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A criterion for box-counting measurability
John A. Rock
Cal Poly [email protected]
Featuring various collaborative efforts of:P. Chau, R. Giza, M. Landeros, M. Lapidus, R. Morales,
C. Knox (nee Sargent), K. Kurianski (nee Dettmers), M. van Frankenhuijsen,E. Voskanian, K. Zaluzec, and D. Zubrinic.
6th Cornell Conference onAnalysis, Probability, and Mathematical Physics on Fractals
June 13th, 2017
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 1 / 33
Motivation
σ
it
0 11/2 log 23
1/3
. . .
2
4
8
x
x
x
1/9
1/27
1/81
Lapidus, van Frankenhuijsen ’97, ’06, ’13
If L is measurable, then M = 21−DLres(ζL(s);DL)
DL(1−DL).
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 2 / 33
Counting boxes
8
1
2
3
4
Figure: An example of NB(A, x) where x = ε−1.
Let A ⊆ Rm and ε > 0. The box-counting function of A, NB(A, ε−1), is
the maximum number of disjoint closed balls B(a, ε) with centers a ∈ Aof radius ε. We consider the range of NB(A, ε
−1) to be a strictlyincreasing (with x = ε−1) sequence of positive integers denoted (Mn)
∞n=1.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 3 / 33
A part of the motivation
For simplicity’s sake, we assume throughout that the box-countingdimension of A, D := dimB A, exists.
Motivation: With x = ε−1, we have NB(A, x) ≈volm(Aε)
εm, so
NB(A, x)
xD≈ volm(Aε)
εm−D. (1)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 4 / 33
A part of the motivation
For simplicity’s sake, we assume throughout that the box-countingdimension of A, D := dimB A, exists.
Motivation: With x = ε−1, we have NB(A, x) ≈volm(Aε)
εm,
so
NB(A, x)
xD≈ volm(Aε)
εm−D. (1)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 4 / 33
A part of the motivation
For simplicity’s sake, we assume throughout that the box-countingdimension of A, D := dimB A, exists.
Motivation: With x = ε−1, we have NB(A, x) ≈volm(Aε)
εm, so
NB(A, x)
xD≈ volm(Aε)
εm−D. (1)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 4 / 33
Box-counting content and measurability
Dettmers, Giza, Knox, Morales, R. ’17The upper and lower box-counting contents of a bounded set A ⊆ Rm aredefined, respectively, by
B∗(A) := lim supx→∞
NB(A, x)
xD, B∗(A) := lim inf
x→∞
NB(A, x)
xD, (2)
where D := dimB A.
If 0 < B∗(A) = B∗(A) <∞, then A is box-counting measurable and thebox-counting content of A is given by
B(A) := limx→∞
NB(A, x)
xD. (3)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 5 / 33
Box-counting content and measurability
Dettmers, Giza, Knox, Morales, R. ’17The upper and lower box-counting contents of a bounded set A ⊆ Rm aredefined, respectively, by
B∗(A) := lim supx→∞
NB(A, x)
xD, B∗(A) := lim inf
x→∞
NB(A, x)
xD, (2)
where D := dimB A.
If 0 < B∗(A) = B∗(A) <∞, then A is box-counting measurable and thebox-counting content of A is given by
B(A) := limx→∞
NB(A, x)
xD. (3)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 5 / 33
The unit interval [0, 1] is box-counting measurable
Figure: A plot of NB([0, 1], x)/x. Note that B([0, 1]) = 1/2.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 6 / 33
Hey John...
Show everyone Kalin’s movies and ask:
“Which is measurable and which is not?”
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 7 / 33
The “Quarter Fractal” Q
1x1x1 4x1/4x1/4 16x1/16x1/16
For the self-similar set Q, nicknamed the “Quarter Fractal”, we haveM1 = 1 (which is always the case for any bounded set A ⊆ Rm) andM2 = 2.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 8 / 33
M3 = NB(Q, (√17/8)−1) = 3
M3 = NB(Q, (√17/8)−1) = 3
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 9 / 33
M4 = NB(Q, (1/2)−1) = 4
M4 = NB(Q, (1/2)−1) = 4
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 10 / 33
M5 = NB(Q, (√2/8)−1) = 8 = 4M2
M5 = NB(Q, (√2/8)−1) = 8 = 4M2
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 11 / 33
Fractal strings, zeta functions, complex dimensions
Lapidus, van Frankenhuijsen ’97, ’06, ’13A fractal string L is a sequence of positive real numbers such that
L = (`j)∞j=1 “=” {ln : ln distinct with multiplicity mn, n ∈ N} (4)
where the lengths `j satisfy 0 < `j+1 ≤ `j ∀j and `j → 0.
