1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu
Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS
2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Dual systems of algebraic iteratedfunction systems

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Hui RAORIMS, Kyoto University

Dec. 20-22, 2011

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu
Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS
2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§1. Algebraic IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Graph-directed Iterated function system

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Let G = (V, Γ) be a directed graph with vertex set V ={1, · · · , N} and directed-edge set Γ.

3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultDefinition (Graph IFS)
A graph-directed IFS is a sequence of of contractions(fγ )γ∈Γwherefγ : Rd 7→ Rd .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(1)1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Graph-directed Iterated function system

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Let G = (V, Γ) be a directed graph with vertex set V ={1, · · · , N} and directed-edge set Γ.

3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultDefinition (Graph IFS)
A graph-directed IFS is a sequence of of contractions(fγ )γ∈Γwherefγ : Rd 7→ Rd .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(1)1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Invariant sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFSΓij : edges from vertex i to j.
(E1 , . . . , EN ) non-empty compact sets satisfying

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Ei =

N [
[

fγ (Ej ), 1 ≤ i ≤ N.

j=1 γ∈Γij

is called the invariant sets of the graph IFS (1).

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(2)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Invariant sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFSΓij : edges from vertex i to j.
(E1 , . . . , EN ) non-empty compact sets satisfying

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Ei =

N [
[

fγ (Ej ), 1 ≤ i ≤ N.

j=1 γ∈Γij

is called the invariant sets of the graph IFS (1).

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(2)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A separation condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Graph IFS (1) is said to satisfy the open set condition (OSC),
if there exist open sets U1 , . . . , UN such that

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsN [[fγ (Uj ) ⊂ Ui ,1 ≤ i ≤ N,

j=1 γ∈Γij

and the left-hand side are disjoint unions.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A separation condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Graph IFS (1) is said to satisfy the open set condition (OSC),
if there exist open sets U1 , . . . , UN such that

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsN [[fγ (Uj ) ⊂ Ui ,1 ≤ i ≤ N,

j=1 γ∈Γij

and the left-hand side are disjoint unions.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A separation condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Graph IFS (1) is said to satisfy the open set condition (OSC),
if there exist open sets U1 , . . . , UN such that

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsN [[fγ (Uj ) ⊂ Ui ,1 ≤ i ≤ N,

j=1 γ∈Γij

and the left-hand side are disjoint unions.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

IFS with algebraic parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Definition (Algebraic IFS)
Graph IFS (1) is called an algebraic graph IFS, if
fγ : R → R has the form

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultfγ (x) =x + bγβ(3)Expanding factor: β > 1 is a real algebraic number.
bγ ∈ Q(β).

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

IFS with algebraic parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Definition (Algebraic IFS)
Graph IFS (1) is called an algebraic graph IFS, if
fγ : R → R has the form

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultfγ (x) =x + bγβ(3)Expanding factor: β > 1 is a real algebraic number.
bγ ∈ Q(β).

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

IFS with algebraic parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Definition (Algebraic IFS)
Graph IFS (1) is called an algebraic graph IFS, if
fγ : R → R has the form

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultfγ (x) =x + bγβ(3)Expanding factor: β > 1 is a real algebraic number.
bγ ∈ Q(β).

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

IFS with algebraic parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Definition (Algebraic IFS)
Graph IFS (1) is called an algebraic graph IFS, if
fγ : R → R has the form

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultfγ (x) =x + bγβ(3)Expanding factor: β > 1 is a real algebraic number.
bγ ∈ Q(β).

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

IFS with algebraic parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§2. Background and motivation

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Background

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The tiling systems arising from
Pisot substitutions: initialed by Rauzy in 1982
β-numeration systems: initialed by Thurston in 1989
Tiling theory, Substitution dynamical system,
Markov partition, Number theory,
Spectral theory, Fractal geometry......

