A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH APPLICATIONS TO COMPLEX DIMENSIONS.

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1 A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH APPLICATIONS TO COMPLEX DIMENSIONS. MICHEL L. LAPIDUS AND ERIN P.J. PEARSE Current (i.e., unfinished) draught of the full version is available at 1

2 K 0 K 1 K 2 K K 4... Figure 1. The Koch curve K (left) and the Koch snowflake Ω (right). K 2

3 Goal: derive a formula for the ε-neighbourhood of the Koch curve (and snowflake). { { Figure 2. The ε-neighbourhood of the Koch curve, for two different values of ε. We want to find a formula for V (ε) = area of shaded region = vol 2 {x Ω : d(x, Ω) < ε}

4 Q: What use is V (ε)? A: A precise formula for V (ε) will help towards extending the theory of fractal strings into higher dimensions. A fractal string is any bounded open subset of R L := {l j } j=1, with l j <. j=1 l 1 l 2 l..., or distinctly (with multiplicity): l 1 > l 2 > l >.... Idea/origin: comes from studying fractal subsets L R. Figure. The Cantor Set Figure 4. The Cantor String l 1 = 1 / l 2 = 1 / 9 l = 1 / 27 4

5 The Cantor String example has lengths { (n+1) } with multiplicities w (n+1) = 2 n. l 1 l 2 l l 4 l 5 l 6 l 7 CS = { 1, 1 9, 1 9, 1 27, 1 27, 1 27, 1 27,...} The geometric zeta function of a string ζ L (s) = lj s = w l l s j=1 l encodes all this information. Example: ζ CS (s) = 2 n (n+1)s = n=0 s 1 2 s. 5

6 Three key things about ζ L : (1) Relates to the dimension of L. (2) Connects spectral and geometric properties. () Gives an explicit formula for V (ε). For (1), recover the Minkowski dimension D L := inf{t 0 : V (ε) = O ( ε 1 t) as ε 0 + } = inf{σ 0 : ζ L (s) < } Generalize and define the complex dimensions: D = {ω C : ζ L has a pole at ω} Theorem 1 (Structure of Complex Dimensions). If L is self-similar, then either (1) D is periodic and L is not measurable, or (2) D is quasiperiodic and L is measurable. Here, L is (Minkowski) measurable iff exists in (0, ). M = M(D; L) = lim ε 0+ V (ε)ε (1 D) 6

7 For (2), a frequency of L is f = λ/π = k l j. The spectral zeta function of L is ζ ν (s) = (k lj 1 ) s = j,k=1 f w f f s Then ζ ν (s) = ζ L (s)ζ(s) relates spectral and geometric information. For (), we have the explicit (distributional) formula V (ε) = ( ζl (s)(2ε) 1 s ) res ; ω + R(ε) s(1 s) ω D L = ( res (ζl ; ω) 2 1 ω ) ε 1 ω + R(ε). ω(1 ω) ω D L We want higher-dimensional analogues of these results. Computing V (ε) for the Koch curve provides a test of how well the theory holds up for Ω R 2 intuition about how to extend into R 2 7

8 First, partition the ε-neighbourhood. overlapping rectangles fringe triangles (from overlap) wedges error rectangles Figure 5. An approximation to the inner ε-neighbourhood of the Koch curve. The level of refinement is based on ] ( ε ( (n+/2), (n+1/2) = (n+1) /, n / ]. Figure 6. Another ε-neighbourhood of the Koch curve, for smaller ε. 8

9 overlapping rectangles fringe triangles (from overlap) wedges error rectangles Count each type of piece: shape number volume(area) rectangles r n = 4 n ε n wedges w n = 2 (4n 1) πε 2 /6 triangles u n = 2 (4n 1) + 2 ε 2 /2 fringe 4 n 9 1 n /160 A preliminary formula is ˆV (ε) = ˆV 1 (ε) + ˆV 2 (ε) ˆV (ε) + ˆV 4 (ε), where ˆV 1 (ε) := 4 n ε n ˆV 2 (ε) := 2 (4n 1) πε2 ( ) 6 2 ˆV (ε) := (4n 1) + 2 ε2 2 ( ) ( n 4 ˆV 2 ) 4 (ε) :=