The dimension DL, geometric zeta function ζL, and complex dimensionsDL of L are respectively given by
DL := inf{t ∈ R :
∑`tj <∞
}, (5)
ζL(s) :=∑
`sj =∑
mnlsn, (6)
DL(W ) := {ω ∈W ⊆ C : ζL has a pole at ω}, (7)
where Re(s) > DL and W is a suitable open region. If W = C, we writeDL for DL(W ).
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 12 / 33
Fractal strings, zeta functions, complex dimensions
Lapidus, van Frankenhuijsen ’97, ’06, ’13A fractal string L is a sequence of positive real numbers such that
L = (`j)∞j=1 “=” {ln : ln distinct with multiplicity mn, n ∈ N} (4)
where the lengths `j satisfy 0 < `j+1 ≤ `j ∀j and `j → 0.
The dimension DL, geometric zeta function ζL, and complex dimensionsDL of L are respectively given by
DL := inf{t ∈ R :
∑`tj <∞
}, (5)
ζL(s) :=∑
`sj =∑
mnlsn, (6)
DL(W ) := {ω ∈W ⊆ C : ζL has a pole at ω}, (7)
where Re(s) > DL and W is a suitable open region. If W = C, we writeDL for DL(W ).
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 12 / 33
Geometric counting functions
Lapidus, van Frankenhuijsen ’97, ’06, ’13For x > 0, the geometric counting function of a fractal string L is given by
NL(x) := #{j ∈ N : `−1j ≤ x} =∑
n∈N, l−1n ≤x
mn. (8)
Theorem (Lapidus, van Frankenhuijsen ’97, ’06, ’13)
For Re(s) > DL we have
ζL(s) =∞∑n=1
mnlsn = s
∫ ∞0
NL(x)x−s−1dx. (9)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 13 / 33
Geometric counting functions
Lapidus, van Frankenhuijsen ’97, ’06, ’13For x > 0, the geometric counting function of a fractal string L is given by
NL(x) := #{j ∈ N : `−1j ≤ x} =∑
n∈N, l−1n ≤x
mn. (8)
Theorem (Lapidus, van Frankenhuijsen ’97, ’06, ’13)
For Re(s) > DL we have
ζL(s) =
∞∑n=1
mnlsn = s
∫ ∞0
NL(x)x−s−1dx. (9)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 13 / 33
Fractals strings from counting boxes
8
m =
m =
m =
m =1
2
3
4
2
1
4
1
1
2
3
4
Figure: NB(Q, x) where x = ε−1.
