complex dimensions of self-similar systems dissertation...
TRANSCRIPT
Complex dimensions of self-similar systems
Dissertation Defense
Erin P.J. Pearse
http://math.ucr.edu/∼erin/
March 14, 2006
References
[CDS] “Tube formulas and complex dimensions of self-similartilings”, M. L. Lapidus and E. P. J. Pearse, preprint, Dec.2005, 34 pages.
[SST] “Canonical self-similar tilings by IFS”, E. P. J. Pearse,preprint, Nov. 2005, 14 pages.
[KTF] “A tube formula for the Koch snowflake curve, with ap-plications to complex dimensions”, M. L. Lapidus and E.P. J. Pearse, J. London Math. Soc. (in press). arXiv:math-ph/0412029
[FGCD] “Fractal Geometry, Complex Dimensions and Zeta Func-tions: Geometry and Spectra of Fractal Strings”, M. L.Lapidus and M. van Frankenhuijsen, Springer-Verlag, 2006.
1
Fractal stringsThe 1-dimensional case: from [FGCD].
A fractal string L ⊆ R is a bounded open subset
L := {`n}∞n=1.
Translation invariance: `n ∈ R, and may assume
`1 ≥ `2 ≥ `3 ≥ . . .
Also assume `n > 0, or else trivial.
Idea: ∂L = F , where F ⊆ R is fractal.
The geometric zeta function ζL is a generatingfunction for the string
ζL(s) :=∑∞
n=1`sn.
The complex dimensions of L are
DL := {poles of ζL}.
2
Example: the Cantor String
l1l2 l3l4 l5 l6 l7
CS ={
13,
19,
19,
127,
127,
127,
127, . . .
}= [0, 1] \ C
∂(CS) = the Cantor set
CS has lk = 13k+1 with wlk
= 2k.
ζCS(s) =∑∞
k=02k3−(k+1)s =
3−s
1 − 2 · 3−s.
DCS(s) = {D +�np
... p = 2πlog 3, n ∈ Z}.
D = log3 2 = Minkowski dim of C.
VCS(ε) = 13 log 3
∑
ω∈DCS
(6ε)1−ω
ω(1−ω)− 2ε
3
0 1D◦
10p ◦
◦
◦
◦
◦
◦
◦
◦
◦
◦Figure 1. The complex dimensions of the Cantor string.D = log3 2 and p = 2π/ log 3. (Graphic from [FGCD])
4
Why call the poles of ζL the “complex dimensions”?First reason: relation to Minkowski/box dimension.
An (inner) tube formula for a set A ⊆ R is
VA(ε) = vol1{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 ... VA(ε) = O(ε1−t) as ε → 0+}.
A is said to be Minkowski measurable iff
M = limε→0+
V (ε)ε−(1−D)
exists, and has a value in (0,∞).
For A ⊆ Rd, replace 1 by d in VA, D, M.
Theorem (Lapidus).
D = inf{σ ≥ 0 ...∑∞
n=1`σn < ∞}.
In fact, DL ∩ R+ = {D}.
5
Why call the poles of ζL the “complex dimensions”?Second reason: relation to classical geometry.
The Steiner formula for ε-nbd of A ⊆ Kd:
VA(ε) =∑
i∈{0,1,...,d−1}ci ε
d−i.
The classical tube formula is summed over the inte-gral dimensions of A.
[FGCD] Tube formula for L:
VL(ε) =∑
ω∈DL∪{0,1}cω ε1−ω.
[New] Tube formula for T :
VT (ε) =∑
ω∈DL∪{0,1,...,d}cω εd−ω.
The fractal tube formula is summed over the inte-gral and complex dimensions of L or T .
6
rectangles
rectangles
error
overlapping
wedges
fringe
triangles
ε
(from overlap)
Figure 2. A neighbourhood
The Koch tube formula [KTF]
An ad-hoc computation, exploiting the lattice na-ture of the Koch snowflake curve.Computes the volume (area) of the ε-neighbourhood
of the curve itself; not a tiling.
