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