On Pisot Substitutions

Bernd Sing

Department of Mathematics

The Open University

Walton Hall

Milton Keynes, Buckinghamshire

MK7 6AA

UNITED KINGDOM

b.sing@open.ac.uk

Substitution Sequences

Given: a finite alphabet A and a rule σ how to substitute letters togenerate a (two-sided) sequence (denote by n = cardA).

©ex Kol(3, 1)-substitution A = a, b, c, aσ7→ abc, b

σ7→ ab, cσ7→ b

b.aσ7→ σ(b.a) = ab.abc

σ7→ . . .σ7→ . . . 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 =

1 1 01 1 11 0 0

Pisot Substitution Sequences

We use the left Perron-Frobenius eigenvector ` of Sσ to the Perron-Frobenius eigenvalue λ to represent the sequence as a tiling with pro-totiles [0 , `i] (i ∈ A).

©ex Kol(3, 1)-substitution: ` = (λ2 − λ, λ, 1) ≈ (2.7, 2.2, 1)

. . . ab.abca . . . 7→ . . . r rc r rc rce r0 . . .

Tiling lines up with substitution σ:Inflating the tiles by λ, one can re-partition the inflated tiles according tothe substitution rule into the original (proto)tiles. Especially, the tilingthat corresponds to the fixed point of the substitution is self-replicating.

σ is a Pisot substitution if Sσ has exactly one dominant (simple) eigen-value λ > 1 and all other eigenvalues λi satisfy 0 < |λi| < 1 (inside unitcircle).

©ex for Kol(3, 1)-substitution λ ≈ 2.206 λ2,3 ≈ −0.103± i · 0.665

An algebraic integer λ > 1 is a Pisot-Vijayaraghavan number (PV-number, Pisot num-

ber) if all its (other) algebraic conjugates λi satisfy |λi| < 1.

An Iterated Function System

On R, we have defined a self-replicating tiling by intervals of lengths `i

(i ∈ A) by

R =n⋃

i=1

[0 , `i] + Λi.

Here, Λi is given by

Λi =n⋃

j=1

λ Λj + Aij, respectively, Λ = Θ(Λ).

The sets Aij (card Aij = (Sσ)ij) are determined by the substitution.

By construction, the (proto-)tiles Ai = [0 , `i] are given as the componentsof the attractor of the iterated function system (IFS)

A = Θ#(A) (where A#ij = 1

λAji).

The set equation for Λ yields an iterated function system on the productof all local fields of Q(λ) where the (Archimedean or non-Archimedean)absolute value of λ is less than 1.

Kol(3, 1): Ω = Ωa∪Ωb∪Ωc

-0.75 -0.5 -0.25 0 0.25 0.5 0.75Re

-1

-0.75

-0.5

-0.25

0

0.25

0.5

Im

Hausdorff dimension ofboundaries ≈ 1.217

a 7→ aaba, b 7→ aa

Z2

-0.5 0 0.5

Hausdorff dimension of boundariesbounded by 1.167

Model Sets

Cut and project scheme:

Gπ1←− G×H

π2−→ H

∪ ∪ ∪ dense

Lbijective←→ L −→ L?

Model set: Λ = Λ(S) =x ∈ L | x? =

(π2 π−1

1

)(x) ∈ S

Here: G = R, H = Rr−1 × Cs ×Qp1× · · · ×Qpk

,

L = 〈`1, . . . `d〉Z, L is diagonal embedding of L (Minkowski).Star map ? : Q(λ)→ H, x? = (σ2(x), . . . , σr+s(x), x, . . . , x),where the σi’s denote Galois automorphisms

R π1←− R×Hπ2−→ H

dense ∪ ∪ ∪ dense

Lbijective←→ L

bijective←→ L?

•π2ooπ1 ²²F •π2oo

π1²²F

•π2oo

π1

²²F

• π2 //

π1

²²F

G//

HOO

Ω

Is a given (multi-component) point set Λ a model set?(in that case, its dynamical/diffractive spectrum is pure point)

An Aperiodic Tiling of H

On R: Λ = Θ(Λ)®

­

©

ªA = Θ#(A)

On H:¨§

¥¦Ω = Θ?(Ω) Υ = Θ#?(Υ )

•¨§

¥¦IFS à unique non-empty compact solution

• Expansive MFS Ã Λ is fixed by the given substitution,

for Υ a possible solution is Υi = Λ([0 , `i[).

Λ is a model set iff Υ + Ω is a tiling of H.[Ito-Rao, Barge-Kwapisz, etc.]