10 Problem: formula contains a discrete variable. Need to convert to continuous: n = n(ε) = [x] = x {x} for x = log (ε ) n( ) Figure 7. The exponent n = n(ε), as a function for ε 0. Now as fn of ε, x = log (ε ): ( ˆV (ε) = ε 2 D 4 {x} {x} + ( ε 2 π ) 9 2 {x} + ( )) π

11 Convert to Fourier series using a {u} = a 1 a β Z e 2πiβu log a + 2πiβ and get ˆV (ε) = ε 2 D n Z + ( 27 e 2πinx 2 9 log 4/9+2πin + 8 ( π 72 ) 8 e 2πinx log 4+2πin ) e 2πinx log 4/+2πin ) (π ε Now: collect the error. error Figure 8. Where the error lies - the bold region is not within ε of K. 11

12 We decompose an error block: w( )= n triangle wedge 2 wedge 1 n+1) w( )= Figure 9. w(ε) gives the width of the block Use elementary methods to find the area A 1 (ε): H b= w( ) A 1 ( ) Figure 10. Finding the height of the central triangle A 1 (ε) = ε w ( w ) ε2 sin 1 ε w ( w ) ε 6 6ε 12

13 The area of the error in this block is B(ε) := ( 2 k 1 ε w(ε) ε 2 sin 1 k k=1 ε w(ε) 2 k 1 ( ) w(ε) 2 k ε ( ) 2 w(ε). 2 k ε Figure 11. Naming the trianglets Apply series expansions for sin 1 u and 1 u 2, and Fubini Thm, to get w(ε) = [x] = {x} ε, 1

14 B(ε) = εw(ε) 2 ( (2m)! w(ε) m+1 (m!) 2 2 ε(m + 1) 1 2m+ 2 m=0 ) (w(ε) ) 2m+1 ε2 2m m+1 2 ε How many such error blocks are there? partial complete Figure 12. Error block formation Some blocks are whole; others form as ε 0. 14

15 We count the error blocks δ(ε) = c n + p n h = c(ε) + p(ε)h(ε) = ε D 6 4 {x} + ε D 6 4 {x} h(ε) + 2 h(ε) 4 where h(ε) is some function indicating what portion of the partial block has formed. ( ε 0 h(ε) = h µ < 1 ) We don t know h(ε) explicitly, but we do know h(ε) = α Z g α e 2πiαx Total error is E(ε) = δ(ε)b(ε) Compute the desired volume formula as V (ε) = ˆV (ε) E(ε) by converting everything into series expansion. 15

16 After 11 pages of calculations... where G 1 (ε) : = 1 log G 2 (ε) : = 1 log V (ε) = G 1 (ε)ε 2 D + G 2 (ε)ε 2, ( a n + ) b ν g n ν ( 1) n ε inp n Z ν Z ( σ n + ) τ ν g n ν ( 1) n ε inp, n Z ν Z are periodic functions of multiplicative period, and a n = 9/2 b n = σ n = τ n = 2 9 (D 2+inp) + /2 2 (D 1+inp) + π /2 m 1/2 (2m 2)! 2 4m+1 (2m+1)m!(m 1)! m=1 ( ) π δ0 n τ n, m=1 2 (D+inp) 1 2 b n, (4 2m+1 ) ( 2m+1 2)(D 2m 1+iνp), (2m)! m+1/2 2 4m 2 (2m+1)m!(m 1)! 1 2m+1 ( 2m+1 2)( 2m 1+iνp). The g α are Fourier coefficients of function which counts the error blocks (actually, it describes how much has formed). 16

17 However, the coefficients are not the interesting part. The formula for V (ε) contains all the complex dimensions! We rewrite as V (ε) = n Z G (ε)ε 2 D inp + n Z G 4 (ε)ε 2 inp. This gives the possible dimensions D Ω = {D + inp : n Z} {inp : n Z}. 2p p D 2 4 C 17

18 " $ " $ Notes on h(ε): The least upper bound of h(ε) is µ = C(ε)/B(ε):! Figure 1. µ is the ratio C(ε)/B(ε). Three essential properties: (i) h(ε) oscillates multiplicatively, (ii) h(ε k ) = lim ϑ 0 h(ε k + ϑ) = 0, (iii) lim ϑ 0 + h(ε k + ϑ) = µ, where ε k = k. Compare to h(ε) = µ { [x] x} ~ h(# ) h(# ) Figure 14. The Cantor-like function h and the approximation h. 18