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Motivation

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Studies of substitutions:
Irreducible Pisot casereducible Pisot casenon-Pisot case
Effort has been made to unify these two approaches: [Berthé
and Siegel: Integer 2005].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Motivation

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Studies of substitutions:
Irreducible Pisot casereducible Pisot casenon-Pisot case
Effort has been made to unify these two approaches: [Berthé
and Siegel: Integer 2005].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Motivation

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Studies of substitutions:
Irreducible Pisot casereducible Pisot casenon-Pisot case
Effort has been made to unify these two approaches: [Berthé
and Siegel: Integer 2005].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Motivation

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Studies of substitutions:
Irreducible Pisot casereducible Pisot casenon-Pisot case
Effort has been made to unify these two approaches: [Berthé
and Siegel: Integer 2005].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Motivation

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Studies of substitutions:
Irreducible Pisot casereducible Pisot casenon-Pisot case
Effort has been made to unify these two approaches: [Berthé
and Siegel: Integer 2005].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Some new results

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Kalle and Steiner: Trans. AMS. 2012 studies tilings arisingfromsymmetric β-expansion: Akiyama and Scheicher 2007.
minimal weight expansion: Frougny and Steiner 2008.
We intend to reformulate the above studies in a new way.
Thanks to: S. Ito, S. Akiyama, W. Steiner.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Some new results

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Kalle and Steiner: Trans. AMS. 2012 studies tilings arisingfromsymmetric β-expansion: Akiyama and Scheicher 2007.
minimal weight expansion: Frougny and Steiner 2008.
We intend to reformulate the above studies in a new way.
Thanks to: S. Ito, S. Akiyama, W. Steiner.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Some new results

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Kalle and Steiner: Trans. AMS. 2012 studies tilings arisingfromsymmetric β-expansion: Akiyama and Scheicher 2007.
minimal weight expansion: Frougny and Steiner 2008.
We intend to reformulate the above studies in a new way.
Thanks to: S. Ito, S. Akiyama, W. Steiner.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Some new results

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Kalle and Steiner: Trans. AMS. 2012 studies tilings arisingfromsymmetric β-expansion: Akiyama and Scheicher 2007.
minimal weight expansion: Frougny and Steiner 2008.
We intend to reformulate the above studies in a new way.
Thanks to: S. Ito, S. Akiyama, W. Steiner.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Some new results

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§3. Dual system of algebraic IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Reverse graph

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Let G = (V, Γ) be a directed graph.
Reverse graph G0 = (V, Γ0 ) of G is obtained by reverse thedirection of every edge in G.

3. Dual systemof algebraic
IFS

b

b’

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result

a

1

2

a’

1

2

c
e

c’
e’

d
3

d’
3

Figure: 1. Left: Graph of Rauzy substitution.
graph.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

Right: Reverse

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Reverse graph

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Let G = (V, Γ) be a directed graph.
Reverse graph G0 = (V, Γ0 ) of G is obtained by reverse thedirection of every edge in G.

3. Dual systemof algebraic
IFS

b

b’

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result

a

1

2

a’

1

2

c
e

c’
e’

d
3

d’
3

Figure: 1. Left: Graph of Rauzy substitution.
graph.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

Right: Reverse

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Reverse graph

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Let G = (V, Γ) be a directed graph.
Reverse graph G0 = (V, Γ0 ) of G is obtained by reverse thedirection of every edge in G.

3. Dual systemof algebraic
IFS

b

b’

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result

a

1

2

a’

1

2

c
e

c’
e’

d
3

d’
3

Figure: 1. Left: Graph of Rauzy substitution.
graph.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

Right: Reverse

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual in field Q(β)

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Algebraic conjugates of β: 1 > |β2 | ≥ · · · |βd |.

3. Dual systemof algebraic
IFS

For

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsd−1
Xx=xj β j ∈ Q(β),

j=0define
∗

x =

d−1
X

xj (β2j , · · · , βdj ).

j=0

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual in field Q(β)

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Algebraic conjugates of β: 1 > |β2 | ≥ · · · |βd |.

3. Dual systemof algebraic
IFS

For

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsd−1
Xx=xj β j ∈ Q(β),

j=0define
∗

x =

d−1
X

xj (β2j , · · · , βdj ).

j=0

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual in field Q(β)

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

Algebraic conjugates of β: 1 > |β2 | ≥ · · · |βd |.

3. Dual systemof algebraic
IFS

For

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsd−1
Xx=xj β j ∈ Q(β),

j=0define
∗

x =

d−1
X

xj (β2j , · · · , βdj ).

j=0

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual map

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

Dual contraction matrix:

β2

β3

B∗ = 
..

.

Separation
condition: afundamental
result



.