Lapidus, R., Zubrinic ’13For A and n ∈ N, let m1 :=M2,mn :=Mn+1 −Mn (n ≥ 2), and
ln := (sup{x ∈ (0,∞) : NB(A, x) =Mn})−1. (10)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 14 / 33
Box-counting zeta functions and complex dimensions
Lapidus, R., Zubrinic ’13The box-counting fractal string LB of A is given by
LB := {ln : ln has multiplicity mn, n ∈ N}. (11)
The box-counting zeta function, dimension and complex dimensions of A,denoted ζB, DB, and DB are respectively given by
ζB := ζLB , (12)
DB := DLB , and (13)
DB := DLB . (14)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 15 / 33
Box-counting zeta functions and complex dimensions
Lapidus, R., Zubrinic ’13The box-counting fractal string LB of A is given by
LB := {ln : ln has multiplicity mn, n ∈ N}. (11)
The box-counting zeta function, dimension and complex dimensions of A,denoted ζB, DB, and DB are respectively given by
ζB := ζLB , (12)
DB := DLB , and (13)
DB := DLB . (14)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 15 / 33
Results from box-counting fractal strings
Proposition (Lapidus, R., Zubrinic ’13)
Let A be an infinite subset of Rm with box-counting fractal string LB andbox-counting function NB(A, x). Then for x ∈ (l−11 ,∞) \ (l−1n )n∈N,
NLB (x) = NB(A, x). (15)
Theorem (Lapidus, R., Zubrinic ’13)
Let A be a bounded infinite subset of Rm. Then
dimBA = DB. (16)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 16 / 33
Results from box-counting fractal strings
Proposition (Lapidus, R., Zubrinic ’13)
Let A be an infinite subset of Rm with box-counting fractal string LB andbox-counting function NB(A, x). Then for x ∈ (l−11 ,∞) \ (l−1n )n∈N,
NLB (x) = NB(A, x). (15)
Theorem (Lapidus, R., Zubrinic ’13)
Let A be a bounded infinite subset of Rm. Then
dimBA = DB. (16)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 16 / 33
Counting functions over complex dimensions
Theorem (Lapidus, van Frankenhuijsen ’97, ’06, ’13)
Let L be a fractal string such that DL(W ) consists entirely of simplepoles. Then, under certain growth conditions on ζL, we have
NL(x) =∑
ω∈DL(W )
xω
ωres(ζL(s);ω) + {ζL(0)}+R(x), (17)
where R(x) is an error term of small order and the term in braces isincluded only if 0 ∈W\DL.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 17 / 33
A criterion for box-counting measurability
Theorem (Giza, Knox, Kurianski, Morales, R. ’17)
Let A ⊆ Rm be a bounded set such that D = dimB A exists. Suppose ζBsatisfies certain growth conditions and has a sufficiently nice meromorphicextension. Then the following are equivalent:
1 D is the only complex dimension with real part D = DB, and it issimple.
2 NB(A, x) = B · xD + o(xD) as x→∞ for some positive constant B.
3 A is box-counting measurable with box-counting content B.
If any of the above conditions holds, then
B = B(A) =res(ζB(A, s);D)
D. (18)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 18 / 33
The Riemann zeta function?
Example (Lapidus, R., Zubrinic ’13)
For the unit interval I = [0, 1]× {0}, we have NB(I, x) = dx/2e (theceiling function of x/2). Hence, the box-counting zeta function of I is
ζB(s) =1
2s+
∞∑n=1
1
(2n)s=
1
2s+
1
2sζ(s), (19)
where ζ(s) is the Riemann zeta function.
Example (Dettmers, Giza, Knox, Morales, R. ’17)
Since res(ζ; 1) = 1 and the meromorphic extension of ζ to C does nothave any other pole with real part 1, we have
B(I) = 1/2 = res(ζB; 1)/1. (20)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 19 / 33
The Riemann zeta function?
Example (Lapidus, R., Zubrinic ’13)
For the unit interval I = [0, 1]× {0}, we have NB(I, x) = dx/2e (theceiling function of x/2). Hence, the box-counting zeta function of I is
ζB(s) =1
2s+
∞∑n=1
1
(2n)s=
1
2s+
1
2sζ(s), (19)
where ζ(s) is the Riemann zeta function.
