An Introduction to the Theory of Complex Dimensions and Fractal …pi.math.cornell.edu/~fractals/6/slides/Lapidus.pdf · 2017-06-16 · An Introduction to the Theory of Complex Dimensions

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

An Introduction to the Theory of ComplexDimensions and Fractal Zeta Functions

Michel L. Lapidus

University of California, RiversideDepartment of Mathematics

http://www.math.ucr.edu/∼lapidus/[email protected]

6th Cornell Conference on Analysis, Probability,and Mathematical Physics on Fractals

Cornell UniversityIthaca NY

June 13, 2017


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


1 Definitions and MotivationsMinkowski Content and Box DimensionSingularities of Functions

2 Fractal stringsZeta Functions of Fractal Strings

3 Distance and Tube Zeta FunctionsDefinitionAnalyticityResidues of Distance Zeta FunctionsResidues of Tube Zeta Functions(α, β)-chirpsMeromorphic Extensions of Fractal Zeta Functions

4 Relative Distance and Tube Zeta Functions

5 References


Goals

Introducing a new class of fractal zeta functions:distance and tube zeta functions associated with boundedfractal sets in Euclidean spaces of arbitrary dimensions.

Developing a higher-dimensional theory of complex fractaldimensions valid for arbitrary compact sets (and eventually,for suitable metric measure spaces).

Merging of aspects of complex, spectral and harmonicanalysis, geometry, and number theory of fractal sets in RN .


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. Radunovic‡, D. Zubrinic‡, FractalZeta Functions and Fractal Drums: Higher-Dimensional Theory ofComplex Dimensions, research monograph, Springer, New York,2017, 684 pages. ([LRZ])

(And nine related papers by the authors of [LRZ]; see thebibliography.)

† of Utah Valley University, USA‡ of the University of Zagreb, Croatia


Book in Preparation:

M. L. Lapidus, Complex Fractal Dimensions, Quantized NumberTheory and Fractal Cohomology: A Tale of Oscillations, Unrealityand Fractality, book in preparation, approx. 350 pages, 2017.







(a) Fractal stalagmites (b) Fractal stalactites

Figure 2: Stalagmites and stalactites in a fractal cave


Figure 3: Other fractal stalagmites


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 supε→0+

|Aε|εN−s

.



Box Dimensions

Lower box dimension of A: dimBA = infs ≥ 0 :Ms∗(A) = 0.

Upper box dimension of A:

dimBA = infs ≥ 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, thenA is Minkowski nondegenerate and

dimB A = N − σ.



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 tobe Minkowski measurable, with Minkowski content Md(A) (oftensimply 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. ThendimB A = 1/(1 + a) and A is Minkowski measurable (L., 1991).



The Triadic Cantor Set|Aε|ε1−d = G (log3 ε

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

0 ε

Md∗ (A)

M∗d(A)

Graph ofε 7→ G (log3 ε

−1)

16

16·3

16·32

16·33

Figure 4: The oscillating nature of the function ε 7→ |Aε|/ε1−d nearε = 0 for the triadic Cantor set A, with d := dimB A = log3 2. Then, A isMinkowski 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 detailedinformation.)


Singularities of Functions

Singular Function Generated by the Cantor Set

0 11/3 2/3

y = d(x ,A)

Figure 5: The graph of the distance function x 7→ d(x ,A), where A isthe classic ternary (or triadic) Cantor set C (1/3).



Singular Function Generated by the Cantor Set

0 11/3 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 theHausdorff dimension of A.



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).



Figure 7: The classic self-similar Sierpinski carpet



(a) The Holder 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).



Figure 9: Fractal stalagmites associated with the Sierpinski carpet.



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



Figure 10: Fractal stalactites associated with the Sierpinski carpet.



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


Zeta Functions of Fractal Strings

Definition of Fractal Strings

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

j `j <∞. [Alternatively, L can be viewed as a sequence ofscales or as the lengths of the connected components (openintervals) of a bounded open set Ω ⊂ R.]

