Log-Minkowski measurability and complex dimensions Goran Radunovi - - PowerPoint PPT Presentation

Log-Minkowski measurability and complex dimensions

Goran Radunovi´ c

University of California, Riverside

14th June 2017 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, Ithaca, NY

Joint work with Michel L. Lapidus, University of California, Riverside, Darko ˇ Zubrini´ c, University of Zagreb

Relative fractal drum (A, Ω)

∅ ̸= A ⊂ RN , Ω ⊂ RN, Lebesgue measurable, i.e., |Ω| < ∞ δ-neighbourhood of A: Aδ = {x ∈ RN : d(x, A) < δ} upper r-dimensional Minkowski content of (A, Ω): Mr(A, Ω) := lim sup

δ→0+

|Aδ ∩ Ω| δN−r upper Minkowski dimension of (A, Ω): dimB(A, Ω) = inf{r ∈ R : Mr(A, Ω) = 0} lower Minkowski content and dimension defined via lim inf

Minkowski measurability

dimB(A, Ω) = dimB(A, Ω) ⇒ ∃ dimB(A, Ω) if ∃D ∈ R such that 0 < MD(A, Ω) = MD(A, Ω) < ∞, we say (A, Ω) is Minkowski measurable; in that case D = dimB(A, Ω) if the above inequalities are not satisfied for D, we call (A, Ω) Minkowski degenerated

The relative distance zeta function

(A, Ω) RFD in RN, s ∈ C and fix δ > 0 the distance zeta function of (A, Ω): ζA,Ω(s; δ) := ∫

Aδ∩Ω

d(x, A)s−N dx dependence on δ is not essential

The relative distance zeta function

(A, Ω) RFD in RN, s ∈ C and fix δ > 0 the distance zeta function of (A, Ω): ζA,Ω(s; δ) := ∫

Aδ∩Ω

d(x, A)s−N dx dependence on δ is not essential the complex dimensions of (A, Ω) are defined as the poles

• f ζA,Ω

The relative distance zeta function

(A, Ω) RFD in RN, s ∈ C and fix δ > 0 the distance zeta function of (A, Ω): ζA,Ω(s; δ) := ∫

Aδ∩Ω

d(x, A)s−N dx dependence on δ is not essential the complex dimensions of (A, Ω) are defined as the poles

• f ζA,Ω

take Ω to be an open neighborhood of A in order to recover the classical ζA

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu]

Theorem

• (A, Ω) RFD in RN:

(a) ζA,Ω(s) is holomorphic on {Re s > dimB(A, Ω)}

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu]

Theorem

• (A, Ω) RFD in RN:

(a) ζA,Ω(s) is holomorphic on {Re s > dimB(A, Ω)} (b) R ∋ s < dimB(A, Ω) ⇒ the integral defining ζA,Ω(s) diverges

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu]

Theorem

• (A, Ω) RFD in RN:

(a) ζA,Ω(s) is holomorphic on {Re s > dimB(A, Ω)} (b) R ∋ s < dimB(A, Ω) ⇒ the integral defining ζA,Ω(s) diverges (c) if ∃D = dimB(A, Ω) < N and MD(A, Ω) > 0, then ζA,Ω(s) → +∞ when R ∋ s → D+

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu]

Theorem

• (A, Ω) RFD in RN:

(a) ζA,Ω(s) is holomorphic on {Re s > dimB(A, Ω)} (b) R ∋ s < dimB(A, Ω) ⇒ the integral defining ζA,Ω(s) diverges (c) if ∃D = dimB(A, Ω) < N and MD(A, Ω) > 0, then ζA,Ω(s) → +∞ when R ∋ s → D+

• we call {Re s = dimB(A, Ω)} the critical line

(Generalized) complex dimensions of an RFD

Definition Let W be a connected open set s.t. {Re s > dimB(A, Ω)} ⊂ W and ζA,Ω is holomorphic on W . The set of visible complex dimensions of (A, Ω) (with respect to W ) is the set of singularities P(ζA,Ω, W ) ⊂ ∂W of ζA,Ω.

(Generalized) complex dimensions of an RFD

Definition Let W be a connected open set s.t. {Re s > dimB(A, Ω)} ⊂ W and ζA,Ω is holomorphic on W . The set of visible complex dimensions of (A, Ω) (with respect to W ) is the set of singularities P(ζA,Ω, W ) ⊂ ∂W of ζA,Ω. principal complex dimensions: dimPC(A, Ω) := {ω ∈ P(ζA,Ω, W ) : Re ω = dimB(A, Ω)}. (1)

(Generalized) complex dimensions of an RFD

Definition Let W be a connected open set s.t. {Re s > dimB(A, Ω)} ⊂ W and ζA,Ω is holomorphic on W . The set of visible complex dimensions of (A, Ω) (with respect to W ) is the set of singularities P(ζA,Ω, W ) ⊂ ∂W of ζA,Ω. principal complex dimensions: dimPC(A, Ω) := {ω ∈ P(ζA,Ω, W ) : Re ω = dimB(A, Ω)}. (1)