VKS(ε) = G1(ε)ε2−D + G2(ε)ε
2,
where G1 and G2 are multiplicatively periodic func-tions
G1(ε) : =1
log 3
∑
n∈Z
(an +
∑
α∈Z
bαgn−α
)(−1)nε− � np
and G2(ε) : =1
log 3
∑
n∈Z
(σn +
∑
α∈Z
ταgn−α
)(−1)nε− � np,
an, bn, σn, and τn are the complex numbers givenby complicated Fourier series.
7
The self-similar case (in 1 dimension)
A self-similar system is Φ = {Φj}Jj=1, where each
Φj is a contractive similarity mapping:
Φj(x) = rjx + tj, 0 < rj < 1.
A set F is self-similar iff F = Φ(F ) =⋃J
j=1 Φj(F ).For any such Φ, ∃!F 6= ∅ and F is compact.
A string L is self-similar iff ∂L is a s-s set.
My dissertation extends this special case to higherdimensions by focusing on Φ.
Other special aspects of self-similar strings:
(1) closed form for ζL,(2) lattice/nonlattice dichotomy.
These are true for higher-dimensional case, too:
8
(1) The self-similar case: geometric zeta function.
Let I = [L] be the convex hull of L.Let L = ζL(1) = vol1(I) be the length of I .
The connected components of int(I) \Φ(I) are thegenerators g1 ≥ · · · ≥ gQ of L.
Theorem. [FGCD]
ζL(s) =Ls∑Q
q=1 gsq
1 −∑J
j=1 rsj
, s ∈ C.
Example: Cantor String CS .
Φ1(x) = x3 , Φ2(x) = x+2
3 .
I = [0, 1] L = 1 g = 13
int(I) \⋃
Φj(I) = (13,
23)
ζCS(s) =3−s
1 − 2 · 3−s.
9
(2) The self-similar case: lattice/nonlattice dichotomy.
(i) Lattice string : all scaling ratios are integral
powers of some fixed number: rj = rkj , kj ∈ Z+.
〈r1, . . . , rJ〉 is a discrete subgroup of R×.
(ii) Nonlattice string : some scaling ratios have ra-tionally independent logarithms.
〈r1, . . . , rJ〉 is a dense subgroup of R×.
Theorem (Structure of DL). [FGCD]s = D is the only real pole of ζL. Dl ≤ Re s ≤ D.
(i) Lattice: DL lies on finitely many vertical lines,periodically distributed. L is not measurable.
(ii) Nonlattice: only D has Re s = D, D ⊆ {Re s},quasiperiodically distributed. L is measurable.
Nonlattice strings can be approximated by latticestrings
10
0 1D
100
700
◦◦ ◦ ◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦
Figure 3. A nonlattice example: the Golden String. This string has
r1 = 2−1, r2 = 2−φ, φ = 1+√
5
2. (Graphic from [FGCD])
ζGS(s) =1
1 − 2−s − 2−φs
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φ≈ 2/11−1
p
50
◦◦◦◦◦◦◦◦◦
◦◦◦◦◦◦◦◦
φ≈ 3/21−1
p
50
◦◦◦◦
◦◦◦◦
◦◦◦◦
φ≈ 5/31−1
p50
◦◦
◦◦◦
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φ≈ 34/211−1
p
100◦
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φ≈ 8/51−1
p
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φ≈ 13/81−1
p
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φ≈ 21/131−1
p
100
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φ≈ 55/341−1
p
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Figure 4. Seven stages of approximating the complex dimensionsof the Golden string. Emergence of the quasiperiodic pattern.(Graphic from [FGCD])
12
Generalized fractal strings
Consider L := {`n}∞n=1 as a measure
η =∑∞
n=1δ1/`n.
Then the geometric zeta function is
ζη(s) =
∫ ∞
0x−sdη
D and D are defined as before.
Advantage: measures are more flexible.
〈η, ϕ〉 =
∫ ∞
0ϕ(x)dη.
