Spectral Properties of the Hata Tree

Transcription

1 Spectral Properties of the Hata Tree Antoni Brzoska University of Connecticut March 20, 2016 Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

2 Table of Contents 1 A Dynamical System for the Computation of Eigenvalues 2 The Sabot Theory 3 Spectral Asymptotics of Kigami s Laplacians Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

3 Let φ 0 (z) = c z and φ 1 (z) = (1 c 2 ) z + c 2, where 0 < c < 1. The Hata tree is the unique compact set K satisfying K = φ 0 (K) φ 1 (K). Let Φ(A) = φ 0 (A) φ 1 (A). Let V 0 = {0, 1, c} and V n = Φ n (V 0 ). The sets of vertices V n can be given a graph structure in a natural way. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

4 A Dynamical System for the Computation of Eigenvalues Definition The probabilistic Laplacian P on a finite graph G = (V, E) is a linear operator on R V defined by Pu(x) := 1 (u(x) u(y)) d x x y where x y if x and y share an edge and d x denotes the degree of x. The normalized Laplacian N is defined by Nu(x) := 1 (u(x) u(y)). dx d x y y Let P (n) and N (n) denote the probabilistic and normalized Laplacians, respectively, on V n. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

5 A Dynamical System for the Computation of Eigenvalues Proposition Let G be a graph. Let N be the normalized Laplacian on G and D(N) its characteristic polynomial. Fix a vertex j in G and let j 1, j 2,..., j k be its neighboring vertices. Then D(N) = (1 λ)d(n j ) k n=1 1 d j d jn D(N j,jn ) Proof: This follows by expanding the determinant 1 λ d j d 1 2 j1 d j d 1 2 j2... d j1 d 1 2 j 1 λ d j1 d 1 2 j2... D(N) = d j2 d 1 2 j d j2 d 1 2 j1 1 λ Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

6 A Dynamical System for the Computation of Eigenvalues We will need the polynomials g (2) (λ) = (1 λ) (1 λ)3 + 2 (1 λ) 9 g (2) u (λ) = (1 λ)4 2 (1 λ)2 3 g (2) w (λ) = (1 λ)4 7 9 (1 λ) g (2) uw (λ) = (1 λ)3 1 (1 λ) 3 g (2) vw (λ) = (1 λ)3 1 (1 λ) 3 and g (2) uvw (λ) = (1 λ)2 g (3) (λ) = (1 λ) (1 λ) (1 λ) (1 λ) (1 λ)3 243 g (3) u (λ) = (1 λ)10 2(1 λ) (1 λ)6 2 (1 λ)4 9 g (3) w (λ) = (1 λ) (1 λ) (1 λ) (1 λ) (1 λ)2 243 g (3) uw (λ) = (1 λ)9 5 3 (1 λ) (1 λ)5 1 (1 λ)3 9 g (3) vw (λ) = (1 λ)9 5 3 (1 λ) (1 λ)5 11 (1 λ)3 81 g (3) uvw (λ) = (1 λ) (1 λ)6 + 1 (1 λ)4 3 Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

7 A Dynamical System for the Computation of Eigenvalues Proposition Let g (j), g u (j), g w (j), g (j) n 4, let uw, g (j) vw, g uvw (j) for j = 2, 3 be defined as above. For g (n) = (1 λ)g (n 1) g (n 2) g (n 2) w g (n) u g (n) w g (n) uw g (n) vw g (n) uvw = (1 λ)g (n 1) g (n 2) g (n 2) uw (n 1) = (1 λ)g w = (1 λ)g (n 1) w = (1 λ)g (n 1) vw = (1 λ)g (n 1) vw g (n 2) g (n 2) w g (n 2) g (n 2) uw g (n 2) g (n 2) w g (n 2) g (n 2) ww u u uw uw uvw uvw g (n 2) g (n 2) w g (n 2) g (n 2) uw g (n 2) g (n 2) w g (n 2) g (n 2) uw g (n 2) g (n 2) w g (n 2) g (n 2) uw g (n 2) u g (n 2) u w w vw vw g (n 2) u g (n 2) u g (n 2) u g (n 2) u g (n 2) w g (n 2) g (n 2) vw g (n 2) uw g (n 2) g (n 2) uvw g (n 2) w g (n 2) uw g (n 2) w g (n 2) uw w g (n 2) g (n 2) vw w g (n 2) g (n 2) uvw vw g (n 2) g (n 2) vw vw g (n 2) g (n 2) uvw Then D(P (n) ) = (1 λ)g (n) g (n 1) 1 6 g u (n) g (n 1) 1 6 g (n) g u (n 1). Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