Example (Dettmers, Giza, Knox, Morales, R. ’17)
Since res(ζ; 1) = 1 and the meromorphic extension of ζ to C does nothave any other pole with real part 1, we have
B(I) = 1/2 = res(ζB; 1)/1. (20)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 19 / 33
Strong separation
1x1x1 4x1/4x1/4 16x1/16x1/16
Let Φ = {ϕj}Nj=1 be a self-similar system with attractor F and scaling
ratios r = (rj)Nj=1. We say F is strongly separated if the ϕj(F ) are
pairwise disjoint and
δ := sup{α : d(x, y) > α, x ∈ ϕj(F ), y ∈ ϕk(F ), j 6= k} (21)
is finite and positive.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 20 / 33
Box-counting functions and strong separation
Lemma (Lalley ’88)
Let F ⊆ Rm be a strongly separated self-similar set. Then for any x > 0,
NB(F, x) =N∑j=1
NB(F, rjx) + L(x) (22)
where L(x) is an integer valued step function that vanishes for x > δ−1.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 21 / 33
Box-counting zeta functions and strong separation
Theorem (Sargent ’14)
Suppose Φ is a strongly separated self-similar system with attractorF ⊆ Rm and scaling ratios rj , n = 1, . . . , N . Let LB be the box-countingfractal string of F with first length `1. Then
ζB(s) =h(s)
1−∑N
j=1 rsj
(23)
where
h(s) := `s1
N∑j=1
(1− rsj )
+ s
∫ ∞`−11
L(x)x−s−1dx (24)
is an entire function.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 22 / 33
ζB and DB of the Quarter Fractal Q
Example (Lapidus, R., Zubrinic ’13; Sargent ’14)
For Q as above we have
ζB(s) =
(√2
2
)s
+
(√2/2)s
+(√
17/8)s
+ (1/2)s
1− 4 · 4−s. (25)
and
DB =
{1 +
2πiz
log 4: z ∈ Z
}. (26)
Q is not box-counting measurable since it has nonreal box-countingcomplex dimensions ω with real part Reω = 1.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 23 / 33
ζB and DB of the Quarter Fractal Q
Example (Lapidus, R., Zubrinic ’13; Sargent ’14)
For Q as above we have
ζB(s) =
(√2
2
)s
+
(√2/2)s
+(√
17/8)s
+ (1/2)s
1− 4 · 4−s. (25)
and
DB =
{1 +
2πiz
log 4: z ∈ Z
}. (26)
Q is not box-counting measurable since it has nonreal box-countingcomplex dimensions ω with real part Reω = 1.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 23 / 33
ζB and DB of the Quarter Fractal Q
Example (Lapidus, R., Zubrinic ’13; Sargent ’14)
For Q as above we have
ζB(s) =
(√2
2
)s
+
(√2/2)s
+(√
17/8)s
+ (1/2)s
1− 4 · 4−s. (25)
and
DB =
{1 +
2πiz
log 4: z ∈ Z
}. (26)
Q is not box-counting measurable since it has nonreal box-countingcomplex dimensions ω with real part Reω = 1.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 23 / 33
Lattice approximation of nonlattice self-similar sets
Theorem (Dettmers, Giza, Knox, Morales, R. ’17)
Suppose Φ is a strongly separated nonlattice self-similar system on Rm
with attractor F , scaling ratios r = (rj)Nj=1, and box-counting complex
dimensions D. Additionally, assume h(s) = 0 if and only if∑N
j=1 rsj = 1.
Then there is a sequence of lattice self-similar systems (ΦM )∞M=1 with,respectively, attractor FM , scaling ratios rM = (rM,j)
Nj=1, and
box-counting complex dimensions DM such that the following hold asM →∞:
rM → r componentwise (via Diophantine approximation);
FM → F in the Hausdorff metric; and
FM is strongly separated for large enough M .
Under additional hypotheses,
DM → D in the sense described in Chapter 3 ofLapidus, van Frankenhuijsen ’97, ’06, ’13.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 24 / 33
Lattice approximation of nonlattice self-similar sets
Theorem (Dettmers, Giza, Knox, Morales, R. ’17)
Suppose Φ is a strongly separated nonlattice self-similar system on Rm
with attractor F , scaling ratios r = (rj)Nj=1, and box-counting complex
dimensions D. Additionally, assume h(s) = 0 if and only if∑N
j=1 rsj = 1.
Then there is a sequence of lattice self-similar systems (ΦM )∞M=1 with,respectively, attractor FM , scaling ratios rM = (rM,j)
Nj=1, and
box-counting complex dimensions DM such that the following hold asM →∞:
rM → r componentwise (via Diophantine approximation);
FM → F in the Hausdorff metric; and
FM is strongly separated for large enough M .
Under additional hypotheses,
DM → D in the sense described in Chapter 3 ofLapidus, van Frankenhuijsen ’97, ’06, ’13.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 24 / 33
Lattice approximation of nonlattice self-similar sets
Theorem (Dettmers, Giza, Knox, Morales, R. ’17)
Suppose Φ is a strongly separated nonlattice self-similar system on Rm
with attractor F , scaling ratios r = (rj)Nj=1, and box-counting complex
dimensions D. Additionally, assume h(s) = 0 if and only if∑N
j=1 rsj = 1.
