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

Transcription

1 An Introduction to the Theory of Complex Dimensions and Fractal Zeta Functions Michel L. Lapidus University of California, Riverside Department of Mathematics lapidus/ 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Cornell University Ithaca NY June 13, 2017

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

3 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

4 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 R N.

5 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. Radunović, D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2017, 684 pages. ([LRŽ]) (And nine related papers by the authors of [LRŽ]; see the bibliography.) of Utah Valley University, USA of the University of Zagreb, Croatia

6 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.

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

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12 Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions (a) Fractal stalagmites Relative Distance and Tube Zeta Functio (b) Fractal stalactites Figure 2: Stalagmites and stalactites in a fractal cave

13 Figure 3: Other fractal stalagmites

14 Minkowski Content and Box Dimension Minkowski Content Let A R N be a nonempty bounded set. ε-neighborhood of A, A ε = {y R N : d(y, A) < ε}. Lower s-dimensional Minkowski content of A, s 0: M s A ε (A) := lim inf ε 0 + ε N s, where A ε is the N-dimensional Lebesgue measure of A ε. Upper s-dimensional Minkowski content of A: M s (A) := lim sup ε 0 + A ε ε N s.

15 Minkowski Content and Box Dimension Box Dimensions Lower box dimension of A: dim B A = inf{s 0 : M s (A) = 0}. Upper box dimension of A: dim B A = inf{s 0 : M s (A) = 0}. dim H A dim B A dim B A N, where dim H A denotes the Hausdorff dimension of A. If dim B A = dim B A, we write dim B A, the box dimension of A. If there is d 0 such that 0 < M d (A) M d (A) <, we say A is Minkowski nondegenerate. Clearly, d = dim B A. If A ε ε σ for all sufficiently small ε and some σ N, then A is Minkowski nondegenerate and dim B A = N σ.

16 Minkowski Content and Box Dimension Minkowski Measurable and Nondegenerate Sets If M s (A) = M s (A) for some s, we denote this common value by M s (A) and call it the s-dimensional Minkowski content of A. Furthermore, if M d (A) (0, ) for some d 0, then A is said to be Minkowski measurable, with Minkowski content M d (A) (often simply denoted by M). Clearly, we then have d = dim B A. Example (The Cantor set) The triadic Cantor set A has box dimension dim B 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 dim B A = 1/(1 + a) and A is Minkowski measurable (L., 1991).

17 Minkowski Content and Box Dimension The Triadic Cantor Set A ε ε 1 d = G(log 3 ε 1 ) + O(ε d ) as ε 0 +. Here, d = log 3 2. M d (A) M d (A) Graph of ε G(log 3 ε 1 ) ε Figure 4: The oscillating nature of the function ε A ε /ε 1 d near ε = 0 for the triadic Cantor set A, with d := dim B A = log 3 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.) 1

18 Singularities of Functions Singular Function Generated by the Cantor Set y = d(x, A) 0 1/3 2/3 1 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).

19 Singularities of Functions Singular Function Generated by the Cantor Set y = d(x, A) γ 0 1/3 2/3 1 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, dim H A = dim B A = log 2/ log 3, where dim H A denotes the Hausdorff dimension of A.

20 Singularities of Functions The Sierpinski carpet A (two iterations are shown); dim H A = dim B A = log 8 log 3, A is Minkowski nondegenerate, but not Minkowski measurable (L., 1993).

21 Singularities of Functions Figure 7: The classic self-similar Sierpinski carpet

22 Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Functio Singularities of Functions (a) The Ho 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).

23 Singularities of Functions Figure 9: Fractal stalagmites associated with the Sierpinski carpet.

24 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.

25 Singularities of Functions Figure 10: Fractal stalactites associated with the Sierpinski carpet.

26 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 = dim B A = log 3 8. For γ > 0, the graph consists of countably many connected components, called stalactites, all of which are unbounded.

27 Zeta Functions of Fractal Strings Definition of Fractal Strings L = (l j ) j 1 a fractal string (L., 1991, L. & Pomerance, 1993): a nonincreasing sequence of positive numbers (l j ) such that j l j <. [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: ζ L (s) = (l j ) s, j=1 for s C with Re s large enough.