A Periodic Tiling of H

LetM = 〈`2 − `1, . . . , `n − `1〉Z, thenM? is lattice in H.

Λ is a model set iff Ω +M? is a tiling of H.[Rauzy, etc.]

The Bigger Picture (for Subsitutions in General)

Geometry Combinatorics

Υ + Ω is tiling of H⋃

i(−Ai)×Ωi is FD of L

“overlaps are coincidences”

regular model setΛ admits algebr. coinc.

Λ + A admits overlap coinc.

“overlap density tends to 1”

torus parametrisation autocorr. hull compact

dynam. spectrum is pp(cont. EF, sep. almost all points)

ε-almost periods dense

autocorr. meas. is almost periodic

diffraction measure is pp

Dynamical System Diffraction

KS

[Ito-Rao, S.]

®¶ks

[Lee, S.]+3

KS[Lee]

®¶

KS

[Host, Queffelec,Solomyak,Lee-Moody--Solomyak]

®¶

KS

[Baake-Moody]®¶

+3

[Moody--Strungaru]ks

-5

mu[Host, Queffelec, Solomyak]

.6

[Lee-Moody-Solomyak,Baake-Lenz, Gouere]

go

[Dworkin],,

%-[Baake-Moody-Lenz]

em RRRRRRRRRRRRRR

RRRRRRRRRRRRRR

ª´

[Baake-Moody--Lenz]

MU

[Hof, Schlottmann]

´´

intrinsically define CPStt t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4 t4

ddd$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$d$

d$

gg g' g' g' g' g' g' g' g'

Ammann-Beenker and Its Dual Partner

Rhombic Penrose and Its Dual Partner

Note: Internal space H = C× Z/5Z

Conch and Nautilius

Note: inflation-factors are not PV-numbers(≈ −0.727− i 0.934 [dominant root of x4 − x + 1 = 0] and≈ 1.019− i 0.603 [dominant root of x4 − x3 + 1 = 0])

Picture removed becauseof size considerations(fractalized version ofpolygonal tiling to theleft)

Picture removed becauseof size considerations(dual partner of abovetiling)

The Conch & Nautilus Tiling were discovered by P. Arnoux, M. Furukado, S. Ito and E.O.

Harriss. Also see the “Tilings Encyclopedia” at http://tilings.math.uni-bielefeld.de/.

Why Watanabe-Ito-Soma Is A Model Set

Internal space H = C×Q2(ξ8) (uniformizer 1 + ξ8)

Dual tiling at ‖x− .001‖Q2(ξ8)≤ 1

8

Picture removed becauseof size considerations(dual partner of abovetiling)

Picture removed becauseof size considerations(variant of picture to theleft)

Lattice Substitution Systems etc.

©ex “Chair Tiling”:

-

¡¡µ

@@R

@@I

¡¡µ @@I

@@R ¡¡ª

= p = q

= s = r

Picture removed because ofsize considerations (window forchair tiling)

The “aperiodic tiling condition” also works for reducible Pisot substitu-tions (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 cir-cle and are nonzero) and therefore in particular also for β-substitutions.

Let β be a PV-number, and let

1 = a1β0 + a1β

−1 + a2β−2 + . . . = a1a2 . . . aqaq+1 . . . aq+p

be the (greedy) expansion of 1 in powers of β. Then, the (possibly reducible)Pisot substitution

1 7→ 1a122 7→ 1a23

...(q + p− 1) 7→ 1aq+p−1(q + p)(q + p) 7→ 1aq+p(q + 1)

is the corresponding β-substitution.

The Bigger Picture (for Pisot)

⋃i(−Ai)×Ωi is FD of L

Υ + Ω is tiling of H

“overlaps are coincidences”

regular model set

Λ admits algebr. coinc.

“overlap density tends to 1”

Λ + A admits overlap coinc.

KS

®¶

KS

®¶

(W)-condition (“weak finiteness”)

mu

β-substitutions[Hollander, Akiyama, etc.] -5cccccccccccccccccccccccccccccccc

cccccccccccccccccccccccccccccccc

modular coincidence+3

lattice substitution systems[Lee-Moody, Lee-Moody-Solomyak]

ks

geometric/super coinc. cond.

iq[Ito-Rao, Barge-Kwapisz]

)1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

“overlap automaton/graph”

go

[Siegel, S.]

'/WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

“stepped surface & polygons”

em

[Ito et al., etc.]

%-SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

M? + Ω is periodic tilingks[Rauzy, Siegel, etc.]

+3

make use of rationalindependence of the `i’s

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.)