Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius number systems
Möbius transformations
Convergence
Möbius
number
systems
Alexandr Kazda
(with thanks to Petr Kůrka)
Examples
Subshifts
admitting a
number
system
Conclusions
Charles University, Prague
NSAC
Novi Sad
August 17–21, 2009
Möbius
number
systems
Outline
Alexandr
Kazda
(with thanks
to Petr Kůrka)
1 Möbius transformations
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
2 Convergence
3 Möbius number systems
4 Examples
Conclusions
5 Subshifts admitting a number system
6 Conclusions
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
• Our goal: To use sequences of Möbius transformations to
represent points on R = R ∪ {∞}.
Möbius transformations
Convergence
• A Möbius transformation (MT) is any nonconstant
function M : C ∪ {∞} → C ∪ {∞} of the form
Möbius
number
systems
M(z) =
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
• Our goal: To use sequences of Möbius transformations to
represent points on R = R ∪ {∞}.
Möbius transformations
Convergence
• A Möbius transformation (MT) is any nonconstant
function M : C ∪ {∞} → C ∪ {∞} of the form
Möbius
number
systems
M(z) =
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
• Our goal: To use sequences of Möbius transformations to
represent points on R = R ∪ {∞}.
Möbius transformations
Convergence
• A Möbius transformation (MT) is any nonconstant
function M : C ∪ {∞} → C ∪ {∞} of the form
Möbius
number
systems
M(z) =
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
• Our goal: To use sequences of Möbius transformations to
represent points on R = R ∪ {∞}.
Möbius transformations
Convergence
• A Möbius transformation (MT) is any nonconstant
function M : C ∪ {∞} → C ∪ {∞} of the form
Möbius
number
systems
M(z) =
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Classifying Möbius transformations
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
M0 (x) = x/2
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
• Hyperbolic, two fixed points.
M1 (x) = x + 1
• Parabolic, one fixed point.
Conclusions
M2 (x) = −
• Elliptic, no fixed points in R.
1
x +1
Möbius
number
systems
Defining convergence
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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.
Möbius
number
systems
Defining convergence
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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.
Möbius
number
systems
Defining convergence
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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.
Möbius
number
systems
Defining convergence
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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.
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Preliminaries from Symbolic
dynamics
Möbius transformations
Convergence
Möbius
number
systems
• 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.
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 .
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Preliminaries from Symbolic
dynamics
Möbius transformations
Convergence
Möbius
number
systems
• 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.
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 .
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Preliminaries from Symbolic
dynamics
Möbius transformations
Convergence
Möbius
number
systems
• 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.
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 .
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Preliminaries from Symbolic
dynamics
Möbius transformations
Convergence
Möbius
number
systems
• 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.
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 .
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Preliminaries from Symbolic
dynamics
Möbius transformations
Convergence
Möbius
number
systems
• 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.
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 .
Möbius
number
systems
What is a Möbius number system?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω
is a Möbius 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.
Möbius
number
systems
What is a Möbius number system?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω
is a Möbius 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.
Möbius
number
systems
What is a Möbius number system?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
Let us have a system of MTs {Fa : a ∈ A}. A subshift Σ ⊂ Aω
is a Möbius 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.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Getting the idea: Binary system
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
• 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 Möbius number system yet, as it is
not surjective. . .
• . . . we will fix that soon.
Möbius
number
systems
Binary signed system
A = {1, 0, 1, 2}
Alexandr
Kazda
(with thanks
to Petr Kůrka)
4
6
5
-4
-5
-6
8
3
-3
Möbius transformations
-2
221
2
-3/
Examples
F0 (x) = x/2
F1 (x) = (x + 1)/2
-10
- 2
21
21
210
1
F2 (x) = 2x
11
1
10
0
-1
-1-1
1-0
10
1
10
0
4
1/
3
1/
0
01
-1
1001
01
001
001-
00 000
0-
/3
1
0
1-0
-1
/4
10-10-1 -1
-100
3
-101 -2/
-1
/2
Conclusions
3/2
211
1-
Subshifts
admitting a
number
system
F1 (x) = (x − 1)/2
2
221
2
-211
Möbius
number
systems
222 22
Convergence
01
/2
2/3
Forbidden words:
20, 02, 12, 12, 11, 11
Möbius
number
systems
Why forbid words?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• We forbid words to get rid of troublesome combinations.
