simplify notation, we consider only the case ρ(v) = v.
We first prove that for every w ∈ Aω and every x ∈ R it is

true that limn→∞ Fw1w2...wn(x) = Φ(w).

This is highly suspicious. . .
The only way we can obtain such pointwise convergence is

when Fv is parabolic (like x → x + 1) for every v nonempty finite word.

But a simple case consideration shows that then all the Fv

have the same fixed point and Φ is a constant map.

SLIDE 46

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Sketch of a proof

To simplify notation, we consider only the case ρ(v) = v.
We first prove that for every w ∈ Aω and every x ∈ R it is

true that limn→∞ Fw1w2...wn(x) = Φ(w).

This is highly suspicious. . .
The only way we can obtain such pointwise convergence is

when Fv is parabolic (like x → x + 1) for every v nonempty finite word.

But a simple case consideration shows that then all the Fv

have the same fixed point and Φ is a constant map.

SLIDE 47

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Sketch of a proof

To simplify notation, we consider only the case ρ(v) = v.
We first prove that for every w ∈ Aω and every x ∈ R it is

true that limn→∞ Fw1w2...wn(x) = Φ(w).

This is highly suspicious. . .
The only way we can obtain such pointwise convergence is

when Fv is parabolic (like x → x + 1) for every v nonempty finite word.

But a simple case consideration shows that then all the Fv

have the same fixed point and Φ is a constant map.

SLIDE 48

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Conclusions

Sequences of MTs can represent numbers.
M¨
bius number systems can emulate more usual means of

number representation.

We can state (and sometimes prove) nontrivial existence

conditions such as the one presented . . .

. . . however, there is a lot of room for improvements.

SLIDE 49

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Conclusions

Sequences of MTs can represent numbers.
M¨
bius number systems can emulate more usual means of

number representation.

We can state (and sometimes prove) nontrivial existence

conditions such as the one presented . . .

. . . however, there is a lot of room for improvements.

SLIDE 50

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Conclusions

Sequences of MTs can represent numbers.
M¨
bius number systems can emulate more usual means of

number representation.

We can state (and sometimes prove) nontrivial existence

conditions such as the one presented . . .

. . . however, there is a lot of room for improvements.

SLIDE 51

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions

Conclusions

Sequences of MTs can represent numbers.
M¨
bius number systems can emulate more usual means of

number representation.

We can state (and sometimes prove) nontrivial existence

conditions such as the one presented . . .

. . . however, there is a lot of room for improvements.

SLIDE 52

M¨

bius

number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨

bius trans-

formations Convergence M¨

bius

number systems Examples Subshifts admitting a number system Conclusions