Substitutions, Rauzy fractals and Tilings

Transcription

1 Substitutions, Rauzy fractals and Tilings Anne Siegel CANT, 2009

2 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?

3 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 ] [0, 1 ] [0, 2 ] [0, 3 ] [0, 2 ] [h 1 πp(1), 1 ] [0, 3 ] [h 1 πp(1), 2 ]

4 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. [γ 1, i 1 ][γ 2, i 2 ] Γ sr, k, δ δ + [γ 1, i 1 ][γ 2, i 2 ] 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

5 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) = v β, e i (extend to finite words) Self-similar equation: βs(i) = σ(i)=pjs S(j) + δ(p).

6 What is the place of a tile? Expanding tiling = projection of a stair Tile in the tiling Tile S = projection of x + [0, 1]e i along the expanding line Analogy: Tile S Segment x + [0, 1]e i Place of the subdivision of 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. E 1 [x, i] = σ(i)=pjs [Mx + P(p), j]

7 Game... E 1 [x, i] = σ(i)=pjs [Mx + P(p), j] σ(1) = 112, σ(2) = 13, σ(3) = 1 Question 1 Draw E 1 on the basic segments? Question 2 What is the dual map of E 1?

8 Super coincidence Action on finite stairs E 1 [x, i] = σ(i)=pjs [Mx + P(p), j] 2D substitution on faces of the discrete plane Ẽ 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] E 1 [ 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 ]

9 Super coincidence (Ito&Rao 06, Barge&Kwaplisz 06) [π(y), j ] E 1 [πx, i ] iff [ x, i] E 1 [ 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 E N 1 [x, i] and E n 1 [y, j] share a common segment. Exercice: which picture does this property give?

10 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 γ 1 and E γ 2 Focus on intersections of tiles in E and E γ

11 Tiling condition? π e [x, i] = S 1 and π e [y, j] = S 2 appears as tiles in the translations of the expanding tiling. S 1 E S 2 E γ S 1 S 2 Tiling condition Expanding every pair S 1 S 2 always provides a synchronization on a full tile when expanding the tiling. β N S 1 S 2 a full tile.

12 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.

13 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 n βn γ) = 1. 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).

14 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.

15 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, w) (σ(w), σ( w) = (w 1 w 2... w n, w 1 w 2... w n )) Exercise. Apply to σ(1) = 112, σ(2) = 13 σ(3) = 1?

16 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.

17 Examples of applications

18 Prove that a fractal has a non trivial fundamental group Is there a hole inside the fractal?

19 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.

20 Non trivial fundamental group? Lemma [Luo&Thuswaldner] Let B 0, B 1, B 2 R 2 be locally connected continuum such that (i) Interiors are disjoints int(b i ) int(b j ) =, i j. (ii) Each B i is the closure of its interior (0 i 2). (iii) R 2 \ int(b i ) is locally connected (0 i 2). (iv) There exist x 1, x 2 B 0 B 1 B 2 with x 1 int(b 0 B 1 B 2 ). Then there exists i {0, 1, 2} such that B i B i+1 has a bounded connected component U with U int(b i+2 ).

21 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

22 Example Finite number of quadruple points; Finite inner triple points? A node in the triple points graph issues in an infinite number of walks.

23 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))

24 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)) E 1 (σ) 4 [0, 2] Pattern [0, 2][0, 3] [π(1, 0, 1), 1]

25 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, i 1, γ 1, T (i 2 ) + γ 2 ] leading away an infinity of walks. There exists three translations vectors such that the three patterns ([v, i], [γ 1 + v, i 1 ], [γ 2 + v, i 2 ]), ([v, i], [γ 1 + v, i 1 ], [γ 2 + v, i 2 ]) et ([v, i], [γ 1 + v, i 1 ], [γ 2 + v, i 2 ]) lie at the boundary of a finite inflation E 1 (σ) K [0, i]. With additional properties, the fundamental group is not free and uncountable.

26 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, 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.

27 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

28 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

29 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.

30 Morality If a mathematical object is contained }{{ into 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.