βd

5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsDefinition (Dual map)
The dual map of f (x) = x+b
is defined to be f ∗ : Rs → Rs asβf ∗ (y) = B∗ y + b∗ .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(4)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual map

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

Dual contraction matrix:

β2

β3

B∗ = 
..

.

Separation
condition: afundamental
result



.

βd

5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsDefinition (Dual map)
The dual map of f (x) = x+b
is defined to be f ∗ : Rs → Rs asβf ∗ (y) = B∗ y + b∗ .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(4)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual map

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

Dual contraction matrix:

β2

β3

B∗ = 
..

.

Separation
condition: afundamental
result



.

βd

5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsDefinition (Dual map)
The dual map of f (x) = x+b
is defined to be f ∗ : Rs → Rs asβf ∗ (y) = B∗ y + b∗ .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(4)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual IFS

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Definition (Dual IFS)Let
x + bγ(fγ )γ∈Γ =βbe an algebraic graph IFS. Let (V, Γ0 ) be the reverse graphof (V, Γ). We call(5)((fγ )∗ )γ 0 ∈Γ0the dual graph IFS.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual IFS

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Definition (Dual IFS)Let
x + bγ(fγ )γ∈Γ =βbe an algebraic graph IFS. Let (V, Γ0 ) be the reverse graphof (V, Γ). We call((fγ )∗ )γ 0 ∈Γ0(5)
the dual graph IFS.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual IFS

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Definition (Dual IFS)Let
x + bγ(fγ )γ∈Γ =βbe an algebraic graph IFS. Let (V, Γ0 ) be the reverse graphof (V, Γ). We call((fγ )∗ )γ 0 ∈Γ0(5)
the dual graph IFS.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual IFS

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§4. An Example

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A system related to Rauzy substitution

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Let β : β 3 = β 2 + β + 1.
The following graph IFS is arising from Rauzy substitution
 βI1 = I1 ∪ (I2 + 1)
βI2 = I1 ∪ (I3 + 1)
βI3 = I1 .
The invariant sets are
I1 = [0, 1], I2 = [0, 1/β + 1/β 2 ], I3 = [0, 1/β].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A system related to Rauzy substitution

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Let β : β 3 = β 2 + β + 1.
The following graph IFS is arising from Rauzy substitution
 βI1 = I1 ∪ (I2 + 1)
βI2 = I1 ∪ (I3 + 1)
βI3 = I1 .
The invariant sets are
I1 = [0, 1], I2 = [0, 1/β + 1/β 2 ], I3 = [0, 1/β].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A system related to Rauzy substitution

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Let β : β 3 = β 2 + β + 1.
The following graph IFS is arising from Rauzy substitution
 βI1 = I1 ∪ (I2 + 1)
βI2 = I1 ∪ (I3 + 1)
βI3 = I1 .
The invariant sets are
I1 = [0, 1], I2 = [0, 1/β + 1/β 2 ], I3 = [0, 1/β].

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A system related to Rauzy substitution

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

The maps corresponding to the edges of Γ are, fc (x) = βx ,fa (x) = βx , fb (x) = x+1
βfd (x) = x+1, fe (x) = βx .
βb4. An Example
5. Algebraic
IFS arisingfromsubstitutionsa1

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tiling

2

c
e

d
3

Figure: 2. Graph of original IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

A system related to Rauzy substitution

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

The maps corresponding to the edges of Γ are, fc (x) = βx ,fa (x) = βx , fb (x) = x+1
βfd (x) = x+1, fe (x) = βx .
βb4. An Example
5. Algebraic
IFS arisingfromsubstitutionsa1

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tiling

2

c
e

d
3

Figure: 2. Graph of original IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

4. An Example

Conjugates: β2 ≈ 0.419 + 0.606i, β3 ≈ 0.419 − 0.606i.
Dual contraction:

0.419 + 0.606i
0
∗
B =
,
0
0.419 − 0.606i

5. Algebraic
IFS arisingfromsubstitutions

or equivalently,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result∗B =

0.419 −0.606
0.606
0.419


.

Dual translations: The dual of 1 is the vector e1 .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

4. An Example

Conjugates: β2 ≈ 0.419 + 0.606i, β3 ≈ 0.419 − 0.606i.
Dual contraction:

0.419 + 0.606i
0
∗
B =
,
0
0.419 − 0.606i

5. Algebraic
IFS arisingfromsubstitutions

or equivalently,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result∗B =

0.419 −0.606
0.606
0.419


.

