Self-similar tilings and their complex dimensions. Erin P.J. Pearse erin/

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1 Self-similar tilings and their complex dimensions. Erin P.J. Pearse erin/ July 6, 2006 The 21st Summer Conference on Topology and its Applications References [SST] Canonical self-similar tilings by IFS, E. P. J. Pearse, preprint, Nov. 2005, 14 pages. arxiv: math.mg/ [CDS] Tube formulas and complex dimensions of self-similar tilings, M. L. Lapidus and E. P. J. Pearse, preprint, Apr. 2006, 51 pages. arxiv: math.ds/ [FGCD] Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, M. L. Lapidus and M. van Frankenhuijsen, Springer-Verlag, [KTF] A tube formula for the Koch snowflake curve, with applications to complex dimensions, M. L. Lapidus and E. P. J. Pearse, J. London Math. Soc. (in press). arxiv: math-ph/

2 Self-similarity A self-similar system is Φ = {Φ j } J j=1, where each Φ j is a contractive similarity mapping, i.e., Φ j (x) = r j A j x + t j, 0 < r j < 1, where A j O(d) is a rotation/reflection and t j R d. A set F is self-similar iff F = Φ(F ) = J j=1 Φ j(f ). For any such Φ,!F and F is compact. [SST] There is a tiling T of the convex hull [F ] which is canonically associated with Φ. Not a typical fractal tiling : Tiles are not typically fractal. Only the region [F ] is tiled, not R d. Tiles may not be fractal, but T is. T contains key geometric/dynamical information about Φ. T describes scaling/geometric oscillations. Curvature of the tiles relates to curvature of F. From T, one can define ζ T, a geometric zeta function associated with Φ. 2

3 Φ 1 (z) = ξz, Φ 2 (z) = (1 ξ)(z 1) + 1, ξ = i 2 3. C 0 = C C o C 1 T 1 C 2 T 2 C 3 T 3 C 4 T 4 C 5 T 5 C 6 T 6 C 7 T 7 Figure 1. The Koch tiling, with unique generator G = T 1. Figure 2. The tiling of [K] K. The tiles exhaust the complement of the Koch curve within its convex hull. 3

4 Fractal strings [FGCD] A fractal string L R is a bounded open subset L := {l n } n=1. Translation invariance: l n R, and may assume l 1 l 2 l 3... Also assume l n > 0, or else trivial. Idea: L = F, where F R is fractal. A self-similar fractal string is when L = F for some Φ with d = 1 and A j = ±1. The geometric zeta function ζ L is a Dirichlet generating function for the string. ζ L (s) := n=1 ls n. The complex dimensions of L are D L := {poles of ζ L }. Important result: a tube formula for strings V A (ε) = vol 1 {x A. dist(x, A) < ε}, = ω D L c ω ε 1 ω + c 1 ε. c ω is defined in terms of res (ζ L ; ω). 4

5 Example: the Cantor String l 1 l 2 l 3 l 4 l 5 l 6 l 7 CS = { 1 3, 1 9, 1 9, 1 27, 1 27, 1 27, 1 27,...} = [0, 1] C (CS) = the Cantor set C CS has l k = 1 3 k+1 with w lk = 2 k. ζ CS (s) = k=0 2k 3 (k+1)s 3 s = s. V CS (ε) = ω 3 log 3 ω(1 ω) ε1 ω 2ε = ω D CS ω D CS {1} c ω ε 1 ω D CS (s) = {D + inp. p = 2π log 3, n Z}. D = log 3 2 = Minkowski dim of C. As a self-similar string, CS is the tiling associated with the self-similar system {Φ 1, Φ 2 }, Φ 1 (x) = 1 3 x, Φ 2(x) = 1 3 x

6 10 p 0 D 1 Figure 3. The complex dimensions of the Cantor string. D = log 3 2 and p = 2π/ log 3. (Plot from [FGCD]) 6

