SymmetricRauzy
fractals
Paul Surer

Symmetric and congruent Rauzy fractals

Motivation
Construction
Congruence

Paul Surer

Symmetry
Problems

Universität für Bodenkultur
Institut für Mathematik
1180 Vienna - AUSTRIA

Paris, May 2018
based on a joint-research with Klaus Scheicher and Víctor Sirvent

SymmetricRauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Motivation

Observation 1
Symmetric
Rauzy
fractals
Paul Surer

Congruence
The Rauzy fractals induced by the substitutions

Motivation

σ : 1 7→ 1211, 2 7→ 311, 3 7→ 1 andConstruction

σ 0 : 1 7→ 1112, 2 7→ 113, 3 7→ 1

Congruence
Symmetry
Problems

over the alphabet A = {1, 2, 3} are congruent (that is theydiffer by an affine transformation only).

Observation 2
Symmetric
Rauzy
fractals
Paul Surer
Motivation

Symmetry (Sellami, Sirvent: 2011, 2012, 2016)
The (original) Rauzy fractal induced by the substitutions
σ : 1 7→ 12, 2 7→ 13, 3 7→ 1

Construction
Congruence
Symmetry
Problems(over the alphabet A = {1, 2, 3}) is central-symmetric withrespect to some point c.

ProblemSymmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Some Questions
Which conditions ensure that Rauzy fractals are congruent?

ProblemSymmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Some Questions
Which conditions ensure that Rauzy fractals are congruent?
Which conditions ensure that a Rauzy fractal is central
symmetric?

ProblemSymmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Some Questions
Which conditions ensure that Rauzy fractals are congruent?
Which conditions ensure that a Rauzy fractal is central
symmetric?
What is the centre of symmetry?

ProblemSymmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Some Questions
Which conditions ensure that Rauzy fractals are congruent?
Which conditions ensure that a Rauzy fractal is central
symmetric?
What is the centre of symmetry?
Are the conditions necessary?

SymmetricRauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Construction of the
Rauzy fractal

Definitions
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Notations
We denoteA
finite set (alphabet) (here A = {1, 2, . . . , m})∗A
set of finite words over A
εempty wordX̃
mirror-word of X ∈ A∗
|X |y number of occurrences of the letter y ∈ A within theword X ∈ A∗l(X ) “Ablianisation” of X ∈ A∗ , i.e.
l(X ) = (|X |1 , . . . , |X |m ) ∈ Zm

Some linear algebra
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Substitution and induced subspaces
Let σ be a primitive unimodular Pisot substitution over A, i.e.
an endomorphism A∗ −→ A∗ such thatMσ := (|σ(x)|y )1≤x,y ≤m is a primitive matrix; the dominant
real eigenvalue θ > 1 of Mσ is a Pisot unit. Let d + 1 be thealgebraic degree of θ. If d + 1 = m then σ is irreducible. We
defineE u subspace spanned by the right eigenvector associatedwith θ (E u ∼
= R).
s
E
subspace spanned by the right eigenvectors associatedwith the Galois conjugates different from θ (E u ∼=Rd ).
E c subspace spanned by the right eigenvectors associatedwith the remaining eigenvalues (E u ∼= Rm−d−1 ).
πprojection of Rm onto E s (along E s and E c )Induced language and Rauzy fractal
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
Let (xj )j≥1 ∈ AN be a periodic word (that is
σ n (x1 )σ n (x2 )σ n (x3 ) · · · = (xj )j≥1 for some n ≥ 1).
The language Lσ induced by σ is the subset of words overA that appear in (xj )j≥1 , i.e.
Lσ = {X ∈ A∗ : ∃1 ≤ i ≤ j : X = xi · · · xj }.

Induced language and Rauzy fractal
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
Let (xj )j≥1 ∈ AN be a periodic word (that is
σ n (x1 )σ n (x2 )σ n (x3 ) · · · = (xj )j≥1 for some n ≥ 1).
The language Lσ induced by σ is the subset of words overA that appear in (xj )j≥1 , i.e.
Lσ = {X ∈ A∗ : ∃1 ≤ i ≤ j : X = xi · · · xj }.
The Rauzy fractal associated with σ is the compact set
Rσ := {π ◦ l(x1 · · · xn ) : n ∈ N} ⊂ E s .

SymmetricRauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

On congruenceA general result
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

TheoremLet σ, σ 0 be irreducible primitive unimodular Pisot substitutions
over the same alphabet A. If Lσ = Lσ0 then Rσ and Rσ0 arecongruent.

A general result
Symmetric
Rauzy
fractals
Paul Surer
Motivation

TheoremLet σ, σ 0 be irreducible primitive unimodular Pisot substitutions
over the same alphabet A. If Lσ = Lσ0 then Rσ and Rσ0 arecongruent.

Construction
Congruence
Symmetry
Problems

RemarkFor reducible substitutions this does not hold in general. For
example, the substitutions σ1 , σ2 , σ3 over A = {1, 2, 3} induce
the same language, but . . .
Substitution
σ1 : 1 7→ 131, 2 7→ 312, 3 7→ 2
σ2 : 1 7→ 13, 2 7→ 1312, 3 7→ 12
σ3 : 1 7→ 12, 2 7→ 1313, 3 7→ 13

Rauzy fractal
-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

Conjugacy
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
Two substitutions σ, σ 0 over A are conjugated (written σ ∼ σ 0 )if there exists a word X ∈ A∗ such that for each y ∈ A we haveX σ(y ) = σ 0 (y )X (or for each y ∈ A we haveσ(y )X = X σ 0 (y )).

Conjugacy
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction

Definition
Two substitutions σ, σ 0 over A are conjugated (written σ ∼ σ 0 )if there exists a word X ∈ A∗ such that for each y ∈ A we haveX σ(y ) = σ 0 (y )X (or for each y ∈ A we haveσ(y )X = X σ 0 (y )).

Congruence
Symmetry
Problems

Lemma
If two substitutions σ, σ 0 over A are conjugated then Lσ = Lσ0
and Mσ = Mσ0 .

Conjugacy
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction

Definition
Two substitutions σ, σ 0 over A are conjugated (written σ ∼ σ 0 )if there exists a word X ∈ A∗ such that for each y ∈ A we haveX σ(y ) = σ 0 (y )X (or for each y ∈ A we haveσ(y )X = X σ 0 (y )).

Congruence
Symmetry
Problems

Lemma
If two substitutions σ, σ 0 over A are conjugated then Lσ = Lσ0
and Mσ = Mσ0 .
TheoremSuppose that σ ∼ σ 0 such that X σ(y )P= σ 0 (y )X holds for ally ∈ A. Then Rσ0 = Rσ + t with t = n≥0 f n ◦ π ◦ l(X ) ∈ E s ,where f is the restriction of the action of Mσ on E s (especially,f is a contraction).

Example
Symmetric
Rauzy
fractals
Paul Surer

Our initial example
The Rauzy fractals induced by the substitutions

Motivation

σ : 1 7→ 1211, 2 7→ 311, 3 7→ 1 andConstruction

σ 0 : 1 7→ 1112, 2 7→ 113, 3 7→ 1

Congruence
Symmetry
Problems

over the alphabet A = {1, 2, 3} differ by a translation only sinceσ ∼ σ 0 (we have 11σ(y ) = σ 0 (y )11 for all y ∈ A. We can easily
calculate the translation vector t.

SymmetricRauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

On symmetry

A general result
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
The language Lσ induced by a primitive substitution σ is called
mirror-invariant if for each X ∈ Lσ we have X̃ ∈ Lσ .

A general result
Symmetric
Rauzy
fractals
Paul Surer

Definition
The language Lσ induced by a primitive substitution σ is called
mirror-invariant if for each X ∈ Lσ we have X̃ ∈ Lσ .

