An Introduction to the Theory of Complex Dimensions and Fractal Zeta - - PowerPoint PPT Presentation

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

An Introduction to the Theory of Complex Dimensions and Fractal Zeta Functions

Michel L. Lapidus

University of California, Riverside Department of Mathematics http://www.math.ucr.edu/∼lapidus/ lapidus@math.ucr.edu 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Cornell University Ithaca NY

June 13, 2017

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Figure 1: The rain of complex dimensions falling from the music of the angel’s fractal harp (or fractal string).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

1

Definitions and Motivations Minkowski Content and Box Dimension Singularities of Functions

2

Fractal strings Zeta Functions of Fractal Strings

3

Distance and Tube Zeta Functions Definition Analyticity Residues of Distance Zeta Functions Residues of Tube Zeta Functions (α, β)-chirps Meromorphic Extensions of Fractal Zeta Functions

4

Relative Distance and Tube Zeta Functions

5

References

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Goals Introducing a new class of fractal zeta functions: distance and tube zeta functions associated with bounded fractal sets in Euclidean spaces of arbitrary dimensions. Developing a higher-dimensional theory of complex fractal dimensions valid for arbitrary compact sets (and eventually, for suitable metric measure spaces). Merging of aspects of complex, spectral and harmonic analysis, geometry, and number theory of fractal sets in RN.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Main References for this Talk: Background Material: M. L. Lapidus, M. van Frankenhuijsen†: Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, research monograph, second revised and enlarged edition (of the 2006 edition), Springer, New York, 2013, 593 pages. ([L-vF]) New Results: M. L. Lapidus, G. Radunovi´ c‡, D. ˇ Zubrini´ c‡, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2017, 684 pages. ([LRˇ Z]) (And nine related papers by the authors of [LRˇ Z]; see the bibliography.) † of Utah Valley University, USA ‡ of the University of Zagreb, Croatia

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Book in Preparation:

• M. L. Lapidus, Complex Fractal Dimensions, Quantized Number

Theory and Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality, book in preparation, approx. 350 pages, 2017.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

(a) Fractal stalagmites (b) Fractal stalactites Figure 2: Stalagmites and stalactites in a fractal cave

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Figure 3: Other fractal stalagmites

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Minkowski Content and Box Dimension

Minkowski Content Let A ⊂ RN be a nonempty bounded set. ε-neighborhood of A, Aε = {y ∈ RN : d(y, A) < ε}. Lower s-dimensional Minkowski content of A, s ≥ 0: Ms

∗(A) := lim inf ε→0+

|Aε| εN−s , where |Aε| is the N-dimensional Lebesgue measure of Aε. Upper s-dimensional Minkowski content of A: M∗s(A) := lim sup

ε→0+

|Aε| εN−s .

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Minkowski Content and Box Dimension

Box Dimensions Lower box dimension of A: dimBA = inf{s ≥ 0 : Ms

∗(A) = 0}.

Upper box dimension of A: dimBA = inf{s ≥ 0 : M∗s(A) = 0}. dimH A ≤ dimBA ≤ dimBA ≤ N, where dimH A denotes the Hausdorff dimension of A. If dimBA = dimBA, we write dimB A, the box dimension of A. If there is d ≥ 0 such that 0 < Md

∗(A) ≤ M∗d(A) < ∞,

we say A is Minkowski nondegenerate. Clearly, d = dimB A. If |Aε| ≍ εσ for all sufficiently small ε and some σ ≤ N, then A is Minkowski nondegenerate and dimB A = N − σ.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Minkowski Content and Box Dimension

Minkowski Measurable and Nondegenerate Sets If Ms

∗(A) = M∗s(A) for some s, we denote this common value by

Ms(A) and call it the s-dimensional Minkowski content of A. Furthermore, if Md(A) ∈ (0, ∞) for some d ≥ 0, then A is said to be Minkowski measurable, with Minkowski content Md(A) (often simply denoted by M). Clearly, we then have d = dimB A. Example (The Cantor set) The triadic Cantor set A has box dimension dimB A = log 2/ log 3. A is not Minkowski measurable (Lapidus & Pomerance, 1993). Example (a-set) Let A = {k−a : k ∈ N} be the a-set, for a > 0. Then dimB A = 1/(1 + a) and A is Minkowski measurable (L., 1991).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Minkowski Content and Box Dimension

The Triadic Cantor Set

|Aε| ε1−d = G(log3 ε−1) + O(εd) as ε → 0+. Here, d = log3 2.

