![Page 1: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/1.jpg)
Self-affine sets with nonempty interior
Timo JolivetJoint work with Jarkko Kari
![Page 2: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/2.jpg)
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
![Page 3: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/3.jpg)
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?
![Page 4: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/4.jpg)
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?
![Page 5: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/5.jpg)
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?
![Page 6: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/6.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 7: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/7.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 8: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/8.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 9: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/9.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 10: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/10.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 11: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/11.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 12: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/12.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 13: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/13.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 14: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/14.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 15: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/15.jpg)
Example 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
![Page 16: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/16.jpg)
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
![Page 17: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/17.jpg)
Necessary conditionI A ∈Md(Z) is expandingI D ⊆ Zd
We will now assume that |D| = det(A)
![Page 18: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/18.jpg)
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
![Page 19: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/19.jpg)
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
![Page 20: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/20.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 21: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/21.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 22: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/22.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 23: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/23.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 24: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/24.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 25: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/25.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 26: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/26.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 27: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/27.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 28: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/28.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 29: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/29.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 30: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/30.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 31: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/31.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 32: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/32.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 33: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/33.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 34: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/34.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 35: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/35.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 36: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/36.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 37: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/37.jpg)
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?
![Page 38: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/38.jpg)
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?
![Page 39: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/39.jpg)
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?
![Page 40: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/40.jpg)
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior(why?)
![Page 41: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/41.jpg)
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
(why?)
![Page 42: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/42.jpg)
Examples with “incomplete” D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior(why?)
![Page 43: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/43.jpg)
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)}
![Page 44: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/44.jpg)
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)}
![Page 45: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/45.jpg)
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)}
![Page 46: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/46.jpg)
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)},
![Page 47: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/47.jpg)
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).
![Page 48: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/48.jpg)
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
![Page 49: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/49.jpg)
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
![Page 50: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/50.jpg)
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?)
![Page 51: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/51.jpg)
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?)
![Page 52: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/52.jpg)
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?)
![Page 53: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/53.jpg)
Affine iterated function systems
![Page 54: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/54.jpg)
Affine iterated function systems
![Page 55: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/55.jpg)
Affine iterated function systems
![Page 56: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/56.jpg)
Affine iterated function systems
![Page 57: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/57.jpg)
Affine iterated function systems
![Page 58: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/58.jpg)
Affine iterated function systems
![Page 59: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/59.jpg)
Affine iterated function systems
![Page 60: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/60.jpg)
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
![Page 61: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/61.jpg)
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
![Page 62: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/62.jpg)
Computational tools: multitape automata
![Page 63: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/63.jpg)
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
![Page 64: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/64.jpg)
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
![Page 65: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/65.jpg)
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
![Page 66: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/66.jpg)
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
![Page 67: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/67.jpg)
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
![Page 68: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/68.jpg)
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
![Page 69: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/69.jpg)
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
![Page 70: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/70.jpg)
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
![Page 71: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/71.jpg)
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
![Page 72: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/72.jpg)
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
)
![Page 73: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/73.jpg)
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
)
![Page 74: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/74.jpg)
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
)
![Page 75: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/75.jpg)
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
)
![Page 76: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/76.jpg)
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
}
![Page 77: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/77.jpg)
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
}
![Page 78: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/78.jpg)
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
}
![Page 79: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/79.jpg)
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
}
![Page 80: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/80.jpg)
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
}
![Page 81: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/81.jpg)
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]}
![Page 82: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/82.jpg)
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]}
![Page 83: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/83.jpg)
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
![Page 84: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/84.jpg)
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
![Page 85: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/85.jpg)
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
![Page 86: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/86.jpg)
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
![Page 87: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/87.jpg)
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
![Page 88: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/88.jpg)
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
![Page 89: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/89.jpg)
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
![Page 90: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/90.jpg)
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
![Page 91: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/91.jpg)
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
![Page 92: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/92.jpg)
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
![Page 93: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/93.jpg)
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
![Page 94: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/94.jpg)
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
![Page 95: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/95.jpg)
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
![Page 96: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/96.jpg)
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
![Page 97: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/97.jpg)
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
![Page 98: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/98.jpg)
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”)
![Page 99: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/99.jpg)
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
![Page 100: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/100.jpg)
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
![Page 101: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/101.jpg)
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
![Page 102: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/102.jpg)
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
![Page 103: TimoJolivet Jointworkwith JarkkoKarijolivet.org/timo/docs/slides_interior.pdf · Affineiteratedfunctionsystems Letf 1,...,f n: Rd →Rd becontractingmaps Theorem(Hutchinson 1981)](https://reader034.vdocument.in/reader034/viewer/2022042017/5e75a4707e00cf1c1c50784e/html5/thumbnails/103.jpg)
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