Motivation
Construction
Congruence
Symmetry
Problems

TheoremLet σ be a primitive unimodular Pisot substitution such that thelanguage Lσ is mirror-invariant. Then the Rauzy fractal Rσ iscentral symmetric (with respect to some centre of symmetry c).

A general result
Symmetric
Rauzy
fractals
Paul Surer

Definition
The language Lσ induced by a primitive substitution σ is called
mirror-invariant if for each X ∈ Lσ we have X̃ ∈ Lσ .

Motivation
Construction
Congruence
Symmetry
Problems

TheoremLet σ be a primitive unimodular Pisot substitution such that thelanguage Lσ is mirror-invariant. Then the Rauzy fractal Rσ iscentral symmetric (with respect to some centre of symmetry c).
ExampleThe (reducible) substitution
σ : 1 7→ 23, 2 7→ 23, 3 7→ 45, 4 7→ 23, 5 7→ 1 overA = {1, 2, 3, 4, 5} induces the (original) Rauzy fractal which iscentral symmetric but Lσ is not mirror-invariant (the words oflength 2 in Lσ are given by {12, 23, 31, 34, 45, 52}).

Necessity
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
A primitive unimodular Pisot substitution σ is said to have thetiling property if Rσ induces a proper lattice tiling with respect
to the lattice{π(z1 , . . . , zm ) : (z1 , . . . , zm ) ∈ Zm , z1 + · · · + zm = 0}.

Necessity
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence

Definition
A primitive unimodular Pisot substitution σ is said to have thetiling property if Rσ induces a proper lattice tiling with respect
to the lattice{π(z1 , . . . , zm ) : (z1 , . . . , zm ) ∈ Zm , z1 + · · · + zm = 0}.

Symmetry
Problems

Conjecture (Pisot conjecture)
Each irreducible primitive unimodular Pisot substitution has thetiling property.

Necessity
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence

Definition
A primitive unimodular Pisot substitution σ is said to have thetiling property if Rσ induces a proper lattice tiling with respect
to the lattice{π(z1 , . . . , zm ) : (z1 , . . . , zm ) ∈ Zm , z1 + · · · + zm = 0}.

Symmetry
Problems

Conjecture (Pisot conjecture)
Each irreducible primitive unimodular Pisot substitution has thetiling property.
TheoremLet σ be a primitive unimodular Pisot substitution with central
symmetric Rauzy fractal Rσ that possesses the tiling property.
Then the language Lσ is mirror-invariant.

Substitutions that are conjugate to their mirror
substitution
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
For a substitution σ we define the mirror-substitution σ̃ byg) for each y ∈ A.
σ̃(y ) := σ(ySubstitutions that are conjugate to their mirror
substitution
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence

Definition
For a substitution σ we define the mirror-substitution σ̃ byg) for each y ∈ A.
σ̃(y ) := σ(ySymmetry
Problems

TheoremLet σ be a primitive unimodular Pisot substitution such thatσ(y )X = X σ̃(y ) holds for all y ∈ A. Then the Rauzy fractal
Rσ is central
symmetric with respect to
1P
c := 2 n≥0 f n ◦ π ◦ l(X ).

Arnoux-Rauzy substitutions
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

DefinitionLet
σ1 :1 7→ 1, 2 7→ 12, 3 7→ 13
σ2 :1 7→ 21, 2 7→ 2, 3 7→ 23
σ3 :1 7→ 31, 2 7→ 32, 3 7→ 3.
Each composition that includes σ1 , σ2 and σ3 at least once is aprimitive, irreducible, unimodular Pisot substitution.

Arnoux-Rauzy substitutions
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

DefinitionLet
σ1 :1 7→ 1, 2 7→ 12, 3 7→ 13
σ2 :1 7→ 21, 2 7→ 2, 3 7→ 23
σ3 :1 7→ 31, 2 7→ 32, 3 7→ 3.
Each composition that includes σ1 , σ2 and σ3 at least once is aprimitive, irreducible, unimodular Pisot substitution.
TheoremIf σ = σi1 ◦ · · · ◦ σin then σ(y )X = X σ̃(y ) for all y ∈ {1, 2, 3}
with
X = σi1 (σi2 (σi3 (· · · (σin−1 (in )in−1 ) · · · )i3 )i2 )i1 .

