Download Products of two Cantor sets and Application to the Labyrinth model

Products of two Cantor sets and Application to the Labyrinth model
Yuki Takahashi
Department of Mathematics • University of California, Irvine
Thickness of Cantor sets
Main results of products of two Cantor sets
Let K ⊂ R be a Cantor (
set. Define the thickness
of
K
as
)
L
R
L
R
U2 − U1 U2 − U1
inf max
,
,
U1<U2
|U1|
|U2|
where inf is taken for all pairs of gaps of K, with at least one
of them being a finite gap (for gap U , U L and U R represent the
left and the right endpoint, respectively). We denote this value
by τ (K).
Let K be a Cantor set. We denote K ∩ (0, ∞) by K+, and
−(K ∩ (−∞, 0)) by K−. We call K a
• 0-Cantor set if K+, K− 6= φ, inf K+ = 0, and inf K− = 0,
and
• 0+-Cantor set if min K = 0.
Theorem (Gap Lemma, Newhouse [6])
Let K, L be 0+-Cantor sets. Then, K · L is an interval if
2τ (K) + 1
2τ (L) + 1
τ (L) ≥
, or τ (K) ≥
.
2
2
τ (K)
τ (L)
This estimate is optimal. In particular, if τ (K) = τ (L), this
is equivalent to
√
1+ 5
τ (K) = τ (L) ≥
.
2
Let K, L be Cantor sets, and suppose that neither K nor
L lie in a complementary domain of the other. Then, if
τ (K) · τ (L) ≥ 1, K ∩ L contains at least one point.
Corollary
Let K and L be Cantor sets with τ (K) · τ (L) ≥ 1. Then
K + L is a disjoint union of finitely many closed intervals.
In addition, if the size of the largest gap of K is not greater
than the diameter of L, and the size of the largest gap of L
is not greater than the diameter of K, then K + L is a closed
interval.
Motivation: Sums of two Cantor sets
Sums of two Cantor sets appear in
• smooth dynamics (homoclinic bifurcations [7]);
• number theory (sums of Cantor sets related to continued
fractions [5]);
• mathematical physics (spectrum of separable two-dimensional
quasicrystal models [1]);
Products of two Cantor sets
• Products
of two Cantor sets arise naturally as the spectrum
of the Labyrinth model.
Let K and L be Cantor sets with τ (K) · τ (L) > 1. Then K · L
is a disjoint union of finitely many or countably many closed
intervals.
Example
Let K be a standard
√ middle-α Cantor set whose convex hull is
[0, 1]. Let −2 + 5 < α < 1/3. Then K · K is a disjoint union
of 0 and countably many closed intervals which accumulate to
0.
1
Figure 1: K · K
Let K, L > 0 be Cantor sets. We use the equality
K · L = exp(log K + log L).
K and L do not contain 0, log K and log L are again
Cantor sets.
• If K contains 0, log(K \ 0) is "stretched to negative infinity".
• If
• Quasicrystals
are materials whose microscopic organization
is neither random, nor periodic, but quasiperiodic.
• D. Shechtman’s discovery of quasicrystals in 1982 led to his
Nobel prize in Chemistry in 2011.
Fibonacci substitution sequence
Theorem (Case 1, T. ’15 [10])
Similarly, we have the following:
The Labyrinth model was suggested in the late 1980s in [9], and
so far this model has been studied mainly by physicists, and their
work is mostly relied on numerics [8].
a −→ ab −→ aba −→ abaab −→ abaababa −→ · · · .
Theorem (T. ’15 [11])
There is a natural way to extend this sequence to the left. Call
the resulting two-sided sequence the Fibonacci substitution sequence, and denote by ω.
Ĥλ1,λ2 is unitarily equivalent to Hλ1 ⊗Hλ2 . Therefore, we have
Let K be a 0-Cantor set, and let L be a 0+-Cantor set. Then,
K · L is an interval if
2τ (K) + 1
τ (L) ≥
.
2
τ (K)
This estimate is optimal. In particular, if τ (K) = τ (L), this
is equivalent to
√
1+ 5
.
τ (K) = τ (L) ≥
2
Off-diagonal model is a one-dimensional quasicrystal model. It
is given by the following bounded self-adjoint operator in `2(Z):
Theorem (Case 3, T. ’15 [10])
Let K, L be 0-Cantor sets. Then, K · L is an interval if
(Hω ψ)(n) = ω(n + 1)ψ(n + 1) + ω(n)ψ(n − 1),
where ω is the Fibonacci substitution sequence.
|a2−b2|
• Define λ := ab , and call this the coupling constant.
• Spectral properties of Hω only depend on λ.
• Denote the spectrum of this operator by Σλ.