Distributional explicit formulas
〈η, ϕ〉 =∑
ω∈Dη
res(ζη(s)ϕ̃(s); ω
)+ 〈R, ϕ〉.
13
Screen and Window
To obtain the distributional explicit formulas in[FGCD], ζη must satisfy some growth bounds.
S(t) is a bounded, Lipschitz continuous functionon R. The screen is
S := {S(t) +�t ... t ∈ R}.
Suppose {Tn}n∈Z has
Tn ↗ ∞, T−n ↘ −∞, and limn→∞
Tn|T−n| = 1.
η (or ζη) is languid iff ∃M,C > 0 with
L1 |ζη(σ +�Tn)| ≤ C·(|Tn| + 1)M ,∀σ ≤ f (Tn),
L2 |ζη(f (t) +�t)| ≤ C·|t|M ,∀|t| ≥ 1,
or strongly languid if also there is A s.t.
L2’ |ζη(fm(t) +�t)| ≤ CA|fm(t)|(|t| + 1)M ,
where {fm} has sup fm → −∞.
Self-similar strings are always strongly languid.
14
The self-similar tiling: extension to higherdimensions
Construct the self-similar tiling and produce asystem of tiles T = {Rn} = {Φw(Gq)}.
string L = {`n} tiling T = {Rn}length `n inradius ρn
ζL =∑
`sn ζs =∑
rsw
DL = {poles of ζL} Ds = {poles of ζs}
Idea: decompose the complement of the attractorF within its convex hull [F ].
1. Find the attractor F of Φ.
2. Take the (closed) convex hull C = [F ].
3. Define the generators Gq to be the connectedcomponents of relint(C) \ Φ(C).Note: Φ(C) ⊆ C.
4. The sets {Φw(Gq)} form a tiling of C \ F .
15
Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, ξ = 1
2+ �
2√
3.
BC0 = C
C1
C2
C3
C4
C5
C6
C7
B1
B2
B3
B4
B5
B6
B7
Figure 5. The Koch tiling, with unique generator G = B1.
Figure 6. The tiles exhaust the complement of the Koch curve within itsconvex hull.
16
Tube formula of a self-similar tilingObtain a tube formula for T using the idea
VT (ε) = 〈η, vε〉 =
∫ ∞
0vε(x) dηg(x).
vε(x) is the volume of the ε-neighbourhood of a tilewith inradius 1/x; i.e., the contribution of an tile atscale r−1
w .
The support of ηs(x) indicates which scales occur.The support of ηg(x) indicates which inradii occur.
Obtain a tube formula
VT (ε) =∑
ω∈DT
cω εd−ω
for DT = Ds ∪ {0, 1, . . . , d}.
17
Contributions to VT (ε) =
∫ ∞
0vε(x) dηg(x)
GG
ε
g
(g−ε)
3(g−ε)
g
3g
g−ε g−ε
g−ε
√3g√3
Figure 7. The volume VG(ε) = vε(1/g) of the generator of the Koch tiling.
1
31/21
2
4
ηs(x)
x3
Figure 8. The Koch scaling measure.
4
21
ηg(x)
x1/g
vε(x)
vε(1/g) v
ε(√3/g) v
ε(3/g)
/g√3 3/g
ε
Figure 9. The Koch geometric measure and contributions to the integral.
18
(a) (b) (c)
ξ
0 1
ξ
0 1
ξ
0 1
Figure 10. Φ1(z) := ξz, Φ2(z) := (1 − ξ)(z − 1) + 1, ξ = 1
2+ �
2√
3.
C1
C2
C3
C4
C5
B1
B2
B3
B4
B5
C1
C2
C3
C4
C5
B1
B2
B3
B4
B5
C0
C1
C2
C3
B
B1
B2
B3
19
C0 C1 C2 C3 C4 C5
B B1 B2 B3 B4 B5
Figure 11. The Sierpinski gasket tiling. The equilateral triangleG = B1 is the unique generator of this tiling.
