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Self-affine sets with nonempty interior
Timo JolivetJoint work with Jarkko Kari
Affine iterated function systemsLet f1, . . . , fn : Rd → Rd be contracting maps
Theorem (Hutchinson 1981)There is a unique nonemtpy compact set X ⊆ Rd such that
X = f1(X) ∪ · · · ∪ fn(X).
I f1, . . . , fn is an iterated function system (IFS)I We will restrict to affine maps fi : x 7→ Aix+ vi
Self-affine tilesI Let A ∈Md(Z) be an expanding matrix (eigenvalues |λi| > 1)I Let D ⊆ Zd
Theorem (Hutchinson 1981)There is a unique nonempty compact X ⊆ Rd such that
X =⋃
d∈DA−1(X + d).
I Question: When does X have nonemtpy interior?
Self-affine tilesI Let A ∈Md(Z) be an expanding matrix (eigenvalues |λi| > 1)I Let D ⊆ Zd
Theorem (Hutchinson 1981)There is a unique nonempty compact X ⊆ Rd such that
X =⋃
d∈DA−1(X + d).
I Question: When does X have nonemtpy interior?
Self-affine tilesI Let A ∈Md(Z) be an expanding matrix (eigenvalues |λi| > 1)I Let D ⊆ Zd
Theorem (Hutchinson 1981)There is a unique nonempty compact X ⊆ Rd such that
X =⋃
d∈DA−1(X + d).
I Question: When does X have nonemtpy interior?
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
We have 4µ(X) = µ(X) + µ(X) + µ(X) = 3µ(X),so µ(X) = 0 and X has empty interior
Necessary conditionI A ∈Md(Z) is expandingI D ⊆ Zd
We will now assume that |D| = det(A)
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
A sufficient condition
Theorem (Bandt 1991)D is a complete set of representatives in Zd/AZd
=⇒ X has nonempty interior
Previous “Sierpiński” examples:
Zd/( 2 0
0 2)Zd =
{( 00),( 1
0),( 0
1), V}
with V =(−1−1),( 3
1),( 1
1).
What happens when D is not complete?
A sufficient condition
Theorem (Bandt 1991)D is a complete set of representatives in Zd/AZd
=⇒ X has nonempty interior
Previous “Sierpiński” examples:
Zd/( 2 0
0 2)Zd =
{( 00),( 1
0),( 0
1), V}
with V =(−1−1),( 3
1),( 1
1).
What happens when D is not complete?
A sufficient condition
Theorem (Bandt 1991)D is a complete set of representatives in Zd/AZd
=⇒ X has nonempty interior
Previous “Sierpiński” examples:
Zd/( 2 0
0 2)Zd =
{( 00),( 1
0),( 0
1), V}
with V =(−1−1),( 3
1),( 1
1).
What happens when D is not complete?
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior(why?)
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
(why?)
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior(why?)
Examples with “incomplete” DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)},
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
Examples with “incomplete” DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)},
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
Examples with “incomplete” DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)},
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
Examples with “incomplete” DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)},
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2),( 2
0),( 3
0),( 2
1),( 3
2),( 0
2),( 1
2),( 0
3),( 1
4),( 2
4),( 3
4),( 2
5),( 3
6)},
Examples with “incomplete” DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)},
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2),( 2
0),( 3
0),( 2
1),( 3
2),( 0
2),( 1
2),( 0
3),( 1
4),( 2
4),( 3
4),( 2
5),( 3
6)},
so 16µ(X) 6 15µ(X).
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
2),( 1
2)}
X has nonempty interior
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
2),( 1
2)}
X has nonempty interior
SummaryOriginal question: when does X have nonempty interior?
I Self-affine tilesI D is complete: sufficient conditionI D is not complete: anything can happen
I There is an algorithm: [Gabardo-Yu 2006] and[Bondarenko-Kravchenko 2011]
I More general families? (When matrices Ai are not equal?)
SummaryOriginal question: when does X have nonempty interior?
I Self-affine tilesI D is complete: sufficient conditionI D is not complete: anything can happenI There is an algorithm: [Gabardo-Yu 2006] and
[Bondarenko-Kravchenko 2011]
I More general families? (When matrices Ai are not equal?)
SummaryOriginal question: when does X have nonempty interior?