ε Md

∗(A)

M∗d(A) Graph of ε → G(log3 ε−1)

1 6 1 6·3 1 6·32 1 6·33

Figure 4: The oscillating nature of the function ε → |Aε|/ε1−d near ε = 0 for the triadic Cantor set A, with d := dimB A = log3 2. Then, A is Minkowski nondegenerate, but is not Minkowski measurable [Lapidus & Pomerance, 1993]. The function G(τ) is log 3-periodic. (See [Lapidus & van Frankenhuijsen, 2000, 2006 & 2013] for much more detailed information.)

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

Singular Function Generated by the Cantor Set

1 1/3 2/3 y = d(x, A) Figure 5: The graph of the distance function x → d(x, A), where A is the classic ternary (or triadic) Cantor set C (1/3).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

Singular Function Generated by the Cantor Set 1 1/3 2/3 y = d(x, A)−γ

Figure 6: For the triadic Cantor set A, the function y = d(x, A)−γ, x ∈ (0, 1), is Lebesgue integrable if and only if γ < 1 − log 2/ log 3. Here, dimH A = dimB A = log 2/ log 3, where dimH A denotes the Hausdorff dimension of A.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

The Sierpinski carpet A (two iterations are shown); dimH A = dimB A = log 8 log 3, A is Minkowski nondegenerate, but not Minkowski measurable (L., 1993).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

Figure 7: The classic self-similar Sierpinski carpet

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

(a) The H¨

• lder case

(b) The Lipschitz case Figure 8: The Sierpinski stalagmites The graph of f (x) = d(x, A)r, where r ∈ (0, 1) or r ≥ 1, respectively. Here, r = 0.5 (a) or r = 1.3 (b).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

Figure 9: Fractal stalagmites associated with the Sierpinski carpet.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

The figure on the previous slide depicts the graph of the distance function y = d(x, A), defined on the unit square, where A is the Sierpinski carpet. Only the first three generations of the countable family of pyramidal tents (called stalagmites) are shown.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

Figure 10: Fractal stalactites associated with the Sierpinski carpet.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Singularities of Functions

The figure on the previous slide shows the graph of the function y = d(x, A)−γ, defined on the unit square, where A is the Sierpinski carpet. Since A is known to be Minkowski nondegenerate, this function is Lebesgue integrable if and only if γ ∈ (−∞, 2 − D), D = dimB A = log3 8. For γ > 0, the graph consists of countably many connected components, called stalactites, all of which are unbounded.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Zeta Functions of Fractal Strings

Definition of Fractal Strings L = (ℓj)j≥1 a fractal string (L., 1991, L. & Pomerance, 1993): a nonincreasing sequence of positive numbers (ℓj) such that

• j ℓj < ∞. [Alternatively, L can be viewed as a sequence of

scales or as the lengths of the connected components (open intervals) of a bounded open set Ω ⊂ R.] The zeta function (geometric zeta function) of the fractal string L is the Dirichlet series: ζL(s) =

∞

• j=1

(ℓj)s, for s ∈ C with Re s large enough.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Zeta Functions of Fractal Strings

Consider the open intervals Ij = (aj, aj−1) for j ≥ 1, where aj :=

• k>j

ℓk, and ℓj := |Ij|. Define A = {aj}. Then A is a bounded set, A ⊂ R, and aj → 0 as j → ∞. The set A = AL is introduced in [LRˇ Z]. dimBA is defined via the upper Minkowski content of A, as usual.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Zeta Functions of Fractal Strings

Theorem (L) The abscissa of convergence of ζL is equal to dimBA : dimBA = inf

• α > 0 :

∞

• j=1

(ℓj)α < ∞

• .