The zeta function (geometric zeta function) of the fractalstring L is the Dirichlet series:

ζL(s) =∞∑j=1

(`j)s ,

for s ∈ C with Re s large enough.



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

aj :=∑k>j

`k , and `j := |Ij |.

Define A = aj. Then A is a bounded set, A ⊂ R, andaj → 0 as j →∞. The set A = AL is introduced in [LRZ].

dimBA is defined via the upper Minkowski content of A, asusual.



Theorem (L)

The abscissa of convergence of ζL is equal to dimBA :

dimBA = inf

α > 0 :

∞∑j=1

(`j)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 ofview of the absolute convergence and the holomorphiccontinuation.

Moreover, if s ∈ R and s → dimBA from the right, thenζL(s)→ +∞.



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, theboundary of any geometric realization of L by a bounded open setΩ ⊂ R.


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; LRZ, 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.


Definition

Remarks:

For s ∈ C such that Re s < N, the function d(x ,A)s−N issingular 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 ofζA = ζA(·,Aδ) does not depend on δ in an essential way. Inparticular, the complex dimensions of A (i.e., the poles of ζA)do not depend on δ.


Definition

Zeta Function of the Set A Associated to a Fractal String L

Let L = (`j) be a fractal string, and A = (aj), aj =∑

k≥j `k .

We would like to compare ζL(s) =∑

j(`j)s and

ζA(s) =

∫ 1+δ

−δd(x ,A)s−1dx , for δ ≥ `1/2.

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 areholomorphic on Re s > 0, and a(s) 6= 0 for all such s. Itfollows that (when they exist) the meromorphic extensions ofζA(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).


Definition

Definition

The complex dimensions of a fractal string L (L & vF, 1996) aredefined as the poles of ζL.

Definition

The complex dimensions of a bounded set A ⊂ RN are defined asthe poles of ζA (L., 2009; LRZ, 2013).

Remark : We assume here that the zeta functions involved have ameromorphic extension (necessarily unique, by the principle ofanalytic continuation) to some suitable region U ⊂ C.

Visible complex dimensions: the poles of ζA in U.


Analyticity

Holomorphy Half-Plane of the Distance Zeta Function

Let A be a nonempty bounded set in RN ; given δ a fixed positivenumber, let ζA(s) =

∫Aδ

d(x ,A)s−Ndx as before.

Theorem (LRZ)

ζA(s) is holomorphic on the right half-plane Re s > dimBA;the lower bound dimBA is optimal from the point of view ofthe 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 lowerbound dimBA is also optimal from the point of view of theholomorphic continuation.

Remark : If s ∈ R and s < dimBA, then ζA(s) = +∞.


Analyticity

Corollary (LRZ)

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 (LRZ)

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.


Analyticity

Definition

The abscissa of holomorphic continuation of ζA is given by

Dhol(ζA) := infα ∈ R : ζA(s) is holomorphic on Re s > α

Furthermore, the open half-plane Re s > Dhol(ζA) is called theholomorphy half-plane of ζA, while Re s > D(ζA) is called thehalf-plane of (absolute) convergence of ζA.

Remark : In general, we have

Dhol(ζA) ≤ D(ζA) = dimBA.


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 meromorphicallyextended to a neighborhood of D := dimB A, and D < N. Wewrite ζA or ζA(·,Aδ), interchangeably.

Theorem (LRZ)

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.


Residues of Distance Zeta Functions

Theorem (LRZ)

If A is Minkowski measurable (i.e., MD(A) exists andMD(A) ∈ (0,∞)), then

res(ζA(·,Aδ),D) = (N − D)MD(A).


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 functiont 7→ |At |, is given by (for some fixed, small δ > 0)

ζA(s) =

∫ δ

0ts−N−1|At |dt,


Remark : The choice of δ is unimportant, from the point of view ofthe theory of complex dimensions. Indeed, changing δ amounts toadding an entire function to ζA.



The next result follows from its counterpart stated earlier for thedistance zeta function ζA (see the sketch of the proof given below):

Corollary (LRZ)

If D = dimB A exists, D < N and ζA has a meromorphic extensionto 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).