Dual translations: The dual of 1 is the vector e1 .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

4. An Example

Conjugates: β2 ≈ 0.419 + 0.606i, β3 ≈ 0.419 − 0.606i.
Dual contraction:

0.419 + 0.606i
0
∗
B =
,
0
0.419 − 0.606i

5. Algebraic
IFS arisingfromsubstitutions

or equivalently,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result∗B =

0.419 −0.606
0.606
0.419


.

Dual translations: The dual of 1 is the vector e1 .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

4. An Example

Conjugates: β2 ≈ 0.419 + 0.606i, β3 ≈ 0.419 − 0.606i.
Dual contraction:

0.419 + 0.606i
0
∗
B =
,
0
0.419 − 0.606i

5. Algebraic
IFS arisingfromsubstitutions

or equivalently,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result∗B =

0.419 −0.606
0.606
0.419


.

Dual translations: The dual of 1 is the vector e1 .

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

The dual graph IFS is:
ga0 (y) = B∗ y, gb0 (y) = B∗ y + e1 , gc0 (y) = B∗ y,
gd0 (y) = B∗ y + e1 , ge0 (y) = B∗ y.

3. Dual systemof algebraic
IFSb’

4. An Example
5. Algebraic
IFS arisingfromsubstitutions

a’

1

c’

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result

2

e’

d’
3

Figure: 3. Graph of dual IFS.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

The dual graph IFS is:
ga0 (y) = B∗ y, gb0 (y) = B∗ y + e1 , gc0 (y) = B∗ y,
gd0 (y) = B∗ y + e1 , ge0 (y) = B∗ y.

3. Dual systemof algebraic
IFSb’

4. An Example
5. Algebraic
IFS arisingfromsubstitutions

a’

1

c’

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
result

2

e’

d’
3

Figure: 3. Graph of dual IFS.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

An equivalent expression of the dual graph IFS is: ∗ −1
 (B ) X1 = X1 ∪ X2 ∪ X3
(B∗ )−1 X2 = X1 + (B∗ )−1 e1 ∗ −1(B ) X3 = X2 + (B∗ )−1 e1 .
The invariant sets are

4. An Example
5. Algebraic
IFS arisingfromsubstitutions

X2

6. Algebraic
IFS arising
fromnumeration
systems

X3
X1

Separation
condition: afundamental
resultRauzyThurston
tilingFigure: 4. Invariant sets
Hui RAO

Dual IFS of algebraic IFS(6)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Dual parameters

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

An equivalent expression of the dual graph IFS is: ∗ −1
 (B ) X1 = X1 ∪ X2 ∪ X3
(B∗ )−1 X2 = X1 + (B∗ )−1 e1 ∗ −1(B ) X3 = X2 + (B∗ )−1 e1 .
The invariant sets are

4. An Example
5. Algebraic
IFS arisingfromsubstitutions

X2

6. Algebraic
IFS arising
fromnumeration
systems

X3
X1

Separation
condition: afundamental
resultRauzyThurston
tilingFigure: 4. Invariant sets
Hui RAO

Dual IFS of algebraic IFS(6)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

One-dimensional IFS arising from substitutions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

5. Algebraic IFS arising from substitutions

5. Algebraic
IFS arisingfromsubstitutions

Simple.

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

One-dimensional IFS arising from substitutions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

5. Algebraic IFS arising from substitutions

5. Algebraic
IFS arisingfromsubstitutions

Simple.

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

One-dimensional IFS arising from substitutions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions

6. Algebraic IFS arising from numeration systems

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Piecewise linear dynamical systems with uniform
expanding ratio

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingA numeration system is typically a piecewise linear map
T from an interval I to itself.
Markov condition: the orbits of the discontinuous pointsof T are all eventually periodic.
Basic interval: Discontinuous points and their obits cut
I into small intervals
I = I1 ∪ I2 ∪ · · · ∪ Im .