Theorem (L, L-vF) ζL(s) is holomorphic on the right half-plane {Re s > dimBA}; The lower bound dimBA is optimal, both from the point of view of the absolute convergence and the holomorphic continuation. Moreover, if s ∈ R and s → dimBA from the right, then ζL(s) → +∞.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Zeta Functions of Fractal Strings

Corollary The abscissa of holomorphic continuation and the abscissa of (absolute) convergence of ζL both coincide with the (upper) Minkowski dimension of L : Dhol(ζL) = D(ζL) = dimBA. Remarks: In the above discussion, A could be replaced by ∂Ω, the boundary of any geometric realization of L by a bounded open set Ω ⊂ R.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Definition

Definition of the Distance Zeta Function Let A ⊂ RN be an arbitrary bounded set, and let δ > 0 be fixed. As before, Aδ denotes the δ-neighborhood of A. Definition (L., 2009; LRˇ Z, 2013) The distance zeta function of A is defined by ζA(s) =

• Aδ

d(x, A)s−Ndx, for s ∈ C with Re s sufficiently large.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Definition

Remarks: For s ∈ C such that Re s < N, the function d(x, A)s−N is singular on A. The inequality δ < δ1 implies that ζA(s; Aδ1) − ζA(s) =

• Aδ,δ1

d(x, A)s−Ndx is an entire function. As a result, the definition of ζA = ζA(·, Aδ) does not depend on δ in an essential way. In particular, the complex dimensions of A (i.e., the poles of ζA) do not depend on δ.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Definition

Zeta Function of the Set A Associated to a Fractal String L Let L = (ℓj) be a fractal string, and A = (aj), aj =

k≥j ℓk.

We would like to compare ζL(s) =

j(ℓj)s and

ζA(s) = 1+δ

−δ

d(x, A)s−1dx, for δ ≥ ℓ1/2. The zeta functions of L and A are ‘equivalent’, ζA(s) ∼ ζL(s), in the following sense: ζA(s) = a(s)ζL(s) + b(s), where a(s) and b(s) are explicit functions which are holomorphic on {Re s > 0}, and a(s) = 0 for all such s. It follows that (when they exist) the meromorphic extensions of ζA(s) and ζL(s) have the same sets of poles in {Re s > 0} (i.e., the same set of visible complex dimensions up to 0).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Definition

Definition The complex dimensions of a fractal string L (L & vF, 1996) are defined as the poles of ζL. Definition The complex dimensions of a bounded set A ⊂ RN are defined as the poles of ζA (L., 2009; LRˇ Z, 2013). Remark: We assume here that the zeta functions involved have a meromorphic extension (necessarily unique, by the principle of analytic continuation) to some suitable region U ⊂ C. Visible complex dimensions: the poles of ζA in U.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Analyticity

Holomorphy Half-Plane of the Distance Zeta Function Let A be a nonempty bounded set in RN; given δ a fixed positive number, let ζA(s) =

• Aδ d(x, A)s−Ndx as before.

Theorem (LRˇ Z) ζA(s) is holomorphic on the right half-plane {Re s > dimBA}; the lower bound dimBA is optimal from the point of view of the convergence of the Lebesgue integral defining ζA. Moreover, if D = dimB A exists, D < N, and MD

∗ (A) > 0,

then ζA(s) → +∞ as s ∈ R and s → D+; so that the lower bound dimBA is also optimal from the point of view of the holomorphic continuation. Remark: If s ∈ R and s < dimBA, then ζA(s) = +∞.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Analyticity

Corollary (LRˇ Z) The abscissa of (absolute) convergence of ζA is equal to dimBA, the (upper) Minkowski dimension of A: D(ζA) := inf

• α ∈ R :
• Aδ

d(x, A)α−Ndx < ∞

• = dimBA.