28 Zeta Functions of Fractal Strings Consider the open intervals I j = (a j, a j 1 ) for j 1, where a j := k>j l k, and l j := I j. Define A = {a j }. Then A is a bounded set, A R, and a j 0 as j. The set A = A L is introduced in [LRŽ]. dim B A is defined via the upper Minkowski content of A, as usual.

29 Zeta Functions of Fractal Strings Theorem (L) The abscissa of convergence of ζ L is equal to dim B A : dim B A = inf { α > 0 : } (l j ) α <. j=1 Theorem (L, L-vF) ζ L (s) is holomorphic on the right half-plane {Re s > dim B A}; The lower bound dim B A is optimal, both from the point of view of the absolute convergence and the holomorphic continuation. Moreover, if s R and s dim B A from the right, then ζ L (s) +.

30 Zeta Functions of Fractal Strings Corollary The abscissa of holomorphic continuation and the abscissa of (absolute) convergence of ζ L both coincide with the (upper) Minkowski dimension of L : D hol (ζ L ) = D(ζ L ) = dim B A. 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.

31 Definition Definition of the Distance Zeta Function Let A R N be an arbitrary bounded set, and let δ > 0 be fixed. As before, A δ denotes the δ-neighborhood of A. Definition (L., 2009; LRŽ, 2013) The distance zeta function of A is defined by ζ A (s) = d(x, A) s N dx, A δ for s C with Re s sufficiently large.

32 Definition Remarks: For s C such that Re s < N, the function d(x, A) s N is singular on A. The inequality δ < δ 1 implies that ζ A (s; A δ1 ) ζ A (s) = d(x, A) s N dx A δ,δ1 is an entire function. As a result, the definition of ζ A = ζ A (, A δ ) does not depend on δ in an essential way. In particular, the complex dimensions of A (i.e., the poles of ζ A ) do not depend on δ.

33 Definition Zeta Function of the Set A Associated to a Fractal String L Let L = (l j ) be a fractal string, and A = (a j ), a j = k j l k. We would like to compare ζ L (s) = j (l j) s and ζ A (s) = 1+δ δ d(x, A) s 1 dx, for δ l 1 /2. The zeta functions of L and A are equivalent, ζ A (s) ζ L (s), in the following sense: ζ A (s) = a(s)ζ L (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 ζ A (s) and ζ L (s) have the same sets of poles in {Re s > 0} (i.e., the same set of visible complex dimensions up to 0).

34 Definition Definition The complex dimensions of a fractal string L (L & vf, 1996) are defined as the poles of ζ L. Definition The complex dimensions of a bounded set A R N are defined as the poles of ζ A (L., 2009; LRŽ, 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 ζ A in U.

35 Analyticity Holomorphy Half-Plane of the Distance Zeta Function Let A be a nonempty bounded set in R N ; given δ a fixed positive number, let ζ A (s) = A δ d(x, A) s N dx as before. Theorem (LRŽ) ζ A (s) is holomorphic on the right half-plane {Re s > dim B A}; the lower bound dim B A is optimal from the point of view of the convergence of the Lebesgue integral defining ζ A. Moreover, if D = dim B A exists, D < N, and M D (A) > 0, then ζ A (s) + as s R and s D + ; so that the lower bound dim B A is also optimal from the point of view of the holomorphic continuation. Remark: If s R and s < dim B A, then ζ A (s) = +.

36 Analyticity Corollary (LRŽ) The abscissa of (absolute) convergence of ζ A is equal to dim B A, the (upper) Minkowski dimension of A: { } D(ζ A ) := inf α R : d(x, A) α N dx < A δ = dim B A. Corollary (LRŽ) Assume that D = dim B A exists, D < N, and M D (A) > 0. Then the abscissa of holomorphic continuation and the abscissa of (absolute) convergence of ζ A both coincide with the (upper) Minkowski dimension of A : D hol (ζ A ) = D(ζ A ) = dim B A.

37 Analyticity Definition The abscissa of holomorphic continuation of ζ A is given by D hol (ζ A ) := inf{α R : ζ A (s) is holomorphic on Re s > α} Furthermore, the open half-plane {Re s > D hol (ζ A )} is called the holomorphy half-plane of ζ A, while {Re s > D(ζ A )} is called the half-plane of (absolute) convergence of ζ A. Remark: In general, we have D hol (ζ A ) D(ζ A ) = dim B A.