8 A Dynamical System for the Computation of Eigenvalues Blow-Up of Graph Approximations to the Hata Tree Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

9 A Dynamical System for the Computation of Eigenvalues Decomposition of the spectrum of P (n) In this analysis it is more convenient to work with the eigenvectors of P (n). 2 n 1 eigenvalues of multiplicity one that are eigenvalues of multiplicity one of P (n 1). 2 n new eigenvalues of multplicity one that are not eigenvalues of P (n 1). 2 n eigenvalues equal to one. Total: 2 n eigenvalues Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

10 A Dynamical System for the Computation of Eigenvalues Distribution of Eigenvalues Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

11 A Dynamical System for the Computation of Eigenvalues Spectrum on the Infinite Blow-Up We can define a Laplacian operator P ( ) on the infinite blow-up in a pointwise manner. There is a natural extension of P (n) to the blow-up. It can be shown that the spectrum of P ( ) is the limit of the spectrum of P (n). Depending on the blow-up, we can restrict P (n) and P ( ) to the interior of the lattice. The spectrum of this Dirichlet operator on the lattice is equal to the limit of the spectrum of the approximating operators. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

12 A Dynamical System for the Computation of Eigenvalues Back to the Hata Tree There is some work left in constructing the Green s function and Laplacian operator on the Hata tree itself and to make conclusions about its spectrum. One complication is that resistances between any pair of points evolves as a Fibonacci sequence as opposed to being scaled by certain fixed factors. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

13 The Sabot Theory In Spectral properties of self-similar lattices and iteration of rational maps (2003), Sabot works with dynamical systems that can be used to compute the spectrum of Laplacian operators on lattices that satisfy certain symmetry conditions. The Hata tree does not satisfy these symmetry conditions. Nonetheless, it is possible to apply some aspects of the theory. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

14 The Sabot Theory Let A (0) = ( a d f ) d b e, f e c be an operator on V 0 and let A (n) be formed by placing a copy of A (0) on each cell of V n isomorphic to V 0. In particular, we have ) A (1) = ( c e f 0 0 e b d 0 0 f d a+b d e 0 0 d a f 0 0 e f c. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

15 The Sabot Theory Define T : Sym 3 Sym 3 Q (Q (1) ) V 1 Here, Q (1) denotes the operator on V 1 constructed in the same way as A (1). In matrix notation, we have ( a d f ) ( ) a d b e (1) d (1) f (1) f e c where a (1) = c af 2 a 2 + ab d 2, b(1) = b ad 2 a 2 + ab d 2, d (1) b (1) e (1) f (1) e (1) c (1) d(1) = e adf a 2 + ab d 2, d( ae + df ) e(1) = a 2 + ab d 2 c (1) = a2 c + cd 2 + f ( 2de + bf ) + a( bc + e 2 + f 2 ), f (1) f ( ae + df ) = a 2 ab + d 2 a 2 + ab d 2 Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

16 The Sabot Theory It can be proved by induction that (Q (n) ) Vn = T n (Q). In the case of the Hata tree, since there is no edge between 1 and c, we can write the map T as ( ad 2 T (a, b, c, d, e) = c, b a 2 + ab d 2, c ae 2 ) a 2 + ab d 2, e, ade a 2 + ab d 2. Let D(a, b, c, d, e) = abc ae 2 cd 2 be the determinant of the corresponding 3 3 matrix. Observe that D(T n (1 λ, 2 2λ, 1 λ, 1, 1)) will give us the characteristic polynomial of the probabilistic Laplacian P (n). Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

