Connectedness of atomic surfaces generated by invertible - - PowerPoint PPT Presentation

Connectedness of atomic surfaces generated by invertible substitutions

Hiromi EI (Chuo University) ei@ise.chuo-u.ac.jp 30/7/2008 - 1/8 @ London

1

A substitution σ

A substitution σ over {1, 2, · · · , d} is an endomorphism on the free monoid {1, 2, · · · , d}∗. Example A substitution σ of rank 2 An incidnce matrix of σ σ : {1, 2}∗ → {1, 2}∗ 1 → 121 2 → 12 , Lσ =   2 1 1 1   . Assumption 1) Unimodular: det Lσ = ±1, 2) Pisot: the biggest eigenvalue λ is bigger than 1 and the

• thers have modulus less than 1.

3) Primitive: ∃N > 0 such that LN

σ > 0. 2

An atomic surface (in the case of rank 2)

ω = s0s1 · · · : an fixed point of σ (i.e. ω = σ(ω)) f : {1, 2}∗ → Z2 : an abelianzation map Pe (Pc): the expanding (contractive) eigenspace of Lσ π : R2 → Pc : the projection along Pe Atomic surfaces: Xi := {πf(s0 · · · sk−1) | sk = i}(i = 1, 2), X = X1 ∪ X2

σ :    1 → 121 2 → 12 ω = 12112121 · · ·

3

Connectedness of atomic surfaces

Theorem ([Berth´ e-Rao-Ito-E][Ito-E][Lamb]) σ is a primitive unimoduler substitution over {1, 2}. The atomic surfaces X1, X2, X = X1 ∪ X2 are interval ⇔ σ is invertible. Theorem ([Wen-Wen]) Every invertible substitution over {1, 2} is generated by α :    1 → 2 2 → 1 β :    1 → 12 2 → 1 δ :    1 → 21 2 → 1

4

An atomic surface (in the case of rank 3)

Example (Rauzy substitution)

σ = 8 > > < > > : 1 → 12 2 → 13 3 → 1 , Lσ = B B @ 1 1 1 1 1 1 C C A

5

Tiling substitution [Ito-Arnoux][Sano-Arnoux-Ito]

2 3 1 2 3 1 1 2

1

E

2

e e 1

3

e x x

x

L -1 x L -1 x L -1 x

* * * * * * * * * ( ) σ σ σ σ

6

Atomic surfaces generated by a tiling substitution

...

1

1

E* ( )

σ

1

E* ( )

σ

1

E* ( )

σ *

It is known that the atomic surface are given by Xi = limn→∞ Ln

σπE∗ n 1

(i∗) (i = 1, 2, 3).

7

Examples

Invertible and disconnected

σ = 8 > > < > > : 1 → 1213211 2 → 121321 3 → 1132

E∗

1(σ)(1∗)

E∗

1(σ)(2∗)

E∗

1(σ)(3∗)

8

Examples

Not invertible and connected

σ = 8 > > < > > : 1 → 1213121 2 → 123112 3 → 1213

E∗

1(σ)(1∗)

E∗

1(σ)(2∗)

E∗

1(σ)(3∗)

9

Question Is there good property (instead of invertibility) which determines the connectedness of atomic surfaces of rank 3?

10

Reference

[Arnoux-Ito] Pierre ARNOUX and Shunji ITO, Pisot substitutions and Rauzy fractals, Bull.

• Belg. Math. Soc. 8 (2001), 181-207.

[Berthe-Rao-Ito-E] Val´ erie BERTH´ E, Hiromi EI, Shunji ITO, and Hui RAO, On Substitution invariant sturmian words : an application of Rauzy fractals, Theoret. Informatics Appl., 41, no. 3 (2007), 329-349. [Ito-E] Hiromi EI and Shunji ITO, Decomposition theorem on invertible substitutions, OSAKA

• J. Math., 35 (1998), 821-834.

[Lamb] J S W Lamb, On the canonical projection method for one-dimensional quasicrystals and invertible substitution rules, J. Phys., A 31, no. 18 (1998), L331–L336. MR 99d:82075. [Sano-Arnoux-Ito] Yuki SANO, Pierre ARNOUX and Shunji ITO: Higher dimensional extensions of substitutions and their dual maps, Journal D’analyse Math´ ematique 83(2001), 183-206.

11

[Wen-Wen] Zhi-Xiong WEN and Zhi-Ying WEN, Local isomorphisms of invertible substitutions, C.R.Acad.Sci.Paris, t.318, S´ erie I (1994), 299-304.

12