Shigeki Akiyama (Niigata University, Japan) 17 …...The ﬁx point x of ˙ naturally gives rise to...

Pisot conjecture session

Shigeki Akiyama (Niigata University, Japan)

17 February 2010

at MathInfo (at CIRM)

– Typeset by FoilTEX –

Pisot conjecture

Let A+ = 0, 1, . . . , k−1+ be the semi-group generated by

concatenation over k letters. Substitution is a homomorphism

A+ → A+. σ also acts on AN by a1a2 . . . 7→ σ(a1)σ(a2) . . . .

Introduce a natural metric to make AN compact. Assume

σ(0) = 0 . . . and σn(0) becomes infinite. Then we have a fix

point σ(x) = x starting by 0. For e.g.,

σ(0) = 01, σ(1) = 12, σ(2) = 220

we have x = 0112122201222022022001 . . . . Let s be the shift

acting on AN: s(a1a2 . . . ) = a2a3 . . . . We have a topological

dynamics Xσ = sn(x) | n = 0, 1, . . . where s acts.

– Typeset by FoilTEX – 1

The incidence matrix Mσ is the k × k matrix whose (i, j)

entry is the number of i occurs in σ(j): for e.g.,

Mσ =

1 0 1

1 1 0

0 1 2

.

Mσ is clearly non-negative. If this is primitive, (Xσ, s) is

minimal and uniquely ergodic. Therefore we can discuss the

spectral property of isometry Us : f → f s on L2(Xσ, µ). A

Pisot number is a real algebraic integer > 1 that all other

conjugates are less than one in modulus.


Theorem 1 (Bombieri-Taylor[1], Solomyak[8]). (Xσ, s) is not

weakly mixing then the Perron Frobenius root of Mσ must be

a Pisot number.

We say σ is a Pisot substitution if the Perron-Frobenius root

of Mσ is a Pisot number,

Conjecture 1 (Pisot Conjecture). σ is a Pisot substitution

and the characteristic polynomial of Mσ is irreducible then

(Xσ, s) has purely discrete spectrum.


Two well known methods

Geometric realization of the substitution:

Rauzy, Arnoux-Ito, . . .

Harmonic Analysis :

Solomyak, Moody, Lee . . .

Coincidence condition plays a crucial role in both

methods. A variety of coincidences: Dekking’s coincidence,

super coincidence, strong coincidence, geometric coincidence,

balanced pairs, modular coincidence, overlap coincidence,

algebraic coincidence.


The fix point x of σ naturally gives rise to a 1-dimensional

self-affine tiling T by giving length to each letter. The lengths

are coordinates of a left eigen vector of Mσ. Taking closure of

translations of R we get a different dynamical system (XT ,R).

Now we have two systems: (Xσ, s) and (XT ,R). Spectral

property of (Xσ, s) is rather intricate. Sadun-Clark showed that

if we ask whether pure discrete or not, two dynamics behave

the same.


Assume that Perron Frobenius root of Mσ is Pisot. Then

Solomyak showed that self-affine tiling dynamical system is

pure discrete iff

density T ∩ (T −Qnv) → 1,

where Q is the Perron Frobenius root of Mσ and v be a return

vector. It is equivalent to overlap coincidence.

In higher dimensional case, the assumption on Pisot is

substituted to Meyer property of the associated Delone set and

we take d linearly independent return vectors over R.


Self affine tilings and multi Delone sets

A Delone set in Rd ⇔ relatively dense and uniformly

discrete. A set Λ in Rd is a Meyer set if Λ and Λ − Λ are

Delone. (Λi)ki=1 is a substitutive Delone set if Λi is Delone and

Λi =

k∪j=1

QΛj +Dij

for an expanding matrix Q. An adjoint system is

QAj =

k∪i=1

Ai +Dij,


which gives a unique non empty compact solution Ai. By

these two equations, a tiling and a point set becomes dual.Tiling dynamical system can be defined in both sides and they

are isomorphic. Assume that its incidence matrix (#Dij) is

primitive. Then the system is minimal and uniquely ergodic.


Question 1. Under which parameters Q and Dij, we have

a tiling ?

Lagarias-Wang condition: | det(Q)| = PF root of (#Dij).

Question 2. Under which parameters Q and Dij, it is pure

discrete ? If pure, make precise the group translation.

Pisot family condition: Any conjugate of the dominant

eigenvalue of Q of modulus > 1 must be an eigenvalue of Q.


Possible Pisot Conjecture in Higher dimension (?)

