A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH

APPLICATIONS TO COMPLEX DIMENSIONS.

MICHEL L. LAPIDUS AND ERIN P.J. PEARSE

To appear in the Proceedings of the London Mathematical Society.Current (i.e., unfinished) draught of the full version is available at

http://math.ucr.edu/~epearse/koch.pdf.

References

[FGNT] “Fractal Geometry and Number Theory — Complex Dimensions and Zeros of ZetaFunctions”, M. L. Lapidus and M. van Frankenhuijsen, Birkhauser, 2000. (2nd re-vised and enlarged edition to appear shortly.)

“Characterizing the measurability of fractal strings” (a primer for [FGNT]), E.P.J.P.,available on my web page (listed as my Oral Examination).

http://math.ucr.edu/∼epearse/

1

What’s Been Done (FGNT)

geometric object, F ⊆ R,

F =⋃K

k=1 Φk(F )

fractal string,L = lj∞j=1

lj =length of aconnected component(open interval) of L

originally: ∂L = Fnow: L = X\F

(geometric) zeta function,ζL(s) =

∑∞j=1 l

sj

vvnnnnnnnnnnnnnnnnnnnnn

((PPPPPPPPPPPPPPPPPPPPPPPPP

dimM F is the abcissaof convergence of ζL

connections btwn–geometry ζL(s)–spectrum ζν(s)–dynamics ζ(s)

inner tube formula

D = dimM F[Lap]

))SSSSSSSSSSSSSSSSSS

ζν(s) = ζL(s) · ζ(s) V (ε) =∑

ω∈DL

cwε1−ω

uukkkkkkkkkkkkkkk

complex dimensionsDL = poles of ζL

iiSSSSSSSSSSSSSSSSSS

55kkkkkkkkkkkkkkk

dimM F = infσ ≥ 0...V (ε) = O(ε1−t), ε→ 0+

D = infσ ≥ 0...ζL(σ) <∞.

The sum for V (ε) is taken over DL. ζν is the spectral zeta fn; the Mellintransform of the eigenvalue counting function. ζL is the Mellin transform ofthe geometric counting function.

2

Example: the Cantor String

CS =

3−(n+1)

with multiplicities

w3−(n+1) = 2n.

l1

l2 l3

l4 l5 l6 l7

CS =

13 ,

19,

19,

127,

127,

127,

127, . . .

The geometric zeta function of a string

ζL (s) =∞∑

j=1

lsj =∑

l

wlls

encodes all this information:

ζCS(s) =

∞∑

n=0

2n3−(n+1)s =3−s

1 − 2 · 3−s.

DCS(s) = log3 4 + inp...p = 2π

log 3, n ∈ Z.

3

What Needs to Be Done

Need a zeta function for fractal subsets of Rd. Then:

geometric object, F ⊆ Rd,

F =⋃K

k=1 Φk(F )

fractal string,L = ρj∞j=1

ρj =inradius(Rj)

R = X\F =⋃

Rj

(geometric) zeta function,ζL(s) = ?

xxqqqqqqqqqqqqqqqqqqqqqqqqq

power toolsLLLLLLLLLL

&&LLLLLLLLLL

dimM F is the abcissaof convergence of ζL

relations withspectrum & dynamics

V (ε) =∑

ω∈DL

cwεd−ω

Strategy:

(1) Compute V (ε) via brute force methods, for a couple of simple (or atleast well-studied) sample objects.

(2) Get an idea of what ζL ought to be in the higher-dimensional case.(3) Apply the power tools of [FGNT] to check the result.

First sample object: the Koch curve.

4

K0

K1

K2

K4

K3

K

...

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

5

Goal: derive a formula for the ε-neighbourhood of theKoch 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

= vol2x ∈ Ω...d(x, ∂Ω) < ε

6

First, partition the ε-neighbourhood.

rectangles

rectangles

error

overlapping

wedges

fringe triangles

(from overlap)

Figure 3. An approximation to the inner ε-neighbourhood of the Koch curve;ε ∈ I2.