38 Residues of Distance Zeta Functions Residue of the Distance Zeta Function at D = dim B A We assume that ζ A (s) = ζ A (s, A δ ) can be meromorphically extended to a neighborhood of D := dim B A, and D < N. We write ζ A or ζ A (, A δ ), interchangeably. Theorem (LRŽ) If M D (A) <, then s = D is a simple pole of ζ A and (N D)M D (A) res(ζ A (, A δ ), D) (N D)M D (A). The value of res(ζ A (, A δ ), D) does not depend on δ > 0. Remark: For the triadic Cantor set, we have strict inequalities.

39 Residues of Distance Zeta Functions Theorem (LRŽ) If A is Minkowski measurable (i.e., M D (A) exists and M D (A) (0, )), then res(ζ A (, A δ ), D) = (N D)M D (A).

40 Residues of Tube Zeta Functions Tube Zeta Function of a Fractal Set Let A R N be an arbitrary bounded set, and let δ > 0 be fixed. Definition The tube zeta function of A associated with the tube function t A t, is given by (for some fixed, small δ > 0) ζ A (s) = δ for s C with Re s sufficiently large. 0 t s N 1 A t dt, 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 ζ A.

41 Residues of Tube Zeta Functions The next result follows from its counterpart stated earlier for the distance zeta function ζ A (see the sketch of the proof given below): Corollary (LRŽ) If D = dim B A exists, D < N and ζ A has a meromorphic extension to a neighborhood of s = D, then M D (A) res( ζ A, D) M D (A). In particular, if A is Minkowski measurable, then res( ζ A, D) = M D (A).

42 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 = dim B A: ζ A (s, A δ ) = δ s N A δ + (N s) ζ A (s). Remark: It follows from the above equation that if D < N, then ζ A has a meromorphic extension to a given domain U C iff ζ A does. In particular, ζ A and ζ A have the same (visible) complex dimensions; that is, the same poles within U.

43 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) = dim B C (a) = log 1/a 2. Moreover, ( 2D ) 1 D, and M D (A) = 1 D 1 D ( ) 1 D 1 M D (A) = 2(1 a) 2 a, res( ζ A, D) = ( ) 2 1 D log 2 2 a.

44 Residues of Tube Zeta Functions Example (1 continued) For all a (0, 1/2), we have M D (A) < res( ζ A (s), D) < M D (A). Also, res(ζ A, D) = (1 D) res( ζ A, 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.

45 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 = (l j ) j 1, l j = j a (j + 1) a. We have: D(a) = dim B A = a, res( ζ A, D) = M D (A) = 21 D D(1 D) ad, and res(ζ A, D) = (1 D)M D (A) = 21 D a D D, res(ζ L, D) = a D.

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

47 (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 of the standard (α, β)-chirp): Γ(α, β) = k N{k 1/β } (0, k α/β ).

48 (α, β)-chirps Distance Zeta Function of Geometric Chirps The distance zeta function of Γ(α, β) can be computed as follows: ζ Γ(α,β) (s) 1 k α β (1+ 1 β )(s 1), s 1 where, as before, we define k=1 ζ A (s) f (s) f (s) = a(s)ζ A (s) + b(s), with a(s), b(s) holomorphic on {Re s > r}, for some r < dim B A, and a(s) 0 for all such s. The series converges iff Re s > max{1, 2 1+α 1+β }; hence, dim B Γ(α, β) = max{1, α 1 + β }. This is the analog of Tricot s formula (which was originally proved for the standard (α, β)-chirp).

49 Meromorphic Extensions of Fractal Zeta Functions Minkowski Measurable Sets Theorem (LRŽ (Minkowski measurable case)) Given A R N, assume that there exist α > 0, M (0, ) and D 0 such that the tube function t A t satisfies A t = t N D (M + O(t α )) as t 0 +. Then A is Minkowski measurable, and we have: dim B A = D, M D (A) = M, and D( ζ A ) = D. Furthermore, ζ A has a (unique) meromorphic extension to (at least) {Re s > D α}. Moreover, the pole s = D is unique, simple, and res( ζ A, D) = M.