ζs(s) =1
1 − 3 · 2−s, Ds = {log2 3 + � np ... p = 2π
log 2}n∈Z.
vε(x) =
{33/2
(−ε2 + 2εgx
), ε < g
33/2g2x2, ε ≥ g.
ζSG(s) = gsζs(s) =gs
1 − 3 · 2−s, g = 1
4√
3,
κ(ε) = 33/2 [−1, 2,−1] ,
E(ε, s) =
[1
s,
1
s − 1,
1
s − 2
]ε2−s
VSG(ε) =√
316 log 2
∑
n∈Z
(− 1
D+ � np+ 2
D−1+ � np− 1
D−2+ � np
)(εg
)2−D− � np
+ 33/2
2 ε2 − 3ε.
20
C0 C1 C2 C3 C4
B B1 B2 B3 B4
Figure 12. The Sierpinski carpet tiling. The square G = B1 is theunique generator of this tiling.
ζs(s) =1
1 − 8 · 3−s, Ds = {log3 8 + � np ... p = 2π
log 3}n∈Z
vε(x) =
{4(−ε2 + 2εx
), ε < g
4x2, ε ≥ g.
ζSC(s) = gsζs(s) =6−s
1 − 8 · 3−sg = 1
6,
κ(ε) = 4 [−1, 2,−1] ,
E(ε, s) =[
1s, 1
s−1, 1
s−2
]ε2−s
VSC(ε) = 136 log 3
∑
n∈Z
(− 1
D+ � np+ 2
D−1+ � np− 1
D−2+ � np
)(εg
)2−D− � np
+ 47ε
2 − 245 ε
21
C5
B5
C0 C1 C2 C3 C4
B B1 B2 B3B4
Figure 13. The Pentagasket tiling. This tiling has 6 genera-tors; one pentagon G1, and five congruent isoceles trianglesG2, . . . , G6. B1 = G1 ∪ G2 ∪ · · · ∪ G6.
C0 C1 C2
B B1 B2
Figure 14. The Menger Sponge tiling. This tiling has the uniquegenerator G = B1.
22
Q: What’s different in Rd?
A: The generators.
Φj(x) = rjAjx + tj, 0 < rj < 1.
New: Aj ∈ SO(d) is a rotation/reflection.
• Since generators may have different shapes, mustdistinguish between tile of different type, butwith the same inradius.
• Different generators may have different innertube formulas. (Intervals all have 2ε.)
Must distinguish scales from sizes: ηs vs. ηg.
ηs =∑
w∈W δr−1w
ηg = [ηs(x/gq)]Qq=1
ηgq =∑∞
n=1δρn(Gq)−1 =
∑w∈W δ(gqrw)−1
Must calculate each congruency class of generatorsseparately. Cannot integrate against the same mea-sure for each.
23
Q: What’s the same in Rd?
A: The scaling ratios.
Properties of r1, . . . , rJ are still of key importance.
ζs is formally identical to ζη for η with g = 1.
ζs(s) =∑
w∈Wrsw =
1
1 −∑Jj=1 rs
j
.
Ds depends entirely on r1, . . . , rJ .
Lattice/nonlattice dichotomy still holds.
The structure theorem for Ds is the same.
The distributional explicit formulas which appliedto η in the 1-dimensional case apply to each ηgq inthe d-dimensional case.
24
In 1 dimensionRecall the form of VL given earlier:
VL(ε) =∑
ω∈DL∪{0,1}cω ε1−ω.
[FGCD, Th. 8.1] The full form of VL:
VL(ε) =∑
ω∈DL\{0}
res(ζL(s);ω)ω(1−ω)
(2ε)1−ω
+ {2εζL(0)}0/∈DL.
Also: we can write (for L = 1)
ζL(s) =
∑Qq=1 gs
q
1 −∑J
j=1 rsj
= ζs(s)∑
gsq.
Not true in d dimensions. For multiple generatorsG1, . . . , GQ:
ζT (s) = [gs1, . . . , g
sq ]ζs(s).