Then there is a sequence of lattice self-similar systems (ΦM )∞M=1 with,respectively, attractor FM , scaling ratios rM = (rM,j)
Nj=1, and
box-counting complex dimensions DM such that the following hold asM →∞:
rM → r componentwise (via Diophantine approximation);
FM → F in the Hausdorff metric; and
FM is strongly separated for large enough M .
Under additional hypotheses,
DM → D in the sense described in Chapter 3 ofLapidus, van Frankenhuijsen ’97, ’06, ’13.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 24 / 33
Lattice approximation of nonlattice self-similar sets
Theorem (Dettmers, Giza, Knox, Morales, R. ’17)
Suppose Φ is a strongly separated nonlattice self-similar system on Rm
with attractor F , scaling ratios r = (rj)Nj=1, and box-counting complex
dimensions D. Additionally, assume h(s) = 0 if and only if∑N
j=1 rsj = 1.
Then there is a sequence of lattice self-similar systems (ΦM )∞M=1 with,respectively, attractor FM , scaling ratios rM = (rM,j)
Nj=1, and
box-counting complex dimensions DM such that the following hold asM →∞:
rM → r componentwise (via Diophantine approximation);
FM → F in the Hausdorff metric; and
FM is strongly separated for large enough M .
Under additional hypotheses,
DM → D in the sense described in Chapter 3 ofLapidus, van Frankenhuijsen ’97, ’06, ’13.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 24 / 33
Lattice approximation of nonlattice self-similar sets
Theorem (Dettmers, Giza, Knox, Morales, R. ’17)
Suppose Φ is a strongly separated nonlattice self-similar system on Rm
with attractor F , scaling ratios r = (rj)Nj=1, and box-counting complex
dimensions D. Additionally, assume h(s) = 0 if and only if∑N
j=1 rsj = 1.
Then there is a sequence of lattice self-similar systems (ΦM )∞M=1 with,respectively, attractor FM , scaling ratios rM = (rM,j)
Nj=1, and
box-counting complex dimensions DM such that the following hold asM →∞:
rM → r componentwise (via Diophantine approximation);
FM → F in the Hausdorff metric; and
FM is strongly separated for large enough M .
Under additional hypotheses,
DM → D in the sense described in Chapter 3 ofLapidus, van Frankenhuijsen ’97, ’06, ’13.
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 24 / 33
A (generic) nonlattice self-similar set
Figure: Constructing a strongly separated (generic) nonlattice self-similar set Swith scaling ratios 1/2, 1/3, and 1/6.
Let D denote the set of roots of the corresponding Moran equation
3∑j=1
rsj =1
2s+
1
3s+
1
6s= 1. (27)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 25 / 33
A (generic) nonlattice self-similar set
Figure: Constructing a strongly separated (generic) nonlattice self-similar set Swith scaling ratios 1/2, 1/3, and 1/6.
Let D denote the set of roots of the corresponding Moran equation
3∑j=1
rsj =1
2s+
1
3s+
1
6s= 1. (27)
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 25 / 33
A first lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 26 / 33
A second lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 27 / 33
A third lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 28 / 33
A fourth lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 29 / 33
A fifth lattice approximation of D
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A sixth lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 31 / 33
A final lattice approximation of D
John A. Rock (Cal Poly Pomona) A criterion for box-counting measurability June 13th, 2017 32 / 33
References
K. Dettmers (now Kurianski), R. Giza, C. Knox, R. Morales, andJ. A. Rock, A survey of complex dimensions, measurability, and thelattice/nonlattice dichotomy, Discrete Contin. Dyn. Syst. Ser. S (2)10 (2017), 213–240.
S. P. Lalley, Packing and covering functions of some self-similarfractals, Indiana Univ. Math. J. 37 (1988), 699–709.
M. L. Lapidus, J. A. Rock, and D. Zubrinic, Box-counting fractalstrings, zeta functions, and equivalent forms of Minkowski dimension,in: Fractal Geometry and Dynamical Systems in Pure and AppliedMathematics, Part 1, Contemporary Mathematics, Amer. Math. Soc.,Providence, RI, 2013.
C. Sargent (now Knox), Box-counting zeta functions of self-similarsets, Master’s thesis, Cal Poly Pomona, 2014.
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