50 Meromorphic Extensions of Fractal Zeta Functions Remark: Provided D < N, the exact same results hold for ζ A, the distance zeta function of A. Then, we have instead res(ζ A, D) = (N D)M.

51 Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (LRŽ (Minkowski nonmeasurable case)) Given A R N, assume that there exist D 0, a nonconstant periodic function G : R R with minimal period T > 0, and α > 0, such that A t = t N D ( G(log t 1 ) + O(t α ) ) as t 0 +. Then we have: dim B A = D, M D (A) = min G, M D (A) = max G, and D( ζ A ) = D. Furthermore, ζ A (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 ζ A ) is given by { P( ζ A ) = s k = D + 2π T ik : Ĝ 0 ( k } T ) 0, k Z ;

52 Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (... continued) they are all simple. Here, Ĝ 0 (s) := T 0 e 2πis τ G(τ) dτ. For all s k P( ζ A ), res( ζ A, s k ) = 1 T Ĝ0( k T ). We have and res( ζ A, s k ) 1 T Moreover, T 0 G(τ) dτ, res( ζ A, D) = 1 T T 0 lim res( ζ A, s k ) = 0. k ± G(τ) dτ M D (A) < res( ζ A, D) < M D (A). In particular, A is not Minkowski measurable.

53 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 A ε /ε N D ) exists and is given by res( ζ A, D) = M = 1 T T 0 G(τ)dτ. Provided D < N, an entirely analogous theorem holds for ζ A (instead of ζ A ), the distance zeta function of A, except for the fact that the residues take different values. In particular, res(ζ A, D) = (N D) M.

54 Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Example If A is the ternary Cantor set, we have (see [L-vF, 2000]) A t = t 1 D G(log t 1 ) as t 0 +, where D = log 3 2 and the nonconstant function G is log 3-periodic: G(τ) = 2 1 D ( 2 { τ log 2 log 3 } + ( ) 3 { τ log 2 }) log 3, 2 where {x} := x x is the fractional part of x and x is the integer part of x. Conclusion: ζ A and ζ A have a (unique) meromorphic extension to {Re s > D α} for any α > 0, and hence to all of C.

55 Meromorphic Extensions of Fractal Zeta Functions Definition The principal complex dimensions of a bounded set A in R N are given by P(ζ A ) := dim C A {Re s = D(ζ A )}, where dim C A denotes the set of (visible) complex dimensions of A. (Recall that D(ζ A ) = D( ζ A ) = dim B A.) The vertical line {Re s = D(ζ A )} is called the critical line. Remark: For the ternary Cantor set A, we have P(ζ A ) = dim C A = log π log 3 iz.

56 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 R N, such that Ω < and there exists δ > 0 such that Ω A δ. 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) = lim ε 0 + A ε Ω ε N t.

57 Definition The upper box dimension of the relative fractal drum (A, Ω) is given by dim B (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 [LRŽ] for details.)

58 Analyticity of Relative Zeta Functions Definition (Relative distance zeta function of (A, Ω), LRŽ) ζ A (s, Ω) = d(x, A) s N dx, for s C with Re s sufficiently large. Ω Theorem (LRŽ) ζ A (s, Ω) is holomorphic for Re s > dim B (A, Ω); the lower bound Re s > dim B (A, Ω) is optimal. Hence, the abscissa of convergence of ζ A (, Ω) is equal to dim B (A, Ω), the relative upper box dimension of (A, Ω). Assume that D = dim B (A, Ω) exists and M D (A, Ω) > 0. Then, if s R and s D +, we have ζ A (s, Ω) +.

59 Definition (Relative tube zeta function of (A, Ω), LRŽ) ζ A (s, Ω) = δ 0 t s N 1 A t Ω dt, for s C with Re s sufficiently large. (Here, δ > 0 is fixed.) Remark: The above theorem is valid without change for ζ A (, Ω) (instead of ζ A (, Ω)). In particular, dim B (A, Ω) = the abscissa of convergence of ζ A (, Ω).

