On Pisot Substitutions

Transcription

1 On Pisot Substitutions Bernd Sing Department of Mathematics The Open University Walton Hall Milton Keynes, Buckinghamshire MK7 6AA UNITED KINGDOM

2 Substitution Sequences Given: a finite alphabet A and a rule σ how to substitute letters to generate a (two-sided) sequence (denote by n = card A). ex Kol(3, 1)-substitution A = {a, b, c}, a σ abc, b σ ab, c σ b b.a σ σ(b.a) = ab.abc σ σ cabbabcab.abcabbabc... Define (n n)-substitution matrix Sσ where (Sσ) ij = #i s in σ(j) = # i (σ(j)). ex for Kol(3, 1)-substitution Sσ =

3 Pisot Substitution Sequences We use the left Perron-Frobenius eigenvector l of Sσ to the Perron- Frobenius eigenvalue λ to represent the sequence as a tiling with prototiles [0, l i ] (i A). ex Kol(3, 1)-substitution: l = (λ 2 λ, λ, 1) (2.7, 2.2, 1)... ab.abca Tiling lines up with substitution σ: Inflating the tiles by λ, one can re-partition the inflated tiles according to the substitution rule into the original (proto)tiles. Especially, the tiling that corresponds to the fixed point of the substitution is self-replicating. σ is a Pisot substitution if Sσ has exactly one dominant (simple) eigenvalue λ > 1 and all other eigenvalues λ i satisfy 0 < λ i < 1 (inside unit circle). ex for Kol(3, 1)-substitution λ λ 2, ± i An algebraic integer λ > 1 is a Pisot-Vijayaraghavan number (PV-number, Pisot number) if all its (other) algebraic conjugates λ i satisfy λ i < 1.

4 An Iterated Function System On R, we have defined a self-replicating tiling by intervals of lengths l i (i A) by n R = [0, l i ] + Λ i. Here, Λ i is given by n Λ i = λ Λ j + A ij, respectively, Λ = Θ(Λ). j=1 i=1 The sets A ij (card A ij = (Sσ) ij ) are determined by the substitution. By construction, the (proto-)tiles A i = [0, l i ] are given as the components of the attractor of the iterated function system (IFS) A = Θ # (A) (where A # ij = 1 λ A ji ). The set equation for Λ yields an iterated function system on the product of all local fields of Q(λ) where the (Archimedean or non-archimedean) absolute value of λ is less than 1. Kol(3, 1): Ω = Ω a Ω b Ω c a aaba, b aa Im Z Hausdorff dimension of boundaries Re Hausdorff dimension of boundaries bounded by 1.167

5 Model Sets Cut and project scheme: G π 1 G H π 2 H dense L bijective L L Model set: Λ = Λ(S) = { x L x = ( ) } π 2 π1 1 (x) S Here: G = R, H = R r 1 C s Q p1 Q pk, L = l 1,... l d Z, L is diagonal embedding of L (Minkowski). Star map : Q(λ) H, x = (σ 2 (x),..., σ r+s (x), x,..., x), where the σ i s denote Galois automorphisms R π 1 π R H 2 H dense dense L bijective L bijective L H π π 2 2 π 2 π 2 π Ω 1 π1 π π 1 1 G Is a given (multi-component) point set Λ a model set? (in that case, its dynamical/diffractive spectrum is pure point)

6 An Aperiodic Tiling of H On R: Λ = Θ(Λ) A = Θ # (A) On H: Ω = Θ (Ω) Υ = Θ # (Υ ) IFS unique non-empty compact solution Expansive MFS Λ is fixed by the given substitution, for Υ a possible solution is Υ i = Λ([0, l i [). Λ is a model set iff Υ + Ω is a tiling of H. [Ito-Rao, Barge-Kwapisz, etc.]

7 A Periodic Tiling of H Let M = l 2 l 1,..., l n l 1 Z, then M is lattice in H. Λ is a model set iff Ω + M is a tiling of H. [Rauzy, etc.]

8 The Bigger Picture (for Subsitutions in General) Geometry Combinatorics Υ + Ω is tiling of H i ( A i ) Ω i is FD of L overlaps are coincidences [Ito-Rao, S.] Λ admits algebr. coinc. [Lee, S.] regular model set [Lee] Λ + A admits overlap coinc. intrinsically define CPS overlap density tends to 1 [Baake-Moody- [Hof, Schlottmann] -Lenz] [Host, Queffelec, Solomyak, Lee-Moody- -Solomyak] [Moody- -Strungaru] torus parametrisation autocorr. hull compact ε-almost periods dense [Host, Queffelec, Solomyak] [Baake-Moody-Lenz] [Baake-Moody] [Dworkin] dynam. spectrum is pp autocorr. meas. is almost periodic (cont. EF, sep. almost all points) diffraction measure is pp [Lee-Moody-Solomyak, Baake-Lenz, Gouéré] Dynamical System Diffraction

9 Ammann-Beenker and Its Dual Partner

10 Rhombic Penrose and Its Dual Partner Note: Internal space H = C Z/5Z

11 Conch and Nautilius Note: inflation-factors are not PV-numbers ( i [dominant root of x 4 x + 1 = 0] and i [dominant root of x 4 x = 0]) Picture removed because of size considerations (fractalized version of polygonal tiling to the left) Picture removed because of size considerations (dual partner of above tiling) The Conch & Nautilus Tiling were discovered by P. Arnoux, M. Furukado, S. Ito and E.O. Harriss. Also see the Tilings Encyclopedia at

12 Why Watanabe-Ito-Soma Is A Model Set Internal space H = C Q 2 (ξ 8 ) (uniformizer 1 + ξ 8 ) Dual tiling at x.001 Q2 (ξ 8 ) 1 8 Picture removed because of size considerations (dual partner of above tiling) Picture removed because of size considerations (variant of picture to the left)

13 Lattice Substitution Systems etc. ex Chair Tiling : = p = q = s = r Picture removed because of size considerations (window for chair tiling) The aperiodic tiling condition also works for reducible Pisot substitutions (i.e., where the dominant eigenvalue of the substitution matrix S σ is a PV-number but not necessarily all eigenvalues lie inside the unit circle and are nonzero) and therefore in particular also for β-substitutions. Let β be a PV-number, and let 1 = a 1 β 0 + a 1 β 1 + a 2 β = a 1 a 2... a q a q+1... a q+p be the (greedy) expansion of 1 in powers of β. Then, the (possibly reducible)pisot substitution is the corresponding β-substitution. 1 1 a a 2 3. (q + p 1) 1 a q+p 1(q + p) (q + p) 1 a q+p(q + 1)

14 The Bigger Picture (for Pisot) β-substitutions Υ + Ω is tiling of H [Hollander, Akiyama, etc.] (W)-condition ( weak finiteness ) i ( A i ) Ω i is FD of L [Ito-Rao, Barge-Kwapisz] overlaps are coincidences [Siegel, S.] geometric/super coinc. cond. [Ito et al., etc.] overlap automaton/graph stepped surface & polygons regular model set [Rauzy, Siegel, etc.] M + Ω is periodic tiling make use of rational independence of the l i s Λ + A admits overlap coinc. Λ admits algebr. coinc. overlap density tends to 1 lattice substitution systems [Lee-Moody, Lee-Moody-Solomyak] modular coincidence Pictures removed because of size considerations (iteration of polygons for Kol(3, 1)) Pictures removed because of size considerations (iteration of polygons for nonunimodular ex.)