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, thenζA has a meromorphic extension to a given domain U ⊂ C iff ζAdoes. In particular, ζA and ζA have the same (visible) complexdimensions; that is, the same poles within U.



Residues of Fractal Zeta Functions of Generalized Cantor Sets

Example (1)

For the generalized Cantor sets A = C (a), a ∈ (0, 1/2), we haveD(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

.



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 justC (1/3). For any a ∈ (0, 1/2), the generalized Cantor set C (a) isconstructed in much the same way as C (1/3), by removing open“middle a-intervals” at each stage of the construction.



Residues and Minkowski Contents for a-strings, a > 0

Example (2)

The a-string associated with A := k−a : k ∈ N, a > 0, is givenby

L = (`j)j≥1, `j = 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,

andres(ζL,D) = aD .


(α, β)-chirps

Definition (1)

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


(α, β)-chirps

Definition of the (α, β)-chirp

Definition

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

Γ(α, β) =⋃k∈Nk−1/β × (0, k−α/β).


(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−αβ−(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) 6= 0 for all such s.

The series converges iff Re s > max1, 2− 1+α1+β ; hence,

dimBΓ(α, β) = max1, 2− 1 + α

1 + β.

This is the analog of Tricot’s formula (which was originally provedfor the standard (α, β)-chirp).


Meromorphic Extensions of Fractal Zeta Functions

Minkowski Measurable Sets

Theorem (LRZ (Minkowski measurable case))

Given A ⊂ RN , assume that there exist α > 0, M∈ (0,∞) andD ≥ 0 such that the tube function t 7→ |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 (atleast) Re s > D − α.

Moreover, the pole s = D is unique, simple, and res(ζA,D) =M.



Remark : Provided D < N, the exact same results hold for ζA, thedistance zeta function of A. Then, we have instead

res(ζA,D) = (N − D)M.



Minkowski Nonmeasurable Sets

Theorem (LRZ (Minkowski nonmeasurable case))

Given A ⊂ RN , assume that there exist D ≥ 0, a nonconstantperiodic function G : R→ R with minimal period T > 0, andα > 0, such that

|At | = tN−D(G (log t−1) + O(tα)

)as t → 0+.

Then we have:

dimB A = D,MD∗ (A) = minG ,M∗D(A) = maxG , and D(ζA) = D.

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

P(ζA) =

sk = D +

2π

Tik : G0(

k

T) 6= 0, k ∈ Z

;




Theorem (. . . continued)

they are all simple. Here, G0(s) :=∫ T

0 e−2πis·τG (τ) dτ.

For all sk ∈ P(ζA), res(ζA, sk) = 1T G0( k

T ). We have

| res(ζA, sk)| ≤ 1

T

∫ T

0G (τ) dτ, lim

k→±∞res(ζA, sk) = 0.

Moreover,

res(ζA,D) =1

T

∫ T

0G (τ)dτ

and

MD∗ (A) < res(ζA,D) <M∗D(A).

In particular, A is not Minkowski measurable.



Remarks:

Under the assumptions of the theorem, the average Minkowskicontent M of A (defined as a suitable Cesaro logarithmicaverage of |Aε|/εN−D) exists and is given by

res(ζA,D) = M =1

T

∫ T

0G (τ)dτ.

Provided D < N, an entirely analogous theorem holds for ζA(instead of ζA), the distance zeta function of A, except for thefact that the residues take different values. In particular,

res(ζA,D) = (N − D)M.




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 2log 3

+

(3

2

)− τ−log 2log 3

),

where x := x − bxc is the fractional part of x and bxc is theinteger part of x .

Conclusion: ζA and ζA have a (unique) meromorphic extension toRe s > D − α for any α > 0, and hence to all of C.



Definition

The principal complex dimensions of a bounded set A in RN aregiven 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 lineRe 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.


Distance Zeta Functions of Relative Fractal Drums

Definition

A relative fractal drum is a pair (A,Ω) of nonempty subsets A andΩ (open subset) of RN , such that |Ω| <∞ and there exists δ > 0such 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 Arelative to Ω is given by

M∗t(A,Ω) = limε→0+|Aε ∩ Ω|εN−t

.


