Substitutions, Rauzy fractals and Tilings
Anne Siegel
CANT, 2009
Reminder....
Pisot fractals: projection of the stair of a Pisot substitution
Self-replicating substitution multiple tiling: replace faces of a
discrete plane by subtiles
Question: Find an efficient condition to ensure tiling?
Tiling condition
Rauzy fractal= fixed point of a set equation.
T (1) = hT (1) ∪ hT (2) ∪ hT (3)
T (2) = hT (1) + πP(1)
T (3) = hT (2) + πP(1)
[0, 1∗ ] 7→ [0, 1∗ ] ∪ [0, 2∗ ] ∪ [0, 3∗ ]
[0, 2∗ ] 7→ [h−1 πP(1), 1∗ ]
[0, 3∗ ] 7→ [h−1 πP(1), 2∗ ]
→
→
→
Tiling condition ?
Theorem
The multiple tiling is a tiling iff any pair of faces in the discrete
plane appears in the image of the same face under the 2D
substitution, up to a translation.
∗
∀[γ1 , i1∗ ][γ2 , i2∗ ] ∈ Γsr , ∃k, δ | δ + [γ1 , i1∗ ][γ2 , i2∗ ] ⊂ ẼN
1 [0, k ]
Checking the property
reduce the condition to an acceptance window (Corridor)
trace pre-images by graphs.
Problems : computations...
Two dimensional substitutions are UGLY to
manipulate
Solution : coming back to words and lines...
Create intervals fixed by a set equation from a symbolic sequence
Tiles: Projections of images of unit vectors.
Order of tiles: follow the periodic point of the substitution
In other words: take the periodic point of the substitution and
replace letters by intervals with appropriate length
δ(i) = hvβ , ei i (extend to finite words)
Self-similar equation:
βS(i) =
[
σ(i)=pjs
S(j) + δ(p).
What is the place of a tile ?
Expanding tiling = projection of a stair
Tile in the tiling
Tile S = projection of x + [0, 1]ei along the expanding line
Analogy: Tile S ' Segment x + [0, 1]ei
Place of the subdivision of S
βS in the tiling ?
S = S(i) + δ(w ) =⇒ βS = σ(i)=pjs S(j) + βδ(w ) + +δ(p)
Formal equation for vectors: δ(w ) is the projection of a vector
x on the expanding line.
S
E1 [x, i] = σ(i)=pjs [Mx + P(p), j]
Game...
E1 [x, i] =
S
σ(i)=pjs [Mx
+ P(p), j]
σ(1) = 112, σ(2) = 13, σ(3) = 1
Question 1 Draw E1 on the basic segments ?
Question 2 What is the dual map of E1 ?
Super coincidence
Action on finite stairs S
E1 [x, i] = σ(i)=pjs [Mx + P(p), j]
2D substitution onS
faces of the discrete plane
E˜1 [πx, i ∗ ] = σ(j)=pis [πM−1 (x + πP(p)), j ∗ ]
Duality relation: the 2D substitution is very near from the
dual of the action on finite stairs
[π(y), j ∗ ] ∈ Ẽ1 [πx, i ∗ ] iff [−x, i] ∈ E1 [−y, j]
Question: what the the dual-equivalent of the tiling condition.
Tiling condition For every pair of faces [πx, i ∗ ][πy, j ∗ ] in the
discrete plane, there exists a translation vector z and a face [0, k ∗ ]
such that
∗
[πx, i ∗ ][πy, j ∗ ] + πz ∈ ẼN
1 [0, k ]
Super coincidence (Ito&Rao’06, Barge&Kwaplisz’06)
[π(y), j ∗ ] ∈ E1∗ [πx, i ∗ ] iff [−x, i] ∈ E1 [−y, j]
Tiling condition For every pair of faces [πx, i ∗ ][πy, j ∗ ] in the
discrete plane, there exists a translation vector z and a face [0, k ∗ ]
such that
∗
[πx, i ∗ ][πy, j ∗ ] + πz ∈ ẼN
1 [0, k ]
Stair equivalent For every pair of segment [x, i],[y, j] with common
point along the expanding direction there exists k such that
E1 N [x, i] and En1 [y, j] share a common segment.