• includes poles, essential and nonisolated singularities

(accumulation of poles, natural boundaries)

(Generalized) complex dimensions of an RFD

Definition Let W be a connected open set s.t. {Re s > dimB(A, Ω)} ⊂ W and ζA,Ω is holomorphic on W . The set of visible complex dimensions of (A, Ω) (with respect to W ) is the set of singularities P(ζA,Ω, W ) ⊂ ∂W of ζA,Ω. principal complex dimensions: dimPC(A, Ω) := {ω ∈ P(ζA,Ω, W ) : Re ω = dimB(A, Ω)}. (1)

• includes poles, essential and nonisolated singularities

(accumulation of poles, natural boundaries)

• branching points (W is then a subset of the appropriate

Riemann surface) and

(Generalized) complex dimensions of an RFD

Definition Let W be a connected open set s.t. {Re s > dimB(A, Ω)} ⊂ W and ζA,Ω is holomorphic on W . The set of visible complex dimensions of (A, Ω) (with respect to W ) is the set of singularities P(ζA,Ω, W ) ⊂ ∂W of ζA,Ω. principal complex dimensions: dimPC(A, Ω) := {ω ∈ P(ζA,Ω, W ) : Re ω = dimB(A, Ω)}. (1)

• includes poles, essential and nonisolated singularities

(accumulation of poles, natural boundaries)

• branching points (W is then a subset of the appropriate

Riemann surface) and also “mixed singularities”

Fractal tube formulas for relative fractal drums

An asymptotic formula for the tube function t → |At ∩ Ω| as t → 0+ in terms of ζA,Ω .

Fractal tube formulas for relative fractal drums

An asymptotic formula for the tube function t → |At ∩ Ω| as t → 0+ in terms of ζA,Ω . Theorem (Simplified pointwise formula with error term)

• α < dimB(A, Ω) < N; ζA,Ω satisfies suitable rational growth

conditions (d-languidity) on the half-plane W := {Re s > α}, then: |At ∩ Ω| = ∑

ω∈P(ζA,Ω,W)

res ( tN−s N−s ζA,Ω(s), ω ) + O(tN−α).

Fractal tube formulas for relative fractal drums

An asymptotic formula for the tube function t → |At ∩ Ω| as t → 0+ in terms of ζA,Ω . Theorem (Simplified pointwise formula with error term)

• α < dimB(A, Ω) < N; ζA,Ω satisfies suitable rational growth

conditions (d-languidity) on the half-plane W := {Re s > α}, then: |At ∩ Ω| = ∑

ω∈P(ζA,Ω,W)

res ( tN−s N−s ζA,Ω(s), ω ) + O(tN−α). if we allow polynomial growth of ζA,Ω, in general, we get a tube formula in the sense of Schwartz distributions

The Minkowski measurability criterion

Theorem (Minkowski measurability criterion)

• (A, Ω) is such that ∃D := dimB(A, Ω) and D < N
• ζA,Ω is d-languid on a suitable domain W ⊃ {Re s = D}

Then, the following is equivalent: (a) (A, Ω) is Minkowski measurable. (b) D is the only pole of ζA,Ω located on the critical line {Re s = D} and it is simple. MD(A, Ω) = res(ζA,Ω, D) N − D

The Minkowski measurability criterion

(a) ⇒ (b) : from the distributional tube formula and the Uniqueness theorem for almost periodic distributions due to Schwartz (b) ⇒ (a) : a consequence of a Tauberian theorem due to Wiener and Pitt (conditions can be considerably weakened) the assumption D < N can be removed by appropriately embedding the RFD in RN+1

Figure: The Sierpi´ nski gasket

an example of a self-similar fractal spray with a generator G being an open equilateral triangle and with scaling ratios r1 = r2 = r3 = 1/2 (A, Ω) = (∂G, G) ⊔ ⊔3

j=1(rjA, rjΩ)

Fractal tube formula for The Sierpi´ nski gasket

ζA(s; δ) = 6( √ 3)1−s2−s s(s − 1)(2s − 3) + 2πδs s + 3 δs−1 s − 1 i i

i

Fractal tube formula for The Sierpi´ nski gasket

ζA(s; δ) = 6( √ 3)1−s2−s s(s − 1)(2s − 3) + 2πδs s + 3 δs−1 s − 1 P(ζA) = {0, 1} ∪ ( log2 3 + 2π log 2iZ ) i

i

Fractal tube formula for The Sierpi´ nski gasket

ζA(s; δ) = 6( √ 3)1−s2−s s(s − 1)(2s − 3) + 2πδs s + 3 δs−1 s − 1 P(ζA) = {0, 1} ∪ ( log2 3 + 2π log 2iZ ) By letting ωk := log2 3 + pki and p := 2π/ log 2 we have that

i

Fractal tube formula for The Sierpi´ nski gasket

ζA(s; δ) = 6( √ 3)1−s2−s s(s − 1)(2s − 3) + 2πδs s + 3 δs−1 s − 1 P(ζA) = {0, 1} ∪ ( log2 3 + 2π log 2iZ ) By letting ωk := log2 3 + pki and p := 2π/ log 2 we have that |At| = ∑

