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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.