Exercice: which picture does this property give?
Projection on the expanding line ?
Place [x, i] in the stair of the periodic stair.
Translate a copy of the stair so that it contains [y, j].
unify Project both stairs on the expanding line. : Two copies
of the expanding tiling E : E − γ1 and E − γ2
Focus on intersections of tiles in E and E − γ
Tiling condition ?
πe [x, i] = S1 and πe [y, j] = S2 appears as tiles in the
translations of the expanding tiling.
S1 ∈ E
S2 ∈ E − γ
S1 ∩ S2 6= ∅
Tiling condition Expanding every pair S1 ∩ S2 always provides
a synchronization on a full tile when expanding the tiling.
β N S1 ∩ S2 ⊃ a full tile.
The problem:
Shifting process Both tilings are not synchronized on at least one
tile.
The distance between the tiling does not correspond to a prefix of
the periodic point.
Towards a single point condition
Density of E − γ: density of coincidence between E and E − γ
Theorem (Solomyak)
The super coincidence condition is satisfied iff for every γ obtained
as a difference between two prefixes of the periodic point,
lim Density(E − β n γ) = 1.
n→∞
Proof
Main lemma. The number of type of of overlaps between the
tilings (E,E − β n γ is bounded for a fixed γ.
Spread overlaps uniformly.
Corrolary Tiling iff the super-coincidence condition is satisfied for a
prefix p of the periodic point and σ k (p).
How to check the synchronisation ?
Baring Put bars as soon as there are synchronizations.
Balanced pairs Part of the tilings that correspond to the same
letters of the alphabet up to the order of letter
Apply the substitution and put new bars
Super coincidence between overlaps in E − γ and E iff
the number of bars is finite
the decomposition of every balanced pairs leads to a balanced
letter.
Balanced pairs algorithm
Fix a prefix p of the fix point.
Put bars between u and u shifted by p.
Decompose the images of all balanced words
Iterate the process until the set of balanced words is stabilized.
(w , w̃ ) → (σ(w ), σ(w̃ ) = (w1 w2 . . . wn , w˜1 w˜2 . . . w˜n ))
Exercise. Apply to σ(1) = 112, σ(2) = 13 σ(3) = 1 ?
The best condition for tilings
Theorem
The tiling condition is satisfied iff the balanced pairs algorithm
terminates with a coincidence for at least one w prefix of the
periodic point.
Proof: uniform recurrence + repetitivity + GIFS equation
References mix of Barge&Kwaplicz’06 and Solomyak’02.
Easy to implement
Semi-effective
What we really do: explicitely follow edges in a subgraph of
the pairs ancestor graph in which edges have nice
interpretations.
Examples of applications
Prove that a fractal has a non trivial fundamental group
Is there a hole inside the fractal ?
Condition to have a hole ?
Having a hole ? A connected component of the complement is
bounded.
Condition to have a bounded connected component?
Find three “suitable” sets
that intersect simultaneously at least twice
That have one triple point in the interior of the union
Then a part of one set is surrounded by the two others.
Non trivial fundamental group ?
Lemma [Luo&Thuswaldner] Let B0 , B1 , B2 ⊂ R2 be locally
connected continuum such that
(i) Interiors are disjoints int(Bi ) ∩ int(Bj ) = ∅, i 6= j.
(ii) Each Bi is the closure of its interior (0 ≤ i ≤ 2).
(iii) R2 \ int(Bi ) is locally connected (0 ≤ i ≤ 2).
(iv) There exist x1 , x2 ∈ B0 ∩ B1 ∩ B2 with x1 ∈ int(B0 ∪ B1 ∪ B2 ).
Then there exists i ∈ {0, 1, 2} such that Bi ∪ Bi+1 has a bounded
connected component U with U ∩ int(Bi+2 ) 6= ∅.
How to use the lemma
Find an intersection between three tiles that is an inner point
of the union.