Corollary (LRˇ Z) Assume that D = dimB A exists, D < N, and MD

∗ (A) > 0. Then

the abscissa of holomorphic continuation and the abscissa of (absolute) convergence of ζA both coincide with the (upper) Minkowski dimension of A : Dhol(ζA) = D(ζA) = dimBA.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Analyticity

Definition The abscissa of holomorphic continuation of ζA is given by Dhol(ζA) := inf{α ∈ R : ζA(s) is holomorphic on Re s > α} Furthermore, the open half-plane {Re s > Dhol(ζA)} is called the holomorphy half-plane of ζA, while {Re s > D(ζA)} is called the half-plane of (absolute) convergence of ζA. Remark: In general, we have Dhol(ζA) ≤ D(ζA) = dimBA.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Distance Zeta Functions

Residue of the Distance Zeta Function at D = dimB A We assume that ζA(s) = ζA(s, Aδ) can be meromorphically extended to a neighborhood of D := dimB A, and D < N. We write ζA or ζA(·, Aδ), interchangeably. Theorem (LRˇ Z) If M∗D(A) < ∞, then s = D is a simple pole of ζA and (N − D)MD

∗ (A) ≤ res(ζA(·, Aδ), D) ≤ (N − D)M∗D(A).

The value of res(ζA(·, Aδ), D) does not depend on δ > 0. Remark: For the triadic Cantor set, we have strict inequalities.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Distance Zeta Functions

Theorem (LRˇ Z) If A is Minkowski measurable (i.e., MD(A) exists and MD(A) ∈ (0, ∞)), then res(ζA(·, Aδ), D) = (N − D)MD(A).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

Tube Zeta Function of a Fractal Set Let A ⊂ RN be an arbitrary bounded set, and let δ > 0 be fixed. Definition The tube zeta function of A associated with the tube function t → |At|, is given by (for some fixed, small δ > 0) ˜ ζA(s) = δ ts−N−1|At| dt, for s ∈ C with Re s sufficiently large. Remark: The choice of δ is unimportant, from the point of view of the theory of complex dimensions. Indeed, changing δ amounts to adding an entire function to ˜ ζA.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

The next result follows from its counterpart stated earlier for the distance zeta function ζA (see the sketch of the proof given below): Corollary (LRˇ Z) If D = dimB A exists, D < N and ˜ ζA has a meromorphic extension to a neighborhood of s = D, then MD

∗ (A) ≤ res(˜

ζA, D) ≤ M∗D(A). In particular, if A is Minkowski measurable, then res(˜ ζA, D) = MD(A).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

The proof of the previous corollary rests on the following identity, which is valid on {Re s > D}, where D = dimBA: ζA(s, Aδ) = δs−N|Aδ| + (N − s)˜ ζA(s). Remark: It follows from the above equation that if D < N, then ˜ ζA has a meromorphic extension to a given domain U ⊂ C iff ζA

• does. In particular, ˜

ζA and ζA have the same (visible) complex dimensions; that is, the same poles within U.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

Residues of Fractal Zeta Functions of Generalized Cantor Sets Example (1) For the generalized Cantor sets A = C (a), a ∈ (0, 1/2), we have D(a) = dimB C (a) = log1/a 2. Moreover, MD

∗ (A)

= 1 D 2D 1 − D 1−D , M∗D(A) = 2(1 − a) 1 2 − a D−1 , and res(˜ ζA, D) = 2 log 2 1 2 − a D .