Each T(Ij ) is a union of basic intervals, and this gives us
a one-dimensional graph-directed IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Piecewise linear dynamical systems with uniform
expanding ratio

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingA numeration system is typically a piecewise linear map
T from an interval I to itself.
Markov condition: the orbits of the discontinuous pointsof T are all eventually periodic.
Basic interval: Discontinuous points and their obits cut
I into small intervals
I = I1 ∪ I2 ∪ · · · ∪ Im .

Each T(Ij ) is a union of basic intervals, and this gives us
a one-dimensional graph-directed IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Piecewise linear dynamical systems with uniform
expanding ratio

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingA numeration system is typically a piecewise linear map
T from an interval I to itself.
Markov condition: the orbits of the discontinuous pointsof T are all eventually periodic.
Basic interval: Discontinuous points and their obits cut
I into small intervals
I = I1 ∪ I2 ∪ · · · ∪ Im .

Each T(Ij ) is a union of basic intervals, and this gives us
a one-dimensional graph-directed IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Piecewise linear dynamical systems with uniform
expanding ratio

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingA numeration system is typically a piecewise linear map
T from an interval I to itself.
Markov condition: the orbits of the discontinuous pointsof T are all eventually periodic.
Basic interval: Discontinuous points and their obits cut
I into small intervals
I = I1 ∪ I2 ∪ · · · ∪ Im .

Each T(Ij ) is a union of basic intervals, and this gives us
a one-dimensional graph-directed IFS.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Symmetric β-expansions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

In Akiyama and Scheicher (2007) introduced symmetric βtransformation for 1 < β < 3 as
Sβ : [−1/2, 1/2] → [−1/2, 1/2],
1Sβ (x) = βx − bβx + c.
2

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(7)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Symmetric β-expansions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

In Akiyama and Scheicher (2007) introduced symmetric βtransformation for 1 < β < 3 as
Sβ : [−1/2, 1/2] → [−1/2, 1/2],
1Sβ (x) = βx − bβx + c.
2

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS(7)

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Symmetric β-expansions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example

Example: β is same as in Rauzy substitution.
The discontinues orbits of the transformation cut the interval
into 7 basic intervals:
I1 , I2 , I3 , I0 , J3 , J2 , J1
from left to right.
1/2

5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

1/2

1/2β

Separation
condition: afundamental
resultRauzyThurston
tiling

−1/2β

−1/2

−1/2

Figure: 5. Symmetric β-transformation
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Symmetric β-expansions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tiling

The set equations are:βI0 = I3 ∪ I0 ∪ J3
and I1 , I2 , I3 , J1 , J2 , J3 satisfying the set equations

βI1 = (J3 ∪ J2 ∪ J1 ) − 1
βI2 = I1
βI3 = I2
βJ1 = (I1 ∪ I2 ∪ I3 ) + 1
βJ = J1 2
βJ3 = J2 .

(8)

We omit I0 since it is a non-primitive member and its dual isa single point.
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Symmetric β-expansions

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

The six invariant sets of the dual IFS are
22

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

A2 1

1

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

B3

B1
-1

-0.5

0.5

1

1.5

2
-2

-1.5

-1

-0.5

0.5

A1

A3
-1

-1

B2

-2
-2

Separation
condition: afundamental
resultFigure: 6. Tiles of symmetric β-transformation.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

11. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The minimal weight expansion is introduced by Frougny andSteiner (2008).
ββ
Wβ is defined on the interval I = [− β+1
, β+1
] as

β
1

 βx + 1, if x ∈ [− β+1 , − β+1 ]11βx,
if x ∈ [− β+1, β+1]
Wβ (x) = βx − 1, if x ∈ [ 1 , − β ].
β+1
β+1

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The minimal weight expansion is introduced by Frougny andSteiner (2008).
ββ
Wβ is defined on the interval I = [− β+1
, β+1
] as

β
1

 βx + 1, if x ∈ [− β+1 , − β+1 ]11βx,
if x ∈ [− β+1, β+1]
Wβ (x) = βx − 1, if x ∈ [ 1 , − β ].
β+1
β+1

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

The minimal weight expansion is introduced by Frougny andSteiner (2008).
ββ
Wβ is defined on the interval I = [− β+1
, β+1
] as

β
1

 βx + 1, if x ∈ [− β+1 , − β+1 ]11βx,
if x ∈ [− β+1, β+1]
Wβ (x) = βx − 1, if x ∈ [ 1 , − β ].
β+1
β+1

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

The partition is:
I = I1 ∪ I2 ∪ I0 ∪ J2 ∪ J1 .