I Self-affine tilesI D is complete: sufficient conditionI D is not complete: anything can happenI There is an algorithm: [Gabardo-Yu 2006] and
[Bondarenko-Kravchenko 2011]I More general families? (When matrices Ai are not equal?)
Affine iterated function systems
Affine iterated function systems
Affine iterated function systems
Affine iterated function systems
Affine iterated function systems
Affine iterated function systems
Affine iterated function systems
Graph-directed IFS (GIFS)More general: replace X = f1(X) ∪ · · · ∪ fk(X) by a system ofseveral unknowns X1, . . . , Xn.
X1 = f1(X1) ∪ f2(X2) ∪ f3(X3)X2 = f4(X1)X3 = f5(X2)
X1
X3
X2
f1
f3
f2
f4f5
GRAPH IFSwith 3 states
Graph-directed IFS (GIFS)More general: replace X = f1(X) ∪ · · · ∪ fk(X) by a system ofseveral unknowns X1, . . . , Xn.X1 = f1(X1) ∪ f2(X2) ∪ f3(X3)X2 = f4(X1)X3 = f5(X2)
X1
X3
X2
f1
f3
f2
f4f5
GRAPH IFSwith 3 states
Computational tools: multitape automata
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
000
1010001000 . . .
11
00011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
000
1010001000 . . .
11
00011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
000
1010001000 . . .
11
00011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
0001
010001000 . . .
110
0011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
000101
0001000 . . .
11000
11011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
000101000
1000 . . .
1100011
011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
0001010001
000 . . .
11000110
11 . . . ∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
0001010001000
. . .
1100011011
. . .∈ AN = ({0, 1} × {0, 1})N
Multitape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted configurations:
0001010001000 . . .1100011011 . . . ∈ AN = ({0, 1} × {0, 1})N
2-tape automaton 7−→ 2-dimensional IFS
Transitionu | v
Tape alphabets A1, A2
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated IFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
2-tape automaton 7−→ 2-dimensional IFS
Transitionu | v
Tape alphabets A1, A2
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)
Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated IFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
2-tape automaton 7−→ 2-dimensional IFS
Transitionu | v
Tape alphabets A1, A2
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated IFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
2-tape automaton 7−→ 2-dimensional IFS
Transitionu | v
Tape alphabets A1, A2
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated IFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
2-tape automaton 7−→ 2-dimensional IFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
Fractal associated with automatonM
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM
}
2-tape automaton 7−→ 2-dimensional IFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
Fractal associated with automatonM
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM
}
2-tape automaton 7−→ 2-dimensional IFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)
Key correspondenceFractal associated with automatonM
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM
}
2-tape automaton 7−→ 2-dimensional IFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
Fractal associated with automatonM
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM
}
Example: Sierpiński triangle0|0
1|0 0|1
12(
xy
)+(0
0
)12(
xy
)+(0
1/2
) 12(
xy
)+(1/2
0
)Automaton language Iterating IFS maps
={(0.x1x2 . . .
0.y1y2 . . .
): (xn, yn) 6= (1, 1), ∀n > 1
}
Multitape automata ←→ fractals
Language theoretical properties of accepted language←→
Topological properties of fractal set
Example (in 2D) [Dube 1993, original idea]
Automaton accepts one word of the form(0.x1x2 . . .
0.x1x2 . . .
)⇐⇒ X intersects the diagonal {(x, x) : x ∈ [0, 1]}
Multitape automata ←→ fractals
Language theoretical properties of accepted language←→
Topological properties of fractal set
Example (in 2D) [Dube 1993, original idea]
Automaton accepts one word of the form(0.x1x2 . . .
0.x1x2 . . .
