A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, …pi.math.cornell.edu/~erin/docs/talk2.pdfA TUBE...
Transcript of A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, …pi.math.cornell.edu/~erin/docs/talk2.pdfA TUBE...
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/
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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.
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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.
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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.
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K0
K1
K2
K4
K3
K
...
Figure 1. The Koch curve K (left) and the Koch snowflake Ω (right).
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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
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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.
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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
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V3(ε) :=
(
2
3(4n − 1) + 2
)
· ε2√
3
2
V4(ε) :=
(
4
9
)n(
32√
3
5 · 25
)
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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.
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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
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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.
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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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