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 } ∗ 2 1 . , L σ = 1 �→ 121 1 1 2 �→ 12 Assumption 1) Unimodular: det L σ = ± 1, 2) Pisot: the biggest eigenvalue λ is bigger than 1 and the others have modulus less than 1. 3) Primitive: ∃ N > 0 such that L N σ > 0. 2
An atomic surface (in the case of rank 2) ω = s 0 s 1 · · · : an fixed point of σ (i.e. ω = σ ( ω )) f : { 1 , 2 } ∗ → Z 2 : an abelianzation map P e ( P c ): the expanding (contractive) eigenspace of L σ π : R 2 → P c : the projection along P e Atomic surfaces: X i := { πf ( s 0 · · · s k − 1 ) | s k = i } ( i = 1 , 2) , X = X 1 ∪ X 2 1 → 121 σ : ω = 12112121 · · · 2 → 12 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 X 1 , X 2 , X = X 1 ∪ X 2 are interval ⇔ σ is invertible. Theorem ([Wen-Wen]) Every invertible substitution over { 1 , 2 } is generated by 1 → 2 1 → 12 1 → 21 α : β : δ : 2 → 1 2 → 1 2 → 1 4
An atomic surface (in the case of rank 3) Example (Rauzy substitution) 8 0 1 1 → 12 1 1 1 > > < B C σ = L σ = 2 → 13 , 1 0 0 B C @ A > > 3 → 1 0 1 0 : 5
Tiling substitution [Ito-Arnoux][Sano-Arnoux-Ito] L -1 x e 3 σ * ( ) E σ * 3 1 * 1 x * 1 * 2 e 2 * 2 * 1 x L -1 x σ e 1 x * 3 * 2 L -1 x σ 6
Atomic surfaces generated by a tiling substitution E * ( ) E * ( ) E * ( ) σ σ σ ... 1 * 1 1 1 It is known that the atomic surface are given by X i = lim n →∞ L n σ πE ∗ n ( i ∗ ) ( i = 1 , 2 , 3) . 1 7
Examples 8 1 → 1213211 > > Invertible and disconnected < σ = 2 → 121321 > > 3 → 1132 : 1 ( σ )(1 ∗ ) 1 ( σ )(2 ∗ ) 1 ( σ )(3 ∗ ) E ∗ E ∗ E ∗ 8
Examples 8 1 → 1213121 > > Not invertible and connected < σ = 2 → 123112 > > 3 → 1213 : 1 ( σ )(1 ∗ ) 1 ( σ )(2 ∗ ) 1 ( σ )(3 ∗ ) E ∗ E ∗ E ∗ 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. erie BERTH´ [Berthe-Rao-Ito-E] Val´ 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
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