Example
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

A specific Arnoux-Rauzy substitution
Let σ = σ2 ◦ σ1 ◦ σ2 ◦ σ2 ◦ σ3 . Then for each y ∈ {1, 2, 3} wehave σ(y )X = X σ̃(y ) with
X = σ2 (σ1 (σ2 (σ2 (3)2)2)1)2 = 2122122123212212212.
Therefore,
P Rσ is central symmetric with respect to
c := 12 n≥0 f n ◦ π ◦ l(X ).
0.5

0.0

c
!0.5

!1.0
!1.0

!0.5

0.0

0.5

1.0

SymmetricRauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Problems and open
Questions

The class P-conjecture
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Definition
The language Lσ induced by a primitive substitution σ is called
palindromic if it contains infinitely many palindromes.

The class P-conjecture
Symmetric
Rauzy
fractals
Paul Surer

Definition
The language Lσ induced by a primitive substitution σ is called
palindromic if it contains infinitely many palindromes.

Motivation
Construction
Congruence
Symmetry
Problems

Conjecture (Hof-Knill-Simon: 1995, Labbé: 2014,
Harju-Vesti-Zamboni: 2015)
Let σ be a primitive substitution such that Lσ is palindromic.
Then there exist a primitive substitution σ 0 with σ 0 ∼ σ̃ 0 (theclass P) such that Lσ = Lσ0 .

The class P-conjecture
Symmetric
Rauzy
fractals
Paul Surer

Definition
The language Lσ induced by a primitive substitution σ is called
palindromic if it contains infinitely many palindromes.

Motivation
Construction
Congruence
Symmetry
Problems

Conjecture (Hof-Knill-Simon: 1995, Labbé: 2014,
Harju-Vesti-Zamboni: 2015)
Let σ be a primitive substitution such that Lσ is palindromic.
Then there exist a primitive substitution σ 0 with σ 0 ∼ σ̃ 0 (theclass P) such that Lσ = Lσ0 .
RemarkThe conjecture is solved for the 2-letter case (Tan: 2007) andfor a class of substitutions related with interval exchangetransformations (Masáková-Pelantová-Starosta: 2017).

The class P-conjecture in context with symmetric
Rauzy fractals
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry

The example from above
The substitutions σ1 , σ2 , σ3 over A = {1, 2, 3} induce the samelanguage which is palindromic, but only σ3 is conjugate to its
mirror-substitution..

Problems

Substitution
σ1 : 1 7→ 131, 2 7→ 312, 3 7→ 2
σ2 : 1 7→ 13, 2 7→ 1312, 3 7→ 12
σ3 : 1 7→ 12, 2 7→ 1313, 3 7→ 13

Rauzy fractal
-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

Palindomicity vs. mirror-invariance
Symmetric
Rauzy
fractals
Paul Surer

Proposition

Motivation

A palindomic language is always mirror invariant.

Construction
Congruence
Symmetry
Problems

Palindomicity vs. mirror-invariance
Symmetric
Rauzy
fractals
Paul Surer

Proposition

Motivation

A palindomic language is always mirror invariant.

Construction
Congruence
Symmetry
Problems

QuestionIs there a primitive substitution σ such that Lσ ismirror-invariant but not palindromic?

Palindomicity vs. mirror-invariance
Symmetric
Rauzy
fractals
Paul Surer

Proposition

Motivation

A palindomic language is always mirror invariant.

Construction
Congruence
Symmetry
Problems

QuestionIs there a primitive substitution σ such that Lσ ismirror-invariant but not palindromic?
Partial answer
In the two-letter case palindomicity and mirror-invariance are
equivalent (Tan: 2007).

Thanks
Symmetric
Rauzy
fractals
Paul Surer
Motivation
Construction
Congruence
Symmetry
Problems

Thank you for yourattention