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

Example (1 continued) For all a ∈ (0, 1/2), we have MD

∗ (A) < res(˜

ζA(s), D) < M∗D(A). Also, res(ζA, D) = (1 − D) res(˜ ζA, D). Remark: With this notation, the classic ternary Cantor set is just C (1/3). For any a ∈ (0, 1/2), the generalized Cantor set C (a) is constructed in much the same way as C (1/3), by removing open “middle a-intervals” at each stage of the construction.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Residues of Tube Zeta Functions

Residues and Minkowski Contents for a-strings, a > 0 Example (2) The a-string associated with A := {k−a : k ∈ N}, a > 0, is given by L = (ℓj)j≥1, ℓj = j−a − (j + 1)−a. We have: D(a) = dimB A = 1 1 + a, res(˜ ζA, D) = MD(A) = 21−D D(1 − D)aD, res(ζA, D) = (1 − D)MD(A) = 21−DaD D , and res(ζL, D) = aD.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions (α, β)-chirps

Definition (1) Let α > 0 and β > 0. The standard (α, β)-chirp is the graph of y = xα sin x−β near the origin (here α = 1/2, β = 1):

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions (α, β)-chirps

Definition of the (α, β)-chirp Definition Let α > 0 and β > 0. The geometric (α, β)-chirp is the following countable union of vertical intervals in the plane (‘approximation’

• f the standard (α, β)-chirp):

Γ(α, β) =

• k∈N

{k−1/β} × (0, k−α/β).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions (α, β)-chirps

Distance Zeta Function of Geometric Chirps The distance zeta function of Γ(α, β) can be computed as follows: ζΓ(α,β)(s) ∼ 1 s − 1

∞

• k=1

k− α

β −(1+ 1 β )(s−1),

where, as before, we define ζA(s) ∼ f (s) ⇔ f (s) = a(s)ζA(s) + b(s), with a(s), b(s) holomorphic on {Re s > r}, for some r < dimBA, and a(s) = 0 for all such s. The series converges iff Re s > max{1, 2 − 1+α

1+β }; hence,

dimBΓ(α, β) = max{1, 2 − 1 + α 1 + β }. This is the analog of Tricot’s formula (which was originally proved for the standard (α, β)-chirp).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Minkowski Measurable Sets Theorem (LRˇ Z (Minkowski measurable case)) Given A ⊂ RN, assume that there exist α > 0, M ∈ (0, ∞) and D ≥ 0 such that the tube function t → |At| satisfies |At| = tN−D (M + O(tα)) as t → 0+. Then A is Minkowski measurable, and we have: dimB A = D, MD(A) = M, and D(˜ ζA) = D. Furthermore, ˜ ζA has a (unique) meromorphic extension to (at least) {Re s > D − α}. Moreover, the pole s = D is unique, simple, and res(˜ ζA, D) = M.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Remark: Provided D < N, the exact same results hold for ζA, the distance zeta function of A. Then, we have instead res(ζA, D) = (N − D)M.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Minkowski Nonmeasurable Sets Theorem (LRˇ Z (Minkowski nonmeasurable case)) Given A ⊂ RN, assume that there exist D ≥ 0, a nonconstant periodic function G : R → R with minimal period T > 0, and α > 0, such that |At| = tN−D G(log t−1) + O(tα)

• as t → 0+.

Then we have: dimB A = D, MD

∗ (A) = min G, M∗D(A) = max G, and D(˜

ζA) = D. Furthermore, ˜ ζA(s) has a (unique) meromorphic extension to (at least) {Re s > D − α}. The set of all (visible) complex dimensions of A (i.e., the poles of ˜ ζA) is given by P(˜ ζA) =

• sk = D + 2π

T ik : ˆ G0( k T ) = 0, k ∈ Z

• ;

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Minkowski Nonmeasurable Sets Theorem (. . . continued) they are all simple. Here, ˆ G0(s) := T

0 e−2πis·τG(τ) dτ.