ω∈P(ζA)

res ( t2−s 2 − s ζA(s; δ), ω )

i

Fractal tube formula for The Sierpi´ nski gasket

ζA(s; δ) = 6( √ 3)1−s2−s s(s − 1)(2s − 3) + 2πδs s + 3 δs−1 s − 1 P(ζA) = {0, 1} ∪ ( log2 3 + 2π log 2iZ ) By letting ωk := log2 3 + pki and p := 2π/ log 2 we have that |At| = ∑

ω∈P(ζA)

res ( t2−s 2 − s ζA(s; δ), ω ) = t2−log2 3 6 √ 3 log 2

+∞

∑

k=−∞

(4 √ 3)−ωkt−pki (2 − ωk)(ωk − 1)ωk + ( 3 √ 3 2 + π ) t2, valid pointwise for all t ∈ (0, 1/2 √ 3).

Gauge Minkowski content [HeLap]

If (A, Ω) is Minkowski degenerate, ∃D := dimB(A, Ω) and |At ∩ Ω| = tN−D(F(t) + o(1)) as t → 0+, (2) where F(t) = h(t) or F(t) = 1/h(t) for h : (0, ε0) → (0, +∞) , h(t) → +∞ as t → 0+ and h ∈ O(tβ) for ∀β < 0 .

Gauge Minkowski content [HeLap]

If (A, Ω) is Minkowski degenerate, ∃D := dimB(A, Ω) and |At ∩ Ω| = tN−D(F(t) + o(1)) as t → 0+, (2) where F(t) = h(t) or F(t) = 1/h(t) for h : (0, ε0) → (0, +∞) , h(t) → +∞ as t → 0+ and h ∈ O(tβ) for ∀β < 0 . h is called a gauge function of slow growth to +∞ at 0+ 1/h is called a gauge function of slow decay 0 at 0+ typical gauge functions: ( log◦k t−1)a for a ∈ R∗, k ∈ N

Gauge Minkowski content [HeLap]

If (A, Ω) is Minkowski degenerate, ∃D := dimB(A, Ω) and |At ∩ Ω| = tN−D(F(t) + o(1)) as t → 0+, (2) where F(t) = h(t) or F(t) = 1/h(t) for h : (0, ε0) → (0, +∞) , h(t) → +∞ as t → 0+ and h ∈ O(tβ) for ∀β < 0 . h is called a gauge function of slow growth to +∞ at 0+ 1/h is called a gauge function of slow decay 0 at 0+ typical gauge functions: ( log◦k t−1)a for a ∈ R∗, k ∈ N h-Minkowski content: MD(A, Ω, h) = lim

t→0+

|At ∩ Ω| tN−Dh(t).

The fractal nest generated by the a-string

a > 0, aj := j−a, ℓj := j−a − (j + 1)−a, Ω := Ba1(0) ζAa,Ω(s) = 22−sπ s − 1

∞

∑

j=1

ℓs−1

j

(aj + aj+1)

Fractal tube formula for the fractal nest generated by the a-string

Example P(ζAa,Ω) ⊆ { 1, 2 a + 1, 1 a + 1 } ∪ { − m a + 1 : m ∈ N } a ̸= 1, D :=

2 1+a ⇒

|(Aa)t ∩ Ω| = 22−DDπ (2 − D)(D − 1)aD−1t2−D + 2π ( 2ζ(a) − 1 ) t + O ( t2−

1 a+1 )

, as t → 0+

Fractal tube formula for the fractal nest generated by the a-string

Example P(ζAa,Ω) ⊆ { 1, 2 a + 1, 1 a + 1 } ∪ { − m a + 1 : m ∈ N } a ̸= 1, D :=

2 1+a ⇒

|(Aa)t ∩ Ω| = 22−DDπ (2 − D)(D − 1)aD−1t2−D + 2π ( 2ζ(a) − 1 ) t + O ( t2−

1 a+1 )

, as t → 0+ |(A1)t ∩ Ω| = res ( t2−s 2 − s ζA1,Ω(s), 1 ) + o(t) = 2πt(− log t) + const · t + o(t) as t → 0+