Definition

The upper box dimension of the relative fractal drum (A,Ω) isgiven by

dimB(A,Ω) = inft ∈ R :M∗t(A,Ω) = 0.

It may be negative, and even equal to −∞; this is related to theflatness of (A,Ω). (This latter concept is not discussed here; see[LRZ] for details.)


Analyticity of Relative Zeta Functions

Definition (Relative distance zeta function of (A,Ω), LRZ)

ζA(s,Ω) =

∫Ωd(x ,A)s−Ndx ,


Theorem (LRZ)

ζA(s,Ω) is holomorphic for Re s > dimB(A,Ω); the lowerbound Re s > dimB(A,Ω) is optimal. Hence, the abscissa ofconvergence of ζA(·,Ω) is equal to dimB(A,Ω), the relativeupper 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)→ +∞.


Definition (Relative tube zeta function of (A,Ω), LRZ)

ζA(s,Ω) =

∫ δ

0ts−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).


Example (1)

If A = (x , y) : y = xα, 0 < x < 1 with α ∈ (0, 1) andΩ = (−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) andΩ = (x , y) ∈ (0, 1)× R : 0 < y < xα with α > 1, thendimB(A,Ω) = 1− α, which is < 0. Note that Ω is flat at A.


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


In the previous figure (Fig. ), the set of complex dimensions D ofthe relative fractal drum (A,Ω), obtained as a union of relativefractal 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 = 1/2and 2−1 + 4π(log 2)iZ is the set of nonisolated singularities ofζA(·,Ω).

The set D is contained in a union of countably many raysemanating from the origin. The dotted vertical line is theholomorphy critical line Re s = D of ζA(·,Ω), and to the left ofit is the meromorphy critical line Re s = D − α.


Definition

A function G : R→ R is transcendentally quasiperiodic of infiniteorder (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 isTj -periodic in its j-th component, for each j ∈ N (resp., for eachj = 1, · · · ,m), with Tj > 0 as minimal periods, and such that theset of quasiperiods Tj : j ≥ 1 (resp., Tj : j = 1, · · · ,m) isalgebraically independent; i.e., is independent over the ring ofalgebraic numbers.


Definition

A relative fractal drum (A,Ω) in RN is said to be transcendentallyquasiperiodic 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.


Hyperfractals

Using Alan Baker’s theorem (in the theory of transcendentalnumbers) and generalized Cantor sets C (m,a) with two parameters(as in the above example), it is possible to construct atranscendentally quasiperiodic bounded set A in R with infinitelymany algebraically independent quasiperiods.

For this set, we show that ζA(s) has the critical line Re s = D asa 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 aresingularities of ζA(s); the same is true for ζA(s) (instead of ζA(s)).(See [LRZ] for the detailed construction.)

The set A is then said to be (maximally) hyperfractal .


Remark

The above construction of maximally hyperfractal andtranscendentally quasiperiodic sets (and relative fractal drums) ofinfinite order has been applied in [LRZ] in different contexts. Inparticular, it has been applied to prove that certain estimatesobtained by the first author and regarding the abscissae ofmeromorphic continuation of the spectral zeta function of fractaldrums are sharp, in general.

This construction is also relevant to the definition of fractalitygiven in terms of complex dimensions. Recall that in the theory ofcomplex dimensions, an object is said to be “fractal” if it has atleast one nonreal complex dimension (with positive real part) orelse if the associated fractal zeta function has a natural boundary(along a suitable contour). This new higher-dimensional theory ofcomplex dimensions now enables us to define fractality in fullgenerality.


Future Research Directions

1. Fractal tube formulas and geometric complex dimensionsa. Case of self-similar setsb. Devil’s staircase (cf. the definition of fractality)c. Weierstrass functiond. Julia sets and Mandelbrot set

Possible geometric interpretation: fractal curvatures (even forcomplex dimensions)Connections with earlier joint work of the author with Erin Pearseand Steffen Winter for fractal sprays and self-similar tilings.