Theorem (Damanik & Embree &
Gorodetski & Tcheremchantsev ’08 [2],
Damanik & Gorodetski ’11 [3])
The spectrum Σλ is dynamically defined Cantor set for all
λ > 0. Also, we have
2(τ (K) + 1)(τ (L) + 1) ≤ (τ (K)τ (L) − 1)2.
This estimate is optimal. In particular, if τ (K) = τ (L), this
is equivalent to
√
τ (K) = τ (L) ≥ 1 + 2.
We also have the following estimate to guarantee the existence
of 0-Cantor sets K and L such that K · L is a disjoint union of
countably many closed intervals.
Let M, N be positive real numbers. Then, if
M 2 + 3M + 1
(2M + 1)2
a) N <
and N <
, or
2
3
M
M
N 2 + 3N + 1
(2N + 1)2
b) M <
and M <
,
2
3
N
N
there exist 0-Cantor sets K, L such that
1) τ (K) = M, τ (L) = N, and
2) K · L is a disjoint union of countably many closed intervals.
is unitarily equivalent to Hλ1 ⊗ Id + Id ⊗ Hλ2 .
• Therefore, σ(Hλ1,λ2 ) = σ(Hλ1 ) + σ(Hλ2 ).
• σ(Hλ1,λ2 ) is an interval if λ1 and λ2 are sufficiently close to 0,
and a Cantor set of zero Lebesgue measure if λ1 and λ2 are
sufficiently large.
• Hλ1,λ2
Let a, b be positive real numbers. Consider the following substitution.

a −→ ab
P=
b −→ a.
This substitution defines an aperiodic sequence:
Off-diagonal model
When K, L are both 0-Cantor sets, we have the following:
Known results of square tiling
We prove analogous results for the Labyrinth model.
Theorem (Case 2, T. ’15 [10])
Theorem (T. ’15 [10])
0
Quasicrystals
lim τ (Σλ) = ∞ and lim dimH Σλ = 0.
λ→0
• It
Labyrinth model
σ(Ĥλ1,λ2 ) = σ(Hλ1 )σ(Hλ2 ).
The spectrum σ(Ĥλ1,λ2 ) is an interval if λ1 and λ2 are sufficiently close to 0, and a Cantor set of zero Lebesgue measure
if λ1 and λ2 are sufficiently large.
References
[1] D. Damanik, M. Embree, A. Gorodetski, Spectral
properties of Schrödinger operators arising in the study of
quasicrystals, to appear in Mathematics of aperiodic order,
Birkhauser.
[2] D. Damanik, M. Embree, A. Gorodetski, S.
Tcheremchantsev, The fractal dimension of the spectrum of
the Fibonacci Hamiltonian, Commun. Math. Phys. 280
(2008), 499–516.
[3] D. Damanik, A. Gorodetski, Spectral and quantum
dynamical properties of the weakly coupled Fibonacci
Hamiltonian, Commun. Math. Phys. 305 (2011), 221–277.
λ→∞
is easy to see that 0 ∈ Σλ for all λ > 0.
Two-dimensional quasicrystal models
Using two copies of the off-diagonal model (denote the coupling constants of them by λ1 and λ2), we can construct two
types of two-dimensional quasicrystal models (square tiling and
Labyrinth model).
[4] S. Even-Der Mandel, R. Lifshitz, Electronic energy spectra
of square and cubic Fibonacci quasicrystals, Philosophical
Magazine A 88, 2261–2273.
[5] M. Hall, On the sum and product of continued fractions,
Ann. of Math. 48 (1947), 966–993.
[6] S. Newhouse, The abundance of wild hyperbolic sets and
nonsmooth stable sets for diffeomorphisms, Inst. Hautes
Études Sci. Publ. Math. 50 (1979), 101–151.
[7] J. Palis, F. Takens, Hyperbolicity and sensitive chaotic
dynamics at homoclinic bifurcations, Cambridge
University Press, Cambridge, 1993.
[8] S. Rolof, S. Thiem, M. Schreiber, Electronic wave functions
of quasiperiodic systems in momentum space, The
European Physical Journal B 86 (2013).
a b a a b a b a
a b a a b a b a
Figure 2: The square tiling (left) and the Labyrinth model (right).
Let us denote the Schrödinger operator of the square tiling and
the Labyrinth model by Hλ1,λ2 and Ĥλ1,λ2 , respectively.
[9] C. Sire, Electronic spectrum of a 2D quasi-crystal related to
the octagonal quasi-periodic tiling, Europhys. Lett. 10
(1989), 483–488.
[10] Y. Takahashi, Products of two Cantor sets, preprint.
[11] Y. Takahashi, Quantum and spectral properties of the
Labyrinth model, preprint.