Countable fundamental groups of Rauzy fractals
Transcript of Countable fundamental groups of Rauzy fractals
Countable fundamental groups ofRauzy fractals
Timo JolivetUniversité Paris Diderot, FranceUniversity of Turku, Finland
Joint work with Benoît Loridant and Jun Luo
Advances on Fractals and Related TopicsDecember 14, 2012香港中文大學
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1
234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
12
34511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
123
4511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234
511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
12345
11212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
123451
1212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
12345112
12312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
12345112123
12341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
123451121231234
1234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
12345112123123412345
12345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβ
I Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Substitutions
σ :
1 7→ 122 7→ 33 7→ 44 7→ 55 7→ 1
Mσ =
(1 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
)
1234511212312341234512345112345112123451 · · ·
I Cubic Pisot eigenvalue β ≈ 1.325, with β ′, β ′′ ≈ −0.662± 0.562iI Two other eigenvalues ≈ 0.5± 0.866i
Action of Mσ on R5:I Expanding line E spanned by eigenvector uβI Contracting plane P spanned by eigenvectors uβ′ ,uβ′′
I (Supplementary space H spanned by the other eigenvectors)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123 π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)
π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1) + π(e2)
π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1) + π(e2) + π(e3)
π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1) + π(e2) + π(e3) + π(e4)
π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1) + π(e2) + π(e3) + π(e4) + π(e5)
π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)+π(e2)+π(e3)+π(e4)+π(e5)+ · · ·π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)+π(e2)+π(e3)+π(e4)+π(e5)+ · · ·π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)+π(e2)+π(e3)+π(e4)+π(e5)+ · · ·π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)+π(e2)+π(e3)+π(e4)+π(e5)+ · · ·π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Rauzy fractal of 12 7→ 1, 2 7→ 3, 3 7→ 4, 4 7→ 5, 5 7→ 1
12345112123123412345123451123
π(e1)+π(e2)+π(e3)+π(e4)+π(e5)+ · · ·π = projection from R5to P along E⊕H
π(e1)π(e2)
π(e3)
π(e4)
π(e5)
Basic factsI Dynamical properties (domain exchange), number theory, tilings, . . .I Topology:
I Compact, locally connectedI Equal to the closure of the interiorI Graph-directed IFS, self-similar structureI dimH(boundary) ∈ ]1, 2[, computable
What kind of topological properties do we study?I Cut points, connectedness, intersection of tiles, etc.
I Simple connectedness
No cut points
Cut points
What kind of topological properties do we study?I Cut points, connectedness, intersection of tiles, etc.
I Simple connectedness
Connected
Not connected
What kind of topological properties do we study?I Cut points, connectedness, intersection of tiles, etc.I Simple connectedness
Disklike “Many holes”
What kind of topological properties do we study?I Cut points, connectedness, intersection of tiles, etc.I Simple connectedness
Disklike “Many holes”
Most of these properties are decidable [Siegel-Thuswaldner 2010]
Fundamental groupI Topological space X, point x0 ∈ XI π1(X, x0) is the fundamental group of X at x0, where:
I elements: loops based at x0, modulo homotopyI composition law: concatenation of loops
I π1(X) := π1(X, x0) if X path-connected (for any x0)I Examples:
I π1(disc) ∼= {1}I π1(circle) ∼= ZI π1(2D torus) ∼= Z2I π1(∞) ∼= π1(8) ∼= π1(B) ∼= F2 = free group on two elementsI π1(Q) ∼= F6
Examples
There seems to be a dichotomy:
trivial vs. uncountable
easy countable uncountable hard
trivial ?!??!
Examples
There seems to be a dichotomy:
trivial vs. uncountable
easy countable uncountable hard
trivial ?!??!
But. . .
Example with nontrivial countable FG
π1(X) ∼= F1 ∼= Z
1 7→ 11, 1, 3, 8 2 7→ 11, 1, 3, 8
3 7→ 5, 2, 4, 10, 12 4 7→ 6, 9, 7
5 7→ 5, 2, 4, 10, 12 6 7→ 5, 2, 4, 10, 12
7 7→ 6, 9, 7, 11, 1, 3, 8 8 7→ 6, 9, 7, 11, 1, 3, 8
9 7→ 11, 1, 3, 8 10 7→ 11, 1, 3, 8
11 7→ 5, 2, 4, 10, 12 12 7→ 5, 2, 4, 10, 12, 6, 9, 7, 11, 1, 3, 8.