)⇐⇒ X intersects the diagonal {(x, x) : x ∈ [0, 1]}
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Universal: 2D, 1 state, transitions 0|0, 0|1, 1|0, 1|1I Not universal: 2D, 1 state, transitions 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Universal with prefix 1 (but not universal)1D, 1 state, transitions 1, 10, 00
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Universal: 2D, 1 state, transitions 0|0, 0|1, 1|0, 1|1
I Not universal: 2D, 1 state, transitions 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Universal with prefix 1 (but not universal)1D, 1 state, transitions 1, 10, 00
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Universal: 2D, 1 state, transitions 0|0, 0|1, 1|0, 1|1I Not universal: 2D, 1 state, transitions 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Universal with prefix 1 (but not universal)1D, 1 state, transitions 1, 10, 00
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Universal: 2D, 1 state, transitions 0|0, 0|1, 1|0, 1|1I Not universal: 2D, 1 state, transitions 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Universal with prefix 1 (but not universal)1D, 1 state, transitions 1, 10, 00
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Universal: 2D, 1 state, transitions 0|0, 0|1, 1|0, 1|1I Not universal: 2D, 1 state, transitions 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Universal with prefix 1 (but not universal)1D, 1 state, transitions 1, 10, 00
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Main result
Theorem [J-Kari 2013]For 3-state, 2-tape automata:
I universality is undecidableI prefix-universality is undecidable
CorollaryFor 2D affine graph-IFS with 3 states:
I = [0, 1]2 is undecidableI empty interior is undecidable
Main result
Theorem [J-Kari 2013]For 3-state, 2-tape automata:
I universality is undecidableI prefix-universality is undecidable
CorollaryFor 2D affine graph-IFS with 3 states:
I = [0, 1]2 is undecidableI empty interior is undecidable
Proof idea
Post correspondence problem (undecidable)Given n pairs of words (u1, v1), . . . , (un, vn),does there exists i1, . . . , iksuch that ui1 · · ·uik
= vi1 · · · vik?
Examples:I There is a solution:
(u1, v1) = (aa, aab), (u2, v2) = (bb, ba),(u3, v3) = (abb, b)(i1, i2, i3, i4) = (1, 2, 1, 3) is a solution because
u1u2u1u3 = aa bb aa abb = aab ba aab b = v1v2v1v3
I No solution exists:(u1, v1) = (aa, bab), (u2, v2) = (ab, ba)
I No solution exists:(u1, v1) = (a, ab), (u2, v2) = (ba, ab)
Variant: infinite-PCP
Proof idea
Post correspondence problem (undecidable)Given n pairs of words (u1, v1), . . . , (un, vn),does there exists i1, . . . , iksuch that ui1 · · ·uik
= vi1 · · · vik?
Examples:I There is a solution:
(u1, v1) = (aa, aab), (u2, v2) = (bb, ba),(u3, v3) = (abb, b)(i1, i2, i3, i4) = (1, 2, 1, 3) is a solution because
u1u2u1u3 = aa bb aa abb = aab ba aab b = v1v2v1v3
I No solution exists:(u1, v1) = (aa, bab), (u2, v2) = (ab, ba)
I No solution exists:(u1, v1) = (a, ab), (u2, v2) = (ba, ab)
Variant: infinite-PCP
Proof idea
Post correspondence problem (undecidable)Given n pairs of words (u1, v1), . . . , (un, vn),does there exists i1, . . . , iksuch that ui1 · · ·uik
= vi1 · · · vik?
Examples:I There is a solution:
(u1, v1) = (aa, aab), (u2, v2) = (bb, ba),(u3, v3) = (abb, b)(i1, i2, i3, i4) = (1, 2, 1, 3) is a solution because
u1u2u1u3 = aa bb aa abb = aab ba aab b = v1v2v1v3
I No solution exists:(u1, v1) = (aa, bab), (u2, v2) = (ab, ba)
I No solution exists:(u1, v1) = (a, ab), (u2, v2) = (ba, ab)
Variant: infinite-PCP
Proof idea
Theorem [Dube 1993]It is undecidable if the attractor of 2-dimensional IFSintersects the diagonal {(x, x) : x ∈ [0, 1]}
I PCP instance (u1, v1), . . . , (un, vn)7−→ 1-state automatonM with transitions ui|vi
I ∃ infinite-PCP solution⇐⇒ M accepts a configuration
(x1x2 . . .x1x2 . . .
)⇐⇒ attractor contains a point
(0.x1x2 . . .0.x1x2 . . .
)⇐⇒ attractor ∩ diagonal 6= ∅
I So: “attractor ∩ diagonal 6= ∅” is undecidable
Proof idea
Theorem [Dube 1993]It is undecidable if the attractor of 2-dimensional IFSintersects the diagonal {(x, x) : x ∈ [0, 1]}
I PCP instance (u1, v1), . . . , (un, vn)7−→ 1-state automatonM with transitions ui|vi
I ∃ infinite-PCP solution⇐⇒ M accepts a configuration
(x1x2 . . .x1x2 . . .