The level of refinement is based on ε in

In : =(

3−(n+3/2), 3−(n+1/2)]

=(

3−(n+1)/√

3, 3−n/√

3]

Figure 4. Another ε-neighbourhood of the Koch curve, for; ε ∈ I3.

7

rectangles

rectangles

error

overlapping

wedges

fringe triangles

(from overlap)

Count each type of piece:shape number volume(area)

rectangles rn = 4n ε3−n

wedges wn = 23(4n − 1) πε2/6

triangles un = 23(4n − 1) + 2 ε2

√3/2

fringe 4n 91−n√3/160

A preliminary formula is

V (ε) = V1(ε) + V2(ε) − V3(ε) + V4(ε),

whereV1(ε) := 4n · ε3−n

V2(ε) :=2

3(4n − 1) · πε

2

6

V3(ε) :=

(

2

3(4n − 1) + 2

)

· ε2√

3

2

V4(ε) :=

(

4

9

)n(

32√

3

5 · 25

)

8

Now: collect the error.

error Figure 5. Where the error lies - the bold region is not within ε of K.

How many such error blocks are there?

complete

partial

Figure 6. Error block formation

Some blocks are whole; others form as ε→ 0.

9

We count the error blocks ( cn are complete, pn are partial)

δ(ε) = cn + pnh

= c(ε) + p(ε)h(ε)

= ε−D6 4−x + ε−D

6 4−xh(ε) + 23h(ε) − 4

3

where h(ε) is some function indicating what portion ofthe partial block has formed.

0 ≤ h(ε) = h(ε

3

)

≤ µ < 1

We don’t know h(ε) explicitly, but we do know

h(ε) =∑

α∈Z

gαe2πiαx = g(x)

for x = log3

(

1ε√

3

)

.

Total error is (number of blocks) × (area in a block)

E(ε) = δ(ε)B(ε)

Compute the desired volume formula as

V (ε) = V (ε) − E(ε)

by converting everything into series expansion.

10

After a certain amount of calculations . . .

V (ε) = G1(ε)ε2−D −G2(ε)ε

2,

where

G1(ε) : =1

log 3

∑

n∈Z

(

an +∑

α∈Z

bngα

)

(−1)nε−inp

G2(ε) : =1

log 3

∑

n∈Z

(

σn +∑

α∈Z

τngα

)

(−1)nε−inp,

are periodic functions of multiplicative period 3, and

an = − 39/2

28(D−2+inp)+ 33/2

23(D−1+inp)+ π−33/2

23(D+inp)+ 1

2bn,

bn =

∞∑

m=1

3m+3/2(2m−2)!(4−32m+1)

24m+1(m−1)!m!(2m+1)(D−2m−1+iνp),

σn = log 3(

π3 + 2

√3)

δn0 − τn, and

τn =∞∑

m=1

3m+1/2(2m−2)!(1−32m+1)24m−3(m−1)!m!(2m+1)(−2m−1+iνp)

.

The gα are Fourier coefficients of function which countsthe error blocks (actually, it describes how much hasformed).

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However, the coefficients are not the interesting part.The formula for V (ε) contains all the complex dimen-sions! We rewrite as

V (ε) =∑

n∈Z

ϕnε2−D−inp +

∑

n∈Z

ψnε2−inp.

This gives the possible dimensions

D∂Ω = D + inp ...n ∈ Z ∪ inp ...n ∈ Z.

D 2 3 40−1−2−3−4

C

p

2p

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Notes on h(ε):

The least upper bound of h(ε) is µ = C(ε)/B(ε):

Figure 7. µ is the ratio C(ε)/B(ε).

Three essential properties:

(i) h(ε) oscillates multiplicatively,

(ii) h(εk) = limϑ→0−

h(εk + ϑ) = 0,

(iii) limϑ→0+

h(εk + ϑ) = µ,

where εk = 3−k√3. Compare to

h(ε) = µ · −[x] − x

µ

h(ε)

1 1

3 39 3

1

3√

µ

h(ε)~

1 1

3 39 3

1

3√

Figure 8. The Cantor-like function h and the approximation h.

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