Assume 3 conditions:

1. The characteristic polynomial of (#Dij) is irreducible

2. (Λi) has Meyer property (→ Lagarias-Wang condition).

3. Q satisfies Pisot family condition.

Then the tiling dynamical system (XT ,Rd) is pure discrete.


Given a self-affine tiling, we have the overlap algorithm

(Solomyak [8]) to check pure discreteness. It is executed by

polynomial number of overlaps with respect to the input data.

However in practice, this computation often becomes large and

we require computer assistance to check.

The overlap (T, α, S) is defined in topological terms:

(T −α)∩

S = ∅,

it is not easy to tell this to our computer. Especially when the

tile has fractal boundary, it is hard to tell even for us.


Therefore we introduce a weaker notation: (i, α, j) is a

potential overlap if

|xT − α− xS| ≤ 2maxT

diam(T )

where xT ∈ Λi and xS ∈ Λj. Taking control points of tiles T

and S of an overlap, it must give a potential overlap.

For computational reason, we also need to introduce an

overlap graph with multiplicity, in which the graph is made

exactly by the same procedure as the overlap algorithm but

we count multiplicities of occurrences of the same potential

overlap in the inflated overlap.


In other words, a potential overlap graph with multiplicityG has s multiple edges from (i, α, j) to (i′, α′, j′) iff Ω(i, α, j)

contains s copies of (i′, α′, j′).

We say that (i, α, j) is a coincidence if i = j and α = 0.

Let Gcoin be the induced graph of G to the vertices which have

a path leading to a coincidence. Also we define Gres by the

induced graph generated by the complement of such vertices.


Theorem 2. (XT ,Rd) is pure discrete if and only if

ρ(Gcoin) > ρ(Gres).

Moreover the modified Hausdorff dimension of the boundary

of the tile is equal to d log ρ(Gres)/ log ρ(Gcoin).

This algorithm is quite simple, easy to implement and applies

to all self-affine tilings whenever we know the matrix Q and

digits Di,j which defines the multi-color Delone set.

Our current bottle neck is the process to collect all potential

overlaps for d return vectors.


Sufficiency of the inequality.

Consider the real overlap (T, α, S). Applying Ωn we have

µd(Qn((T − α) ∩ S)) = | det(Q)|nµd((T − α) ∩ S)

where µd is the d-dim Lebesgue measure. We must have

| det(Q)| = β where β is the Perron Frobenius root of

substitution matrix (#(Dij)) (Lagarias-Wang [3]). There are

r > 0 and R > 0 such that each overlap (T −α)∩

S contains

a ball of radius r and surrounded by a ball of radius R. After

n-iteration of inflation, the number of overlaps Kn generated


from (T, α, S) is estimated:

c1βn ≤ Kn ≤ c2β

n

with positive constants c1 and c2. As each overlap gives this

growth of overlaps, Gres can not contain an overlap (because

we are taking into account the multiplicities of overlap growth).


Necessity of the inequality.

We prove that if all overlaps leads to a coincidence then

Gres can not have a spectral radius β. Potential overlaps can

be divided into three cases.

• Real overlap: (T −α)∩

S = ∅,

• T − α and S are just touching at their boundaries,

• No intersection: (T − α) ∩ S = ∅.

If (T − α) ∩ S is empty, then the distance becomes larger

by the iteration of Ω. Therefore this potential overlap does not


produce an infinite walk on the produced graph. This case does

not contribute to the spectral radius. However when they are

touching at their boundaries, by repeated inflation, this gives an

infinite walk on this graph and may contribute to the spectral

radius of Gres. Our task is to prove that this contribution is

small.

We recall a:

Conjecture 2 (Urbanski (Solomyak [7])). For d-dim self-

similar tiling, each tile T satisfies d− 1 ≤ dimH(∂(T )) < d.

which is a folklore (every knows the ‘fact’ without proof).


We have to solve partly a version of this conjecture. We

use the recent development to slightly modify the Hausdorff

measure by He and Lau [2]. They introduced a new type of

gauze function, called weak norm w : Rd → R+ corresponding

to Q having key properties:

w(Qx) = |det(Q)|1/dw(x)

and

w(x+ y) ≤ cmax(w(x), w(y))

for some positive constant c. This w induces the same topology

as Euclidean norm. By w, they modified the definition of


Hausdorff measure by:

diamw(U) = supx,y∈U

w(x− y),

Hws,δ(X) = inf

X⊂∪

iUi

∑i

diamw(Ui)s

∣∣∣∣∣ diamw(Ui) < δ

and

Hws (X) = lim

δ↓0Hw

s,δ(X).