17 The Sabot Theory It is possible to reduce this five dimensional dynamical system to two dimensions. In particular, let c n+1 = c n 1 en 2 c n, cn c en 2 n 1(c n 1 e 2 e2 n 3 n 1 e n 3 e n 2 ) en 1 2 e n+1 = c n 1 e n 1 e n. cn c en 2 n 1(c n 1 e 2 e2 n 3 n 1 e n 3 e n 2 ) en 1 2 Then ( en 2 2 D n = c n 1 c n 1 e ) e2 n 3 n 1 c n c n 1 en 2 c n e e n 3 e n 2, 2 n 2 will be the characteristic polynomial of the corresponding Laplacian. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

18 The Sabot Theory By the Sabot theory, we can construct a map R that is the analogue of T on some subset of a projective space that is isomorphic to a Lagrangian Grassmanian. This map can be used to write down a nice compact expression for the density of states (the limiting distribution of eigenvalues).. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

19 Spectral Asymptotics of Kigami s Laplacians In Weyl s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals (1993), Kigami and Lapidus prove some results on the spectral asymptotics on a certain class of Laplacians on p.c.f. fractals, including the Hata tree. It is possible to obtain similar results on the Hata tree by using an alternate construction. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

20 Spectral Asymptotics of Kigami s Laplacians Binary Functions and Orientations on V n Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

21 Spectral Asymptotics of Kigami s Laplacians Mixed Affine Nested Fractals Let A be a finite set. For a A, let ψ a = {ψ a i : i = 1,..., m a } denote a set of m a similtudes in C that determines a unique fixed point S a. Assume the set of fixed points E is the same for each a A. To construct a composite fractal, we can define an address space T and an A-valued function U that will determine the location of cells in approximations to the fractal and the set of similtudes to be applied in the subsequent approximation, respectively. In particular, for i T n, let (S) i = ψ U([i] 0) i(1) ψ U([i] n 1) i(n) (E), and the corresponding mixed affine nested fractal is defined by S = n=0 i Tn (S) i. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

22 Spectral Asymptotics of Kigami s Laplacians Define the set of similtudes ψ a = {ψ1 a, ψa 2, ψa 3 } on C such that ψ a 1 = c 2 z; ψ a 2(z) = (1 c 2 )(z 1) + 1; ψ a 3 = c 2 +i c (1 c 2 )z. Let ψ e denote the set consisting of the identity map. Let T 1 = {{1}, {2}, {3}}, and define the {a, e} valued function U by { a : if [i]m (m) = 0 or 2 U([i] m ) = e : if [i] m (m) = 1 or 3 Let T be the corresponding address space. ψ a, ψ e, T, and U determine a set S that is homeomorphic to half of a Hata tree. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

23 Spectral Asymptotics of Kigami s Laplacians The set S Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

24 Spectral Asymptotics of Kigami s Laplacians The identity map in ψ e is not a contraction map and thus resistances between successive approximations to S to not scale exactly. However, this can be remedied by jumping ahead to the second approximation. Let ψ b = {ψ b j : j = 1,..., 5}, where ψ b j = ψ a 2 ψa j for j = 1, 2, 3, ψ b 4 = ψa 1, and ψ b 5 = ψa 3. Using ψa and ψ b, we can define two affine nested fractals S o and S e such that their approximations are homeomorphic to the odd and even approximations to the Hata tree, respectively. More precisely, if we identify the vertex 0 in S o and S e, then K n is homeomorphic to S e n S o n for n 1. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

25 Spectral Asymptotics of Kigami s Laplacians By applying a multidimensional renewal theorem and the results/methods of Hambly and Nyberg on graph directed fractals, it is possible to obtain the spectral asymptotics for the eigenvalue counting functions. Theorem If ν 1 is non-lattice, then lim z Nx (z)z ds/2 = c 4 (x) lim z Nx 0 (z)z ds/2 = c 5 (x) where c 4, c 5 are constants depending on x. If ν 1 is lattice, then lim z Nx (z)z ds/2 p1(log x z) = 0 where p x 1, px 2 lim z Nx 0 (z)z ds/2 p2(log x z) = 0 are periodic functions depending on x. Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26

26 Spectral Asymptotics of Kigami s Laplacians Thank you!! Antoni Brzoska Spectral Properties of the Hata Tree March 20, / 26