For all sk ∈ P(˜ ζA), res(˜ ζA, sk) = 1

T ˆ

G0( k

T ). We have

| res(˜ ζA, sk)| ≤ 1 T T G(τ) dτ, lim

k→±∞ res(˜

ζA, sk) = 0. Moreover, res(˜ ζA, D) = 1 T T G(τ) dτ and MD

∗ (A) < res(˜

ζA, D) < M∗D(A). In particular, A is not Minkowski measurable.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Remarks: Under the assumptions of the theorem, the average Minkowski content M of A (defined as a suitable Cesaro logarithmic average of |Aε|/εN−D) exists and is given by res(˜ ζA, D) = M = 1 T T G(τ)dτ. Provided D < N, an entirely analogous theorem holds for ζA (instead of ˜ ζA), the distance zeta function of A, except for the fact that the residues take different values. In particular, res(ζA, D) = (N − D) M.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Minkowski Nonmeasurable Sets Example If A is the ternary Cantor set, we have (see [L-vF, 2000]) |At| = t1−DG(log t−1) as t → 0+, where D = log3 2 and the nonconstant function G is log 3-periodic: G(τ) = 21−D

• 2{ τ−log 2

log 3 } +

3 2 −{ τ−log 2

log 3 }

, where {x} := x − ⌊x⌋ is the fractional part of x and ⌊x⌋ is the integer part of x. Conclusion: ˜ ζA and ζA have a (unique) meromorphic extension to {Re s > D − α} for any α > 0, and hence to all of C.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions Meromorphic Extensions of Fractal Zeta Functions

Definition The principal complex dimensions of a bounded set A in RN are given by P(ζA) := dimC A ∩ {Re s = D(ζA)}, where dimC A denotes the set of (visible) complex dimensions of A. (Recall that D(ζA) = D(˜ ζA) = dimBA.) The vertical line {Re s = D(ζA)} is called the critical line. Remark: For the ternary Cantor set A, we have P(ζA) = dimC A = log3 2 + 2π log 3iZ.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Distance Zeta Functions of Relative Fractal Drums Definition A relative fractal drum is a pair (A, Ω) of nonempty subsets A and Ω (open subset) of RN, such that |Ω| < ∞ and there exists δ > 0 such that Ω ⊂ Aδ. Note that A and Ω may be unbounded. We do not assume A ⊆ Ω. Definition Let t ∈ R. Then the upper t-dimensional Minkowski content of A relative to Ω is given by M∗t(A, Ω) = limε→0+ |Aε ∩ Ω| εN−t .

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Definition The upper box dimension of the relative fractal drum (A, Ω) is given by dimB(A, Ω) = inf{t ∈ R : M∗t(A, Ω) = 0}. It may be negative, and even equal to −∞; this is related to the flatness of (A, Ω). (This latter concept is not discussed here; see [LRˇ Z] for details.)

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Analyticity of Relative Zeta Functions Definition (Relative distance zeta function of (A, Ω), LRˇ Z) ζA(s, Ω) =

• Ω

d(x, A)s−Ndx, for s ∈ C with Re s sufficiently large. Theorem (LRˇ Z) ζA(s, Ω) is holomorphic for Re s > dimB(A, Ω); the lower bound Re s > dimB(A, Ω) is optimal. Hence, the abscissa of convergence of ζA(·, Ω) is equal to dimB(A, Ω), the relative upper box dimension of (A, Ω). Assume that D = dimB(A, Ω) exists and MD

∗ (A, Ω) > 0.

Then, if s ∈ R and s → D+, we have ζA(s, Ω) → +∞.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Definition (Relative tube zeta function of (A, Ω), LRˇ Z) ˜ ζA(s, Ω) = δ ts−N−1|At ∩ Ω|dt, for s ∈ C with Re s sufficiently large. (Here, δ > 0 is fixed.) Remark: The above theorem is valid without change for ˜ ζA(·, Ω) (instead of ζA(·, Ω)). In particular, dimB(A, Ω) = the abscissa of convergence of ˜ ζA(·, Ω).

