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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?

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)}


( 2 00 2)

I D ={( 0

0),( 1

0),( 0

1),(−1−1)}


( 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?

Examples with “incomplete” D

A =( 2 0

0 2)

D ={( 0

0),( 1

0),( 0

1),( 1

2)}

X has empty interior(why?)


A =( 2 0

0 2)

D ={( 0

0),( 1

0),( 0

1),( 1

2)}

X has empty interior

(why?)


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)}


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)}


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)},


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).


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?)


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?)


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

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



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


0001

010001000 . . .

110

0011011 . . . ∈ AN = ({0, 1} × {0, 1})N

Multitape automata



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


000101

0001000 . . .

11000

11011 . . . ∈ AN = ({0, 1} × {0, 1})N

Multitape automata



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


000101000

1000 . . .

1100011

011 . . . ∈ AN = ({0, 1} × {0, 1})N

Multitape automata



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


0001010001

000 . . .

11000110

11 . . . ∈ AN = ({0, 1} × {0, 1})N

Multitape automata



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


0001010001000

. . .

1100011011

. . .∈ AN = ({0, 1} × {0, 1})N

Multitape automata



1 × · · · ×A+d )→ Q

X Y01 | 00 1 | 001

000 | 11

1 | 0

A = {0, 1} × {0, 1}Q = {X,Y }


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

)


Transitionu | v



)=( |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

)


Transitionu | v



)=( |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

)


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

}


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

}


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]}

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



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



f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2



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



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

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




(x1x2 . . .x1x2 . . .

)

⇐⇒ attractor contains a point(0.x1x2 . . .

0.x1x2 . . .



Proof idea




(x1x2 . . .x1x2 . . .


(0.x1x2 . . .0.x1x2 . . .

)

⇐⇒ attractor ∩ diagonal 6= ∅I So: “attractor ∩ diagonal 6= ∅” is undecidable

Proof idea




(x1x2 . . .x1x2 . . .


(0.x1x2 . . .0.x1x2 . . .



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

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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