A criterion for box-counting measurabilitymath.cornell.edu/~fractals/6/slides/Rock.pdf · A criterion for box-counting measurability John A. Rock Cal Poly Pomona [email protected] Featuring

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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).


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.


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)





εm,

so

NB(A, x)

xD≈ volm(Aε)

εm−D. (1)





εm, so

NB(A, x)

xD≈ volm(Aε)

εm−D. (1)


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)


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)


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.


Hey John...

Show everyone Kalin’s movies and ask:

“Which is measurable and which is not?”


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.


M3 = NB(Q, (√17/8)−1) = 3

M3 = NB(Q, (√17/8)−1) = 3


M4 = NB(Q, (1/2)−1) = 4

M4 = NB(Q, (1/2)−1) = 4


M5 = NB(Q, (√2/8)−1) = 8 = 4M2

M5 = NB(Q, (√2/8)−1) = 8 = 4M2


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 ).


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 ).


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)


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)


For Re(s) > DL we have

ζL(s) =

∞∑n=1

mnlsn = s

∫ ∞0

NL(x)x−s−1dx. (9)


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)


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)


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)


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)


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)


Counting functions over complex dimensions


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.


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)


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)


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)


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.


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.


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.


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.





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)






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)



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.







j=1 rsj = 1.


Nj=1, and













j=1 rsj = 1.


Nj=1, and













j=1 rsj = 1.


Nj=1, and













j=1 rsj = 1.


Nj=1, and








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)


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)


A first lattice approximation of D


A second lattice approximation of D


A third lattice approximation of D


A fourth lattice approximation of D


A fifth lattice approximation of D


A sixth lattice approximation of D


A final lattice approximation of D


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.


A criterion for box-counting measurabilitymath.cornell.edu/~fractals/6/slides/Rock.pdf · A criterion for box-counting measurability John A. Rock Cal Poly Pomona [email protected] Featuring

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