3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultSet equation isWβ (I1 ) = I0 ∪ J2
 Wβ (I2 ) = I1
Wβ (I0 ) = I2 ∪ I0 ∪ J2
W(J ) = J1 β 2
Wβ (J1 ) = I2 ∪ I0 .

or
βI1 = (I0 ∪ J2 ) − 1 βI2 = I1
βI0 = I2 ∪ I0 ∪ J2βJ
2 = J1βJ1 = (I2 ∪ I0 ) + 1.
(9)RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation

The partition is:
I = I1 ∪ I2 ∪ I0 ∪ J2 ∪ J1 .

3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultSet equation isWβ (I1 ) = I0 ∪ J2
 Wβ (I2 ) = I1
Wβ (I0 ) = I2 ∪ I0 ∪ J2Wβ (J2 ) = J1Wβ (J1 ) = I2 ∪ I0 .

or
βI1 = (I0 ∪ J2 ) − 1 βI2 = I1
βI0 = I2 ∪ I0 ∪ J2βJ
2 = J1βJ1 = (I2 ∪ I0 ) + 1.
(9)RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Minimal weight expansion

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

Minimal weight transformation and invariant sets of its dual
system

2. Backgroundand motivation1β/(β+1)

3. Dual systemof algebraic
IFS

A20.5

4. An Example
5. Algebraic
IFS arisingfromsubstitutions−β/(β+1)−1/(β+1)

1/(β+1)β/(β+1)-1-0.5

6. Algebraic
IFS arising
fromnumeration
systems

0.5

-0.5

A3
−β/(β+1)

Separation
condition: afundamental
resultA1-1

Figure: 7. Minimal weight expansion

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

11. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Open set condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§7. Separation condition:

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Open set condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Theorem (Separation condition)
An algebraic graph IFS and its dual system satisfy the open
set condition simultaneously.
Consequently, a self-affine tiling system is defined by thedual system via dilation and subdivision.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Open set condition

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Theorem (Separation condition)
An algebraic graph IFS and its dual system satisfy the open
set condition simultaneously.
Consequently, a self-affine tiling system is defined by thedual system via dilation and subdivision.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Self-replicating tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§8. Rauzy-Thurston tiling.

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

The ideal generated by digit sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

DenoteZ[β] =( d−1
X)
mj β j : mj ∈ Z .

j=0

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultWe assume that the elements of D are all in Z[β].
Let K be the ideal of Z[β] generated by the elements ofD.
In other words, K is the smallest module which is invariant by multiplying β.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

The ideal generated by digit sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

DenoteZ[β] =( d−1
X)
mj β j : mj ∈ Z .

j=0

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultWe assume that the elements of D are all in Z[β].
Let K be the ideal of Z[β] generated by the elements ofD.
In other words, K is the smallest module which is invariant by multiplying β.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

The ideal generated by digit sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

DenoteZ[β] =( d−1
X)
mj β j : mj ∈ Z .

j=0

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultWe assume that the elements of D are all in Z[β].
Let K be the ideal of Z[β] generated by the elements ofD.
In other words, K is the smallest module which is invariant by multiplying β.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

The ideal generated by digit sets

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

DenoteZ[β] =( d−1
X)
mj β j : mj ∈ Z .

j=0

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultWe assume that the elements of D are all in Z[β].
Let K be the ideal of Z[β] generated by the elements ofD.
In other words, K is the smallest module which is invariant by multiplying β.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzy-Thurston tiling:
J =

N
[

{Xj + b∗ ; b ∈ Ij ∩ K}

(10)

j=1

To show that J is self-replicating or quasi-periodic, we needto assume one of the following conditions:
Ij are finite unions of intervals;∂Ij ∩ K = ∅.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzy-Thurston tiling:
J =

N
[

{Xj + b∗ ; b ∈ Ij ∩ K}

(10)

j=1

To show that J is self-replicating or quasi-periodic, we needto assume one of the following conditions:
Ij are finite unions of intervals;∂Ij ∩ K = ∅.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzy-Thurston tiling:
J =