Convergence
• In the binary signed system F0 and F2 are inverse to each
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
other, 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 Möbius number system.
Möbius
number
systems
Why forbid words?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• We forbid words to get rid of troublesome combinations.
Convergence
• In the binary signed system F0 and F2 are inverse to each
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
other, 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 Möbius number system.
Möbius
number
systems
Why forbid words?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• We forbid words to get rid of troublesome combinations.
Convergence
• In the binary signed system F0 and F2 are inverse to each
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
other, 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 Möbius number system.
Möbius
number
systems
Why forbid words?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• We forbid words to get rid of troublesome combinations.
Convergence
• In the binary signed system F0 and F2 are inverse to each
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
other, 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 Möbius number system.
Möbius
number
systems
Why forbid words?
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• We forbid words to get rid of troublesome combinations.
Convergence
• In the binary signed system F0 and F2 are inverse to each
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
other, 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 Möbius number system.
Möbius
number
systems
Forbidding words is necessary
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
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 Möbius number system then Σ 6= 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 .
Möbius
number
systems
Forbidding words is necessary
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
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 Möbius number system then Σ 6= 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 .
Möbius
number
systems
Forbidding words is necessary
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
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 Möbius number system then Σ 6= 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 .
Möbius
number
systems
Sketch of a proof
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• To simplify notation, we consider only the case ρ(v ) = v .
Convergence
• We first prove that for every w ∈ Aω and every x ∈ R it is
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
true that limn→∞ Fw1 w2 ...wn (x) = Φ(w ).
• This is highly suspicious. . .
• The only way we can obtain such pointwise convergence is
when Fv is parabolic (like x 7→ 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.
Möbius
number
systems
Sketch of a proof
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• To simplify notation, we consider only the case ρ(v ) = v .
Convergence
• We first prove that for every w ∈ Aω and every x ∈ R it is
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
true that limn→∞ Fw1 w2 ...wn (x) = Φ(w ).
• This is highly suspicious. . .
• The only way we can obtain such pointwise convergence is
when Fv is parabolic (like x 7→ 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.
Möbius
number
systems
Sketch of a proof
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• To simplify notation, we consider only the case ρ(v ) = v .
Convergence
• We first prove that for every w ∈ Aω and every x ∈ R it is
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
true that limn→∞ Fw1 w2 ...wn (x) = Φ(w ).
• This is highly suspicious. . .
• The only way we can obtain such pointwise convergence is
when Fv is parabolic (like x 7→ 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.
Möbius
number
systems
Sketch of a proof
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• To simplify notation, we consider only the case ρ(v ) = v .
Convergence
• We first prove that for every w ∈ Aω and every x ∈ R it is
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
true that limn→∞ Fw1 w2 ...wn (x) = Φ(w ).
• This is highly suspicious. . .
• The only way we can obtain such pointwise convergence is
when Fv is parabolic (like x 7→ 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.
Möbius
number
systems
Sketch of a proof
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
• To simplify notation, we consider only the case ρ(v ) = v .
Convergence
• We first prove that for every w ∈ Aω and every x ∈ R it is
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
true that limn→∞ Fw1 w2 ...wn (x) = Φ(w ).
• This is highly suspicious. . .
• The only way we can obtain such pointwise convergence is
when Fv is parabolic (like x 7→ 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.
Möbius
number
systems
Conclusions
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
• Sequences of MTs can represent numbers.
Möbius
number
systems
• Möbius number systems can emulate more usual means of
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Conclusions
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
• Sequences of MTs can represent numbers.
Möbius
number
systems
• Möbius number systems can emulate more usual means of
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Conclusions
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
• Sequences of MTs can represent numbers.
Möbius
number
systems
• Möbius number systems can emulate more usual means of
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Conclusions
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
• Sequences of MTs can represent numbers.
Möbius
number
systems
• Möbius number systems can emulate more usual means of
Examples
Subshifts
admitting a
number
system
Conclusions
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.
Möbius
number
systems
Alexandr
Kazda
(with thanks
to Petr Kůrka)
Möbius transformations
Convergence
Möbius
number
systems
Examples
Subshifts
admitting a
number
system
Conclusions
Thanks for your attention.