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Example (1) If A = {(x, y) : y = xα, 0 < x < 1} with α ∈ (0, 1) and Ω = (−1, 0) × (0, 1), then dimB(A, Ω) = 1 − α, which is < 1. Note that A and Ω are disjoint. Example (2) If A = {(0, 0)} (the origin in R2) and Ω = {(x, y) ∈ (0, 1) × R : 0 < y < xα} with α > 1, then dimB(A, Ω) = 1 − α, which is < 0. Note that Ω is flat at A.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Figure 11: The complex dimensions of a relative fractal drum.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

In the previous figure (Fig. ), the set of complex dimensions D of the relative fractal drum (A, Ω), obtained as a union of relative fractal drums {(Aj, Ωj)}∞

j=1 involving generalized Cantor sets.

Here, D = 4/5 and α = 3/10. Furthermore, D(˜ ζA(·, Ω)) = 4/5, Dmer(˜ ζA(·, Ω)) = D − α = 1/2 and 2−1 + 4π(log 2)iZ is the set of nonisolated singularities of ˜ ζA(·, Ω). The set D is contained in a union of countably many rays emanating from the origin. The dotted vertical line is the holomorphy critical line {Re s = D} of ˜ ζA(·, Ω), and to the left of it is the meromorphy critical line {Re s = D − α}.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Definition A function G : R → R is transcendentally quasiperiodic of infinite

• rder (resp., of finite order m) if it is of the form

G(τ) = H(τ, τ, . . .), where H : R∞ → R (resp., H : Rm → R) is a function which is Tj-periodic in its j-th component, for each j ∈ N (resp., for each j = 1, · · · , m), with Tj > 0 as minimal periods, and such that the set of quasiperiods {Tj : j ≥ 1} (resp., {Tj : j = 1, · · · , m}) is algebraically independent; i.e., is independent over the ring of algebraic numbers.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Definition A relative fractal drum (A, Ω) in RN is said to be transcendentally quasiperiodic if |At ∩ Ω| = tN−D(G(log(1/t)) + o(1)) as t → 0+, where the function G is transcendentally quasiperiodic. In the special case where A ⊆ RN is bounded and Ω = RN, then the set A is said to be transcendentally quasiperiodic.

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Hyperfractals Using Alan Baker’s theorem (in the theory of transcendental numbers) and generalized Cantor sets C (m,a) with two parameters (as in the above example), it is possible to construct a transcendentally quasiperiodic bounded set A in R with infinitely many algebraically independent quasiperiods. For this set, we show that ˜ ζA(s) has the critical line {Re s = D} as a natural boundary, where D = dimB A. (This means that ˜ ζA(s) does not have a meromorphic extension to the left of {Re s = D}.) Moreover, all of the points of the critical line {Re s = D} are singularities of ˜ ζA(s); the same is true for ζA(s) (instead of ˜ ζA(s)). (See [LRˇ Z] for the detailed construction.) The set A is then said to be (maximally) hyperfractal.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Remark The above construction of maximally hyperfractal and transcendentally quasiperiodic sets (and relative fractal drums) of infinite order has been applied in [LR˘ Z] in different contexts. In particular, it has been applied to prove that certain estimates

• btained by the first author and regarding the abscissae of

meromorphic continuation of the spectral zeta function of fractal drums are sharp, in general. This construction is also relevant to the definition of fractality given in terms of complex dimensions. Recall that in the theory of complex dimensions, an object is said to be “fractal” if it has at least one nonreal complex dimension (with positive real part) or else if the associated fractal zeta function has a natural boundary (along a suitable contour). This new higher-dimensional theory of complex dimensions now enables us to define fractality in full generality.

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Future Research Directions

• 1. Fractal tube formulas and geometric complex dimensions
• a. Case of self-similar sets
• b. Devil’s staircase (cf. the definition of fractality)
• c. Weierstrass function
• d. Julia sets and Mandelbrot set

Possible geometric interpretation: fractal curvatures (even for complex dimensions) Connections with earlier joint work of the author with Erin Pearse and Steffen Winter for fractal sprays and self-similar tilings. Added note: A general fractal tube formula has now been

• btained by the authors of [LR˘

Z]. This formula has been applied to a number of self-similar and non self-similar examples, including the Sierpinski gasket and carpet as well as their higher-dimensional

• counterparts. See [LR˘

Z, Chapter 5] and the relevant references in the bibliography.