Added note: A general fractal tube formula has now beenobtained by the authors of [LRZ]. This formula has been applied toa number of self-similar and non self-similar examples, includingthe Sierpinski gasket and carpet as well as their higher-dimensionalcounterparts. See [LRZ, Chapter 5] and the relevant references inthe bibliography.



2. Spectral complex dimensions Determine the complexdimensions of a variety of fractal drums (via the associatedspectral functions) and compare these spectral complex dimensionswith 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 onfractals



4. Box-counting zeta functions(joint work in progress with John Rock and Darko Zubrinic)

5. Spectral zeta functions of relative fractal drums (spectralcomplex dimensions)Connections with geometric fractal zeta functions and complexdimensions


References

M. L. Lapidus, M. van Frankenhuysen, Fractal Geometry andNumber Theory: Complex Dimensions of Fractal Strings and Zerosof Zeta Functions, research monograph, Birkhauser, Boston, 2000,280 pages.

M. L. Lapidus, M. van Frankenhuijsen: Fractal Geometry, ComplexDimensions and Zeta Functions: Geometry and Spectra of FractalStrings, research monograph, second revised and enlarged edition(of the 2006 edition), Springer, New York, 2013, 593 pages.([L-vF])

M. L. Lapidus, G. Radunovic, D. Zubrinic, Fractal Zeta Functionsand Fractal Drums: Higher-Dimensional Theory of ComplexDimensions, research monograph, Springer, New York, 2017, 684pages. ([LRZ])


References

M. L. Lapidus, G. Radunovic and D. Zubrinic, Distance and tubezeta 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; IHESpreprint, M/15/15, 2015.)

M. L. Lapidus, G. Radunovic and D. Zubrinic, Complex dimensionsof 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. Radunovic and D. Zubrinic, Zeta functions andcomplex dimensions of relative fractal drums: Theory, examplesand applications, Dissertationes Mathematicae, in press, 2017, 101pages. (Also: e-print, arXiv:1603.00946v3 [math-ph], 2016.)


References

M. L. Lapidus, G. Radunovic and D. Zubrinic, Fractal tubeformulas and a Minkowski measurability criterion for compactsubsets of Euclidean spaces, Discrete and Continuous DynamicalSystems 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. Radunovic and D. Zubrinic, Fractal tubeformulas for compact sets and relative fractal drums: Oscillations,complex dimensions and fractality, Journal of Fractal Geometry, inpress, 2017, 104 pages. (Also: e-print, arXiv:1604.08014v3[math-ph], 2016.)


References

M. L. Lapidus, G. Radunovic and D. Zubrinic, Fractal zetafunctions and complex dimensions of relative fractal drums, surveyarticle, Journal of Fixed Point Theory and Applications No. 2, 15(2014), 321–378. Festschrift issue in honor of Haim Brezis’ 70thbirthday. (DOI: 10.1007/s11784-014-0207-y.) (Also: e-print,arXiv:1407.8094v3[math-ph], 2014.)


References

M. L. Lapidus, G. Radunovic and D. Zubrinic, Fractal zetafunctions and complex dimensions: A general higher-dimensionaltheory, survey article, in: Fractal Geometry and Stochastics V (C.Bandt, K. Falconer and M. Zahle, eds.), Proc. Fifth Internat. Conf.(Tabarz, Germany, March 2014), Progress in Probability, vol. 70,Birkhauser/Springer Internat., Basel, Boston and Berlin, 2015, pp.229–257; doi:10.1007/978-3-319-18660-3 13. (Based on a plenarylecture given by the first author at that conference.) (Also: e-print,arXiv:1502.00878v3 [math.CV], 2015; IHES preprint,IHES/M/15/16, 2015.)

Documents

An Introduction to the Theory of Complex Dimensions and Fractal …pi.math.cornell.edu/~fractals/6/slides/Lapidus.pdf · 2017-06-16 · An Introduction to the Theory of Complex Dimensions