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

Trivial FG

Examples

Uncountable FG, very hard to describe. . .

Examples

Doesn’t look so bad, but. . .

Examples

Doesn’t look so bad, but. . . uncountable FG

Examples

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

Other examples with nontrivial countable FG

F1 F1

F2 F6

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?Difficulty: control the fractal by working on σ.

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

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 fractalsCompare σ and τ−1στ, with

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)

Uncountable, but manageable FG? — Candidate 1

Uncountable, but manageable FG? — Candidate 2

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

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