7 Why call the poles of ζ L the complex dimensions? First reason: relation to Minkowski/box dimension. Theorem (Lapidus). Assuming l n > 0 for all n, D = inf{σ 0. n=1 lσ n < }. In fact, D L R + = {D}. Recall: An (inner) tube formula for a set A R is V A (ε) = vol 1 {x A. dist(x, A) < ε}, i.e., the volume of the inner ε-neighbourhood of A. The Minkowski dimension of A is then D = inf{t 0. V A (ε) = O(ε 1 t ) as ε 0 + }. A is said to be Minkowski measurable iff M = lim ε 0 + V (ε)ε (1 D) exists, and has a value in (0, ). For A R d, replace 1 by d in V A, D, M. 7

8 Why call the poles of ζ L the complex dimensions? Second reason: relation to classical geometry. The Steiner formula for ε-nbd of A K d : V A (ε) = i {0,1,...,d 1} c i ε d i. The classical (outer) tube formula is summed over the integral dimensions of A. [FGCD] Tube formula for L: V L (ε) = c ω ε 1 ω. ω D L {1} [CDS] Tube formula for T : V T (ε) = ω D s {0,1,...,d} c ω ε d ω. The fractal tube formula is summed over the integral and complex dimensions of L or T. 8

9 Φ 1 (z) = ξz, Φ 2 (z) = (1 ξ)(z 1) + 1, ξ = i 2 3. C 0 = C C o C 1 T 1 C 2 T 2 C 3 T 3 C 4 T 4 C 5 T 5 C 6 T 6 C 7 T 7 Figure 4. The Koch tiling, with unique generator G = T 1. Figure 5. The tiles exhaust the complement of the Koch curve within its convex hull. 9

10 The self-similar tiling: extension to higher dimensions From [SST]: Construct the self-similar tiling and produce a collection of tiles T = {R n } = {Φ w (G q )}. w {1, 2,..., J} k is a (finite) word, like w = Φ 3132 (x) := Φ 2 Φ 3 Φ 1 Φ 3 (x), r 3132 = r 1 r 2 r 2 3. string L = {l n } tiling T = {R n } length l n inradius ρ n ζ L = l s n ζ s = rw s D L = {poles of ζ L } D s = {poles of ζ s } Idea: decompose the complement of the attractor F within its convex hull [F ]. 1. Find the attractor F of Φ. 2. Take the (closed) convex hull C = [F ]. 3. Define the generators G q to be the connected components of relint(c) Φ(C). (Note: Φ(C) C [SST].) 4. The sets {Φ w (G q )} form a tiling of C F. 10

11 ξ ξ ξ (a) (b) (c) Figure 6. Parameters for nonstandard Koch tilings. C 1 C 2 C 3 C 4 C 5 T 1 T 2 T 3 T 4 T 5 C 1 C 2 C 3 C 4 C 5 T 1 T 2 T 3 T 4 T 5 Figure 7. Nonstandard Koch tilings. C 0 C 1 C 2 C 3 C 4 T T 1 T 2 T 3 T 4 Figure 8. The Sierpinski Carpet tiling. 11

12 C 0 C 1 C 2 C 3 C 4 C 5 C o T 1 T 2 T 3 T 4 T 5 Figure 9. The Pentagasket tiling. This tiling has 6 generators; one pentagon G 1, and five congruent isoceles triangles G 2,..., G 6. T 1 = G 1 G 2 G 6. C 0 C 1 C 2 C o T 1 T 2 Figure 10. The Menger Sponge tiling. This tiling has the unique nonconvex generator G = T 1. 12

13 Tube formula of a self-similar tiling Recall the form of V T given earlier: V T (ε) = c ω ε d ω. [CDS] The full form of V T : V T (ε) = ω D s {0,1,...,d} ω D T res (ζ T (ε, s); ω). D T = D s {0, 1,..., d}, where D s = {poles of ζ s }. Once the residues are evaluated, these are the same. For sets with no fine structure (trivially self-similar), ζ s holomorphic = D s = and we recover the inner Steiner formula. In the formula for V T, ζ T is a meromorphic distributionvalued function; a generating function for the geometry of T. 13