New Hausdorff dimension is dimwH(X) = infs|Hw

s (X) = 0.One can treat self-affine attractors almost as easy as self-similar

ones.


Theorem 3.For d-dim self-similar tiling, each tile T satisfies

dimwH(∂T ) < d.

We prove Theorem 2 and 3 simultaneously.

Let us take a piece Y of graph directed set defined by a

strongly connected component of Gres with maximal spectral

radius and put s = d log γ/ log β. Since all overlap leads to a

coincidence, Y must be a part of the boundary of our tile.


GIFS satisfies OSC

This is done according to the method of [2, 4] using

weak norm. Note that the uniform discreteness condition

of inflated digits naturally follows from our setting. Then

by mass distribution principle, we can show Hws (Y ) > 0 (c.f.

[5, 4]). This shows

dimwH(Y ) =

d log(ρ(Gcoin))

log(ρ(Gres)).


Dimension can not be d.

Assume s = d, then this Hws must be a constant

multiple of the d-dim Lebesgue measure, by the uniqueness

of Haar measure. But Praggastis [6] showed that the d-

dim Lebesgue measure of the boundary of self-affine tiles

must be 0 which contradicts Hsw(Y ) > 0. Thus we have

d log ρ(Gcoin)/ log ρ(Gres) < d which completes the proof of

Theorem 2. Noting that potential overlap graph contain all

boundary automata as a subgraph, s = d log γ/ log β < d gives

Theorem 3.


Example by the endomorphism of free group (Dekking)

We consider a self similar tiling is generated a boundary

substitution:

θ(a) = b

θ(b) = c

θ(c) = a−1b−1

acting on the boundary word aba−1b−1, aca−1c−1, bcb−1c−1,

representing three fundamental parallelogram. Associated the


tile equation is

αA1 = A2

αA2 = (A2 − 1− α) ∪ (A3 − 1)

αA3 = A1 − 1

with α ≈ 0.341164+i1.16154 which is a root of the polynomial

x3 + x+ 1.


Figure 1: Tiling by boundary endomorphism

ρ(Gcoin) ≈ 1.46557 and ρ(Gres) ≈ 1.32472. This


shows overlap coincidence and therefore the tiling dynamical

system associated with this tiling has pure point

spectrum. The Hausdorff dimension of the boundary is

2 log(ρ(Gcoin))/ log(ρ(Gres)) = 1.47131.


Example: Arnoux-Furukado-Harriss-Ito tiling

Arnoux-Furukado-Harriss-Ito recently gave an explicit

Markov partition of the toral automorphism for the matrix:0 0 0 −1

1 0 0 0

0 1 0 0

0 0 1 1

which has two dim expanding and two dim contractive planes.

They defined 2-dim substitution of 6 polygons. Let α =

−0.518913 − 0.66661√−1 a root of x4 − x3 + 1. The multi


colour Delone set is given by 6× 6 matrix:

z/α z/α z/α z/α z/α

z/α z/α+ 1− α (z − 1)/α+ α

and the associated tiling for contractive plane is:


Figure 2: AFHI Tiling


Our program saids it is purely discrete. Number of potential

overlap to make the graph is 264. he size of Gcoin becomes 88

and Gres is 178.

ρ(Gcoin) = 1.40127, ρ(Gres) = 1.22074

which shows Overlap coincidence and we have

dimH(∂(T )) = 1.18239.

References


[1] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and

algebraic number theory: some preliminary connections,

Contemp. Math., vol. 64, Amer. Math. Soc., Providence,

RI, 1987, pp. 241–264.

[2] X.-G. He and K.-S. Lau, On a generalized dimension of self-

affine fractals, Math. Nachr. 281 (2008), no. 8, 1142–1158.

[3] J. C. Lagarias and Y. Wang, Substitution delone sets,

Discrete Comput Geom 29 (2003).

[4] J. Luo and Y.-M. Yang, On single-matrix graph-directed

iterated function systems, preprint.


[5] R. D. Mauldin and S. C. Williams, Hausdorff dimension in

graph directed constructions, Trans. Amer. Math. Soc. 309(1988), 811–829.

[6] B. Praggastis, Numeration systems and Markov partitions

from self-similar tilings, Trans. Amer. Math. Soc. 351(1999), no. 8, 3315–3349.

[7] B. Solomyak, Tilings and dynamics, preprint.

[8] , Dynamics of self-similar tilings, Ergodic Theory

Dynam. Systems 17 (1997), no. 3, 695–738.


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