N
[

{Xj + b∗ ; b ∈ Ij ∩ K}

(10)

j=1

To show that J is self-replicating or quasi-periodic, we needto assume one of the following conditions:
Ij are finite unions of intervals;∂Ij ∩ K = ∅.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzy-Thurston tiling:
J =

N
[

{Xj + b∗ ; b ∈ Ij ∩ K}

(10)

j=1

To show that J is self-replicating or quasi-periodic, we needto assume one of the following conditions:
Ij are finite unions of intervals;∂Ij ∩ K = ∅.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzy-Thurston tiling:
J =

N
[

{Xj + b∗ ; b ∈ Ij ∩ K}

(10)

j=1

To show that J is self-replicating or quasi-periodic, we needto assume one of the following conditions:
Ij are finite unions of intervals;∂Ij ∩ K = ∅.

RauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Rauzy-Thurston Tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
4

3. Dual systemof algebraic
IFS
4. An Example

3
2
1
0
−1

5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

−2
−3
−4
−8

−6

−4

−2

0

2

4

6

8

Figure: 8. Rauzy-Thurston Tiling of Rauzy substitution

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Branches of Rauzy-Thurston tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Theorem
If one of the above conditions holds, then there is an m ≥ 1such that almost every point of Rs is covered by exactly mtiles.
TheoremS
J can be written as J = mj=1 Jj , where Jj are tilings of Rs .

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Branches of Rauzy-Thurston tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems

Theorem
If one of the above conditions holds, then there is an m ≥ 1such that almost every point of Rs is covered by exactly mtiles.
TheoremS
J can be written as J = mj=1 Jj , where Jj are tilings of Rs .

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Branches of Rauzy-Thurston tiling

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

The Rauzy-Thurston tiling of the symmetric β-transformation
contains two pages, as indicated below.
664

4. An Example

42
2

5. Algebraic
IFS arisingfromsubstitutions

0
0

−2
−2

−4
−4

6. Algebraic
IFS arising
fromnumeration
systems

−6
−6
−8

−6

−4

−2

0

2

4

6

8

−8

−6

−4

−2

0

2

4

6

8

Figure: 9. Two pages of Rauzy-Thurston tiling

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Pisot Spectrum Conjecture

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions§9. Pisot Spectrum Conjecture

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

Pisot Spectrum Conjecture

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS

Pisot Spectrum Conjecture:
J is a tiling.

4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systemsN
X

L1 (Ij )Ls (Xj ) = norm of K.

j=1

We say the system is tight if the above equality holds.

Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

An reducible substitution

Dual IFS ofalgebraic IFS
Hui RAO

Rauzy substitution and its variation:

1. Algebraic
IFS

σ : 1 7→ 12
2 7→ 13
3 7→ 1,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingΣ : 1 7→ 10 2
2 7→ 10 3
3 7→ 10
10 7→ 120
20 7→ 130
30 7→ 1.

Clearly
I10 = I1 , I20 = I2 , I30 = I3 ,X10 = X1 , X20 = X2 , X30 = X3 .
Hence the Rauzy-Thurston tiling of Σ contains two pages
and these two pages are identical!
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

An reducible substitution

Dual IFS ofalgebraic IFS
Hui RAO

Rauzy substitution and its variation:

1. Algebraic
IFS

σ : 1 7→ 12
2 7→ 13
3 7→ 1,

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions
6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingΣ : 1 7→ 10 2
2 7→ 10 3
3 7→ 10
10 7→ 120
20 7→ 130
30 7→ 1.

Clearly
I10 = I1 , I20 = I2 , I30 = I3 ,X10 = X1 , X20 = X2 , X30 = X3 .
Hence the Rauzy-Thurston tiling of Σ contains two pages
and these two pages are identical!
Hui RAO

Dual IFS of algebraic IFS

1. Algebraic IFS 2. Background and motivation 3. Dual system of algebraic IFS 4. An Example 5. Algebraic IFS arising from substitu

The End

Dual IFS ofalgebraic IFS
Hui RAO
1. Algebraic
IFS

2. Backgroundand motivation
3. Dual systemof algebraic
IFS
4. An Example
5. Algebraic
IFS arisingfromsubstitutions

Thank you!

6. Algebraic
IFS arising
fromnumeration
systems
Separation
condition: afundamental
resultRauzyThurston
tilingHui RAO

Dual IFS of algebraic IFS