14 Ingredients of the Geometric Zeta Function The tiling zeta function is the matrix product ζ T (ε, s) = g(s) κ(ε) E(ε, s). The generator inradii form a Q-vector: g(s) := [g s 1,..., g s Q] ζ s (s). The curvature matrix is the Q (d + 1) matrix κ(ε) := [κ qi (ε)] (κ qi (ε) is the i th curvature of the q th generator; i.e., κ qi (ε) is the coefficient of ε d i in γ q.) The boundary terms compose a (d + 1)-vector E(ε, s) := [ 1 s, 1 s 1,..., 1 s d] ε d s. Then c ω ε d ω is a residue of the matrix product: c ω ε d ω = res (ζ T (ε, s); ω). 14

15 Idea behind the tube formula Obtain a tube formula for T using the idea V T (ε) = η g, γ G = 0 γ G (x, ε) dη g (x). γ G (x, ε) is the volume of the ε-neighbourhood of a tile with inradius 1/x; i.e., the contribution of an tile at scale r 1 w. η g (x) is a measure giving the density of geometric states: it is supported at reciprocal tile inradii. Each generator has its own associated η g : η g (x) = [η s (x/g 1 ),..., η s (x/g Q )] η s (x) is the scaling measure: it is supported at the reciprocal scales. Obtain a tube formula V T (ε) = for D T = D s {0, 1,..., d}. ω D T c ω ε d ω 15

16 Contributions to V T (ε) = 0 γ G (x, ε) dη g (x) G g ε g G 3g 3(g ε) g g ε g ε ε g 3 (g ε) 3 Figure 11. The volume V G (ε) = γ G (1/g, ε) of the generator of the Koch tiling η s (x) 1 3 1/2 3 x Figure 12. The Koch scaling measure. γ T (1/g) γ T ( 3/g) γ T (3/g) γ T (x) ε /g 3/g 3/g η g (x) x Figure 13. The Koch geometric measure and contributions to the integral. 16

17 Q: What s different in R d? A: The generators. Φ j (x) = r j A j x + t j, 0 < r j < 1. New: A j SO(d) is a rotation/reflection. Since generators may have different shapes, must distinguish between tile of different type, but with the same inradius. Different generators may have different inner tube formulas. (Intervals all have 2ε.) Must distinguish scales from sizes: η s vs. η g. η s = w W δ r 1 w η g = [η s (x/g q )] Q q=1 η gq = n=1 δ ρ n (G q ) 1 = w W δ (g q r w ) 1 Must calculate each congruency class of generators separately. Cannot integrate against the same measure for each. 17

18 Q: What s the same in R d? A: The scaling ratios. Properties of r 1,..., r J are still of key importance. ζ s is formally identical to ζ η for η with g = 1. ζ s (s) = w W r s w = 1 1 J j=1 rs j. D s depends entirely on r 1,..., r J. Lattice/nonlattice dichotomy still holds. The structure theorem for D s is the same. The distributional explicit formulas which applied to η in the 1-dimensional case apply to each η gq in the d-dimensional case. 18

19 Convex Geometry The Steiner Formula for convex bodies A K d. V A (ε) = = d 1 i=0 d 1 ( ) d W d i (A)ε d i i µ d i (B d i )µ i (A)ε d i. i=0 W i are the Minkowski functionals. B i is the i-dimensional unit ball. µ i are invariant/intrinsic measures. Homogeneous: µ i (ra) = r i µ i (A), Basic idea: V A (ε) = ω {0,1,...,d 1} c ω ε ω where c ω is related to the curvature of A. 19

20 Curvature measures in convex geometry V A (ε) = d 1 i=0 µ i (A)µ d i (B d i )ε d i. For convex bodies A K d, ( ) d κ i (A) = d µ i (A)µ d i (B d i ) = C i (A). i Here, the C i are curvature measures and the κ i are curvatures (as def d in convex/integral geometry). The κ i are homogeneous and translation invariant because the µ i are. C i (A) is the total curvature of A; a special case of the generalized curvature measure C i (A) := C i (A, R d ) = Θ i (A, R d S d 1 ). Θ i is defined on U(K d ) B(Σ), where U(K d ) is the ring of polyconvex sets of dimension d. 20