60 Example (1) If A = {(x, y) : y = x α, 0 < x < 1} with α (0, 1) and Ω = ( 1, 0) (0, 1), then dim B (A, Ω) = 1 α, which is < 1. Note that A and Ω are disjoint. Example (2) If A = {(0, 0)} (the origin in R 2 ) and Ω = {(x, y) (0, 1) R : 0 < y < x α } with α > 1, then dim B (A, Ω) = 1 α, which is < 0. Note that Ω is flat at A.

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

62 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 {(A j, Ω j )} j=1 involving generalized Cantor sets. Here, D = 4/5 and α = 3/10. Furthermore, D( ζ A (, Ω)) = 4/5, D mer ( ζ A (, Ω)) = D α = 1/2 and π(log 2)iZ is the set of nonisolated singularities of ζ A (, 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 ζ A (, Ω), and to the left of it is the meromorphy critical line {Re s = D α}.

63 Definition A function G : R R is transcendentally quasiperiodic of infinite order (resp., of finite order m) if it is of the form G(τ) = H(τ, τ,...), where H : R R (resp., H : R m R) is a function which is T j -periodic in its j-th component, for each j N (resp., for each j = 1,, m), with T j > 0 as minimal periods, and such that the set of quasiperiods {T j : j 1} (resp., {T j : j = 1,, m}) is algebraically independent; i.e., is independent over the ring of algebraic numbers.

64 Definition A relative fractal drum (A, Ω) in R N is said to be transcendentally quasiperiodic if A t Ω = t N D (G(log(1/t)) + o(1)) as t 0 +, where the function G is transcendentally quasiperiodic. In the special case where A R N is bounded and Ω = R N, then the set A is said to be transcendentally quasiperiodic.

65 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 ζ A (s) has the critical line {Re s = D} as a natural boundary, where D = dim B A. (This means that ζ A (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 ζ A (s); the same is true for ζ A (s) (instead of ζ A (s)). (See [LRŽ] for the detailed construction.) The set A is then said to be (maximally) hyperfractal.

66 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 obtained 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.

67 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 obtained 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.

68 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

69 Future Research Directions 4. Box-counting zeta functions (joint work in progress with John Rock and Darko Žubrinić) 5. Spectral zeta functions of relative fractal drums (spectral complex dimensions) Connections with geometric fractal zeta functions and complex dimensions

70 References M. L. Lapidus, M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, research monograph, Birkhäuser, 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. Radunović, D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2017, 684 pages. ([LRŽ])

71 References M. L. Lapidus, G. Radunović and D. Žubrinić, Distance and tube zeta functions of fractals and arbitrary compact sets, Adv. in Math. 307 (2017), (dx.doi.org/ /j.aim ). (Also: e-print, arxiv: v2 [math-ph], 2016; IHES preprint, M/15/15, 2015.) M. L. Lapidus, G. Radunović and D. Žubrinić, Complex dimensions of fractals and meromorphic extensions of fractal zeta functions, J. Math. Anal. Appl. No. 1, 453 (2017), (doi.org/ /j.jmaa ) (Also: e-print, arxiv: v3 [math-ph], 2016.) M. L. Lapidus, G. Radunović and D. Žubrinić, Zeta functions and complex dimensions of relative fractal drums: Theory, examples and applications, Dissertationes Mathematicae, in press, 2017, 101 pages. (Also: e-print, arxiv: v3 [math-ph], 2016.)

72 References M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces, Discrete and Continuous Dynamical Systems Ser. S, in press, (Also: e-print, arxiv: v4 [math-ph], 2016; IHES preprint, IHES/M/15/17, 2015.) M. L. Lapidus, G. Radunović and D. Žubrinić, 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: v3 [math-ph], 2016.)

73 References M. L. Lapidus, G. Radunović and D. Zubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, survey article, Journal of Fixed Point Theory and Applications No. 2, 15 (2014), Festschrift issue in honor of Haim Brezis 70th birthday. (DOI: /s y.) (Also: e-print, arxiv: v3[math-ph], 2014.)

74 References M. L. Lapidus, G. Radunović and D. Žubrinić, 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. Zähle, eds.), Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, vol. 70, Birkhäuser/Springer Internat., Basel, Boston and Berlin, 2015, pp ; doi: / (Based on a plenary lecture given by the first author at that conference.) (Also: e-print, arxiv: v3 [math.cv], 2015; IHES preprint, IHES/M/15/16, 2015.)