)⇐⇒ attractor contains a point
(0.x1x2 . . .0.x1x2 . . .
)⇐⇒ attractor ∩ diagonal 6= ∅
I So: “attractor ∩ diagonal 6= ∅” is undecidable
Proof idea
Theorem [Dube 1993]It is undecidable if the attractor of 2-dimensional IFSintersects the diagonal {(x, x) : x ∈ [0, 1]}
I PCP instance (u1, v1), . . . , (un, vn)7−→ 1-state automatonM with transitions ui|vi
I ∃ infinite-PCP solution⇐⇒ M accepts a configuration
(x1x2 . . .x1x2 . . .
)
⇐⇒ attractor contains a point(0.x1x2 . . .
0.x1x2 . . .
)⇐⇒ attractor ∩ diagonal 6= ∅
I So: “attractor ∩ diagonal 6= ∅” is undecidable
Proof idea
Theorem [Dube 1993]It is undecidable if the attractor of 2-dimensional IFSintersects the diagonal {(x, x) : x ∈ [0, 1]}
I PCP instance (u1, v1), . . . , (un, vn)7−→ 1-state automatonM with transitions ui|vi
I ∃ infinite-PCP solution⇐⇒ M accepts a configuration
(x1x2 . . .x1x2 . . .
)⇐⇒ attractor contains a point
(0.x1x2 . . .0.x1x2 . . .
)
⇐⇒ attractor ∩ diagonal 6= ∅I So: “attractor ∩ diagonal 6= ∅” is undecidable
Proof idea
Theorem [Dube 1993]It is undecidable if the attractor of 2-dimensional IFSintersects the diagonal {(x, x) : x ∈ [0, 1]}
I PCP instance (u1, v1), . . . , (un, vn)7−→ 1-state automatonM with transitions ui|vi
I ∃ infinite-PCP solution⇐⇒ M accepts a configuration
(x1x2 . . .x1x2 . . .
)⇐⇒ attractor contains a point
(0.x1x2 . . .0.x1x2 . . .
)⇐⇒ attractor ∩ diagonal 6= ∅
I So: “attractor ∩ diagonal 6= ∅” is undecidable
Proof idea
Theorem [J-Kari 2013]For 3-state, 2-tape automata:
I universality is undecidableI prefix-universality is undecidable
I Universality: reduce PCPI Prefix-universality: reduce a variant of PCP (“prefix-PCP”)
Conclusion & perspectivesI Self-affine integer tiles: decidableI Affine 2D GIFS undecidable
I What happens in intermediate cases?I 1D?I One-state GIFS? (i.e., IFS)
I Links automata ↔ topology:
Property of the d-tape automaton Topological property∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Thank you
Conclusion & perspectivesI Self-affine integer tiles: decidableI Affine 2D GIFS undecidableI What happens in intermediate cases?
I 1D?I One-state GIFS? (i.e., IFS)
I Links automata ↔ topology:
Property of the d-tape automaton Topological property∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Thank you
Conclusion & perspectivesI Self-affine integer tiles: decidableI Affine 2D GIFS undecidableI What happens in intermediate cases?
I 1D?I One-state GIFS? (i.e., IFS)
I Links automata ↔ topology:
Property of the d-tape automaton Topological property∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Thank you
Conclusion & perspectivesI Self-affine integer tiles: decidableI Affine 2D GIFS undecidableI What happens in intermediate cases?
I 1D?I One-state GIFS? (i.e., IFS)
I Links automata ↔ topology:
Property of the d-tape automaton Topological property∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Thank you
ReferencesI Lagarias, Wang, Self-affine tiles in Rn, Adv. Math., 1996I Wang, Self-affine tiles, in book Advances in wavelets, 1999
(great survey, available online)I Lai, Lau, Rao, Spectral structure of digit sets ofself-similar tiles on R1, Trans. Amer. Math. Soc., 2013
I Gabardo,Yu, Natural tiling, lattice tiling and lebesguemeasure of integral self-affine tiles, J. London Math. Soc.,2006
I Bondarenko, Kravchenko, On Lebesgue Measure ofIntegral Self-Affine Sets, Disc. Comput. Geom., 2011
I Dube, Undecidable problems in fractal geometry, ComplexSystems, 1993
I Jolivet, Kari, Undecidable properties of self-affine setsand multi-tape automata, arXiv:1401.0705
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