How to prove that π1( )
∼= F1 ?
I Prove that the 12 tiles are disklikeand compute their neighboring graph [Siegel-Thuswaldner 2009]
I Compute a topological complex with the same homotopy type
How to prove that π1( )
∼= F1 ?
I Prove that the 12 tiles are disklikeand compute their neighboring graph [Siegel-Thuswaldner 2009]
I Compute a topological complex with the same homotopy type
How to prove that π1( )
∼= F1 ?
I Prove that the 12 tiles are disklikeand compute their neighboring graph [Siegel-Thuswaldner 2009]
I Compute a topological complex with the same homotopy type
How to prove that π1( )
∼= F1 ?
I Prove that the 12 tiles are disklikeand compute their neighboring graph [Siegel-Thuswaldner 2009]
I Compute a topological complex with the same homotopy type
Countable FGs of Rauzy fractals, in general
Theorem [J-Loridant-Luo]I A nontrivial countable Rauzy fractal FG is always ∼= Fk (k ∈ N).I k can be computed if all the tiles are discs (and intersect well).
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)
I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Can we get all Fk? Possible approach: drill holesI Step 1: cut the tiles into smaller tiles (on σ: “state splittings”)I Step 2: shrink some tiles (on σ: conjugate by free group aut.)
Free group automorphisms ↔ Rauzy fractalsCompare σ and τ−1στ, with σ : 1 7→ 12, 2 7→ 13, 3 7→ 1
and τ : 1 7→ 1, 2 7→ 32, 3 7→ 3
τ−1στ :
1 7→ 1 7→ 12 7→ 132
2 7→ 3−12 7→ 1−113 7→ 3
3 7→ 3 7→ 1 7→ 1
Free group automorphisms ↔ Rauzy fractalsCompare σ and τ−1στ, with σ : 1 7→ 12, 2 7→ 13, 3 7→ 1
and τ : 1 7→ 1, 2 7→ 32, 3 7→ 3
τ−1στ :
1 7→ 1 7→ 12 7→ 132
2 7→ 3−12 7→ 1−113 7→ 3
3 7→ 3 7→ 1 7→ 1
σ
τ−1στ
Free group automorphisms ↔ Rauzy fractals
Compare σ and τ−1στ, with σ = [. . .] and τ :
{i 7→ i if i 6= 1212 7→ 5, 12
σ τ−1στ
Free group automorphisms ↔ Rauzy fractals
Compare σ and τ−1στ, with σ = [. . .] and τ :
{i 7→ i if i 6= 1212 7→ 5, 12
σ τ−1στ
Question [Gähler, Arnoux-Berthé-Hilion-Siegel]Links between σ and τ−1στ?
å The fund. group is not preserved!
Uncountable, but manageable FG?
countable, only Fk uncountable
trivial ?!??!F1 F6F2 F3 F7
?
A Rauzy fractal with uncountable FG that we can describe in detail?
Like:
Hawaiian Earring(Cannon-Conner 2000)
Sierpiński gasket(Akiyama-Dorfer
-Thuswaldner-Winkler 2009)
Uncountable, but manageable FG?
countable, only Fk uncountable
trivial ?!??!F1 F6F2 F3 F7
?
A Rauzy fractal with uncountable FG that we can describe in detail?
Like:
Hawaiian Earring(Cannon-Conner 2000)
Sierpiński gasket(Akiyama-Dorfer
-Thuswaldner-Winkler 2009)
SummaryI Toplogy of Rauzy fractalsI “Inverse” IFS problemsI Operations on σ fractalI New examples
QuestionsI Obtain every Fk.I The case of three letters.I Precise description some
uncountable Rauzy fractal FGs.I bound rank(Fk) by #alphabet?
唔唔唔該該該 for your attention