How ? Finite number of quadruple points but infinite number of triple points
A part of the third tile is inside a hole. Ensure that is it
outside from T (i).
How ? Look at positions of tiles
Example
Finite number of quadruple points;
Finite inner triple points ? A node in the triple points
graph issues in an infinite number of walks.
Example
Finite number of quadruple points;
Finite inner triple points ? A node in the triple points
graph issues in an infinite number of walks.
Consider the node [2, 0, 3, π(1, 0, −1), 1]
corresponds to the intersection T (2) ∩ T (3) ∩ (π(1, 0, −1) + T (1))
Example
Finite number of quadruple points;
Finite inner triple points ? A node in the triple points
graph issues in an infinite number of walks.
Find some configurations of tiles outside from the
iterations of the 2D substitution
Consider the node [2, 0, 3, π(1, 0, −1), 1]
corresponds to the intersection T (2) ∩ T (3) ∩ (π(1, 0, −1) + T (1))
E1 (σ)4 [0, 2]
Pattern
[0, 2][0, 3]
[π(1, 0, −1), 1]
Non trivial fundamental group?
Theorem (S.&Thuswaldner’09)
Assume that d = 3. The fundamental group of each T (i) is
non-trivial as soon as
The tiling property is satisfied;
All T (i)’s are connected;
There are a finite number of quadruple points;
There exists a triple point node [i, i1 , γ1 , T (i2 ) + γ2 ] leading
away an infinity of walks.
There exists three translations vectors such that the three
patterns ([v, i], [γ1 + v, i1 ], [γ2 + v, i2 ]),
0
([v , i], [γ1 + v0 , i1 ], [γ2 + v0 , i2 ]) et
00
([v , i], [γ1 + v00 , i1 ], [γ2 + v00 , i2 ]) lie at the boundary of a
finite inflation E1 (σ)K [0, i].
With additional properties, the fundamental group is not free and
uncountable.
Fractal and beta numeration ?
γ(β) = Infimum of p/q ∈ Q ∩ R+ with a non purely periodic beta-expansion ?
Quadratic case [Schmidt] γ(β) equals 0 or 1 (depending on
the finiteness property )
Cubic case [Akiyama]
γ(smallest Pisot number) = 0, 6666666182
Theorem (Adamczewski,Frougny,S.,Steiner)
β a cubic unit.
γ(β) = 0 iff the finiteness property is not satisfied.
If β as a complex conjugate and (F) is satisfied, then
γ(β) ∈ Q.
Proof
γ(β) lies at the intersection of the Rauzy
fractal and the horizontal line.
The tiling condition is satisfied.
The boundary contains spirals.
Other application: cristals
Manganese have good conductivity properties
Positions of atoms have symmetries (cristals ?) but not
authorized symmetries : quasi-cristals
Theoretical physics What properties should the position of
atoms satisfy ?
repetitivity
Meyer set: cut-and-project scheme
First example Penrose tiling, sturmian sequences
Other class of examples Rauzy fractal that satisfy the tiling
condition
Summary
Combinatorics rule of replacement
→ stairs
→ fractals
→ discrete plane
→ stairs
→ combinatorial condition
Conditions for tiling
Conditions on iterations of two dimensional substitutions
Ugly graphs
Measures of boundaries
Playing with stairs (super coincidence condition)
Combinatorial condition : balanced pairs algorithms
To be continued
Larger family of numeration systems: SRS
(including canonical number system and beta numeration, with
non unit cases)
Fractal shape but no GIFS: fixed point of an infinite tree
(Bratelli diagram)
Theorem[Berthe,S.,Steiner,Surer,Thuswaldner’09]: Finiteness
property implies that the interiors are disjoint.
Main problem: No information on the measure of the
boundary.
Morality
If a mathematical object is contained
|
{zinto itself},
self-replicating
it can be coded by a substitution.
Geometry allows to recover some properties.
Quasi-cristals.
Algebraic Pisot number and beta-numeration : recover finite
expansions.
Plane with an algebraic normal vector.
Tiling spaces : geometric invariants.
Self-induced map.