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Future Research Directions

• 2. Spectral complex dimensions Determine the complex

dimensions of a variety of fractal drums (via the associated spectral functions) and compare these spectral complex dimensions with the geometric complex dimensions discussed in (1) just above.

• 3. Generalization to metric measure spaces or Ahlfors’ spaces

(joint work in progress with Sean Watson) Connections with nonsmooth geometric analysis and analysis on fractals

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

Future Research Directions

• 4. Box-counting zeta functions

(joint work in progress with John Rock and Darko ˇ Zubrini´ c)

• 5. Spectral zeta functions of relative fractal drums (spectral

complex dimensions) Connections with geometric fractal zeta functions and complex dimensions

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References

• M. L. Lapidus, M. van Frankenhuysen, Fractal Geometry and

Number Theory: Complex Dimensions of Fractal Strings and Zeros

• f Zeta Functions, research monograph, Birkh¨

auser, Boston, 2000, 280 pages.

• M. L. Lapidus, M. van Frankenhuijsen: Fractal Geometry, Complex

Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, research monograph, second revised and enlarged edition (of the 2006 edition), Springer, New York, 2013, 593 pages. ([L-vF])

• M. L. Lapidus, G. Radunovi´

c, D. ˇ Zubrini´ c, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2017, 684

• pages. ([LRˇ

Z])

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

References

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Distance and tube zeta functions of fractals and arbitrary compact sets, Adv. in Math. 307 (2017), 1215–1267. (dx.doi.org/10.1016/j.aim2016.11.034). (Also: e-print, arXiv:1506.03525v2 [math-ph], 2016; IHES preprint, M/15/15, 2015.)

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Complex dimensions

• f fractals and meromorphic extensions of fractal zeta functions, J.
• Math. Anal. Appl. No. 1, 453 (2017), 458–484.

(doi.org/10.1016/j.jmaa.2017.03.059.) (Also: e-print, arXiv:1508.04784v3 [math-ph], 2016.)

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Zeta functions and complex dimensions of relative fractal drums: Theory, examples and applications, Dissertationes Mathematicae, in press, 2017, 101

• pages. (Also: e-print, arXiv:1603.00946v3 [math-ph], 2016.)

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

References

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces, Discrete and Continuous Dynamical Systems Ser. S, in press, 2017. (Also: e-print, arXiv:1411.5733v4 [math-ph], 2016; IHES preprint, IHES/M/15/17, 2015.)

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Fractal tube formulas for compact sets and relative fractal drums: Oscillations, complex dimensions and fractality, Journal of Fractal Geometry, in press, 2017, 104 pages. (Also: e-print, arXiv:1604.08014v3 [math-ph], 2016.)

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

References

• M. L. Lapidus, G. Radunovi´

c and D. ˘ Zubrini´ c, Fractal zeta functions and complex dimensions of relative fractal drums, survey article, Journal of Fixed Point Theory and Applications No. 2, 15 (2014), 321–378. Festschrift issue in honor of Haim Brezis’ 70th

• birthday. (DOI: 10.1007/s11784-014-0207-y.) (Also: e-print,

arXiv:1407.8094v3[math-ph], 2014.)

Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functions

References

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Fractal zeta functions and complex dimensions: A general higher-dimensional theory, survey article, in: Fractal Geometry and Stochastics V (C. Bandt, K. Falconer and M. Z¨ ahle, eds.), Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, vol. 70, Birkh¨ auser/Springer Internat., Basel, Boston and Berlin, 2015, pp. 229–257; doi:10.1007/978-3-319-18660-3 13. (Based on a plenary lecture given by the first author at that conference.) (Also: e-print, arXiv:1502.00878v3 [math.CV], 2015; IHES preprint, IHES/M/15/16, 2015.)