Shigeki Akiyama (Niigata University, Japan) 17 …...The fix point x of ˙ naturally gives rise to...
Transcript of 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)
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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.
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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.
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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.
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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.
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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.
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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.
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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,
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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).
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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
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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).
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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
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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.
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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.
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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)).
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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.
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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
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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.
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Figure 1: Tiling by boundary endomorphism
ρ(Gcoin) ≈ 1.46557 and ρ(Gres) ≈ 1.32472. This
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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.
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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
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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:
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Figure 2: AFHI Tiling
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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
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[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.
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[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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