25
In d dimensionsRecall the form of VT given earlier:
VT (ε) =∑
ω∈Ds∪{0,1,...,d}cω εd−ω.
[New] The full form of VL:
VT (ε) =∑
ω∈DT
res(〈ζT (s), E(ε, s)〉κ ; ω
).
When the residues and the product 〈ζT (s), E(ε, s)〉κare evaluated, these are the same.
• ζT is a CQ-valued generating function.
• DT = Ds∪{0, 1, . . . , d}, forDs = {poles of ζT }.
For sets with no fine structure,
ζT holomorphic =⇒ Ds = ∅
and we recover the “inner Steiner formula”.
26
Ingredients of the Tube Formula
The tiling zeta function is
ζT (s) = [ζ1(s), . . . , ζQ(s)]
with components gsqζs(s).
The curvature matrix is the Q × (d + 1) matrix
κ(ε) :=[κqi(ε)
]
κqi(ε) is the “ith curvature” of the qth generator.
The boundary terms compose a (d + 1)-vector
E(ε, s) :=[
1s,
1s−1, . . . ,
1s−d
]εd−s.
Then cω εω is a residue of the matrix product
ZT (ε, s) := 〈ζT (s), E(ε, s)〉κ = ζTT κ E
27
The “boundary term” vector E(ε, s)
VK(ε) =∑
n...ρn≥ε
VRn(ε) +
∑
n...ρn<ε
VRn(ρn)
=N−1∑
k=0
wkVrkG(ε) +∞∑
k=N
wkVrkG(ρk).
=N−1∑
k=0
2k33/2(2εgrk − ε2
)+
∞∑
k=N
2k33/2g2r2k
= 33/2
[(−ε2)
N−1∑
k=0
2k + 2εgN−1∑
k=0
2krk + g2
∞∑
k=N
2k(r2)k
]
= 33/2
[−ε2
(1 − 2N
1 − 2
)+ 2εg
(1 − (2r)N
1 − 2r
)+ g2
(2r2)N
1 − 2r2
]
= 33/2
[ε2
1−22N − 2εg
1−2r(2r)N + g2
1−2r2 (2r2)N]
+ 33/2ε2 + 2·33/2
1−2r εg
u = [u] + {u} =⇒ b[u] = bu−{u}
bN =(
εg
)logr b
b1−{logr ε/g} b = 2, 2r, 2r2
b−{u} =b − 1
b
∑
n∈Z
e2π � nu
log b + 2π � n
=b − 1
b log r
∑
n∈Z
(ε/g)− � np
D + � npD = logr b
(εg
)2−D
(bri)−{u} =b − 1
b log r
∑
ω∈D
(ε/g)2−ω
ω − iD = {D + � np}
28
Convex Geometry
The Steiner Formula for convex bodies A ∈ Kd.
VA(ε) =
d−1∑
i=0
(d
i
)Wd−i(A)εd−i
=
d−1∑
i=0
µd−i(Bd−i)µi(A)εd−i.
•Wi are the Minkowski functionals.
•Bi is the i-dimensional unit ball.
• µi are invariant/intrinsic measures.Homogeneous: µi(rA) = riµi(A),
Basic idea:
VA(ε) =∑
ω∈{0,1,...,d−1}cω εω
where cω is related to the curvature of A.
29
Curvature measures in convex geometry
VA(ε) =
d−1∑
i=0
µi(A)µd−i(Bd−i)εd−i.
For convex bodies A ∈ Kd,
κi(A) = d · µi(A)µd−i(Bd−i) =
(d
i
)Ci(A).
Here, the Ci are curvature measures and the κi arecurvatures (as def’d in convex/integral geometry).
The κi are homogeneous and translation invariantbecause the µi are.
Ci(A) is the total curvature of A; a special caseof the generalized curvature measure
Ci(A) := Ci(A, Rd) = Θi(A, Rd × Sd−1).
Θi is defined on U (Kd) × B(Σ), where U (Kd) isthe ring of polyconvex sets of dimension ≤ d.
30