Fractal tube formula for the fractal nest generated by the a-string

Example P(ζAa,Ω) ⊆ { 1, 2 a + 1, 1 a + 1 } ∪ { − m a + 1 : m ∈ N } a ̸= 1, D :=

2 1+a ⇒

|(Aa)t ∩ Ω| = 22−DDπ (2 − D)(D − 1)aD−1t2−D + 2π ( 2ζ(a) − 1 ) t + O ( t2−

1 a+1 )

, as t → 0+ |(A1)t ∩ Ω| = res ( t2−s 2 − s ζA1,Ω(s), 1 ) + o(t) = 2πt(− log t) + const · t + o(t) as t → 0+

• a pole ω of order m generates terms of type

tN−ω(− log t)k−1 for k = 1, . . . , m in the fractal tube formula

Sufficiency for log-Minkowski measurability via the Wiener-Pitt Tauberian theorem

• m ∈ Z; ζ[m]

A,Ω denotes its the |m|-th derivative if m < 0 and the

m-th primitive if m > 0; i i

Sufficiency for log-Minkowski measurability via the Wiener-Pitt Tauberian theorem

• m ∈ Z; ζ[m]

A,Ω denotes its the |m|-th derivative if m < 0 and the

m-th primitive if m > 0; ζ[0]

A,Ω := ζA,Ω

Theorem

• D := dimB(A, Ω) < N; ∃m ∈ Z, ∃K > 0, s.t. ∀λ > 0

Gx(y) := ζ[m]

A,Ω(x + iy) −

(−1)mK x + iy − D converges in L1(−λ, λ) to a boundary function G(y) as x → D

+.

Sufficiency for log-Minkowski measurability via the Wiener-Pitt Tauberian theorem

• m ∈ Z; ζ[m]

A,Ω denotes its the |m|-th derivative if m < 0 and the

m-th primitive if m > 0; ζ[0]

A,Ω := ζA,Ω

Theorem

• D := dimB(A, Ω) < N; ∃m ∈ Z, ∃K > 0, s.t. ∀λ > 0

Gx(y) := ζ[m]

A,Ω(x + iy) −

(−1)mK x + iy − D converges in L1(−λ, λ) to a boundary function G(y) as x → D

+.

Then, ∃D := dimB(A, Ω) = D and (A, Ω) is h-Minkowski measurable s.t. MD(A, Ω, h) = K N − D , (3) where h(t) := (− log t)m.

Corollary: Case of poles

Theorem

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of poles and

has no accumulation points;

Corollary: Case of poles

Theorem

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of poles and

has no accumulation points;

• D is a pole of order m and all other poles on {Re s = D} are of
• rder strictly less than m.

Corollary: Case of poles

Theorem

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of poles and

has no accumulation points;

• D is a pole of order m and all other poles on {Re s = D} are of
• rder strictly less than m.

Then, ∃D := dimB(A, Ω) = D and (A, Ω) is h-Minkowski measurable: MD(A, Ω, h) = ζA,Ω[D]−m (N − D)(m − 1)!, (4) where h(t) := (− log t)m−1.

• ζA,Ω[D]−m denotes the leading coefficient of the Laurent

expansion of ζA,Ω at D.

Zero-log singularities

Definition

• ψ, ϕ holo. germs at ω ∈ C s.t. ω is a zero of order m of ψ.

We say that the holo. germ f (s) := ψ(s) Log(s − ω) + ϕ(s)

• n the principal branch of Log(s − ω) has a zero-log singularity
• f order m at ω.

Zero-log singularities

Definition

• ψ, ϕ holo. germs at ω ∈ C s.t. ω is a zero of order m of ψ.

We say that the holo. germ f (s) := ψ(s) Log(s − ω) + ϕ(s)

• n the principal branch of Log(s − ω) has a zero-log singularity
• f order m at ω.
• for instance, f (s) = (s − 2)3 Log(s − 2) has a zero-log

singularity of order 3 at ω = 2

• Log s has a zero-log singularity of order 0 at ω = 0, etc.

Corollary: Case of zero-log singularities

Theorem (Case of zero-log singularities)

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of zero-log

singularities and has no accumulation points;

• D is a zero-log sing. of order m

Corollary: Case of zero-log singularities

Theorem (Case of zero-log singularities)

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of zero-log

singularities and has no accumulation points;

• D is a zero-log sing. of order m and all the other zero-log sings.
• f ζA,Ω on {Re s = D} are of strictly higher order than m.

Corollary: Case of zero-log singularities

Theorem (Case of zero-log singularities)

• D := dimB(A, Ω) < N; dimPC(A, Ω) consists only of zero-log

singularities and has no accumulation points;

• D is a zero-log sing. of order m and all the other zero-log sings.
• f ζA,Ω on {Re s = D} are of strictly higher order than m.

Then, ∃D := dimB(A, Ω) = D and (A, Ω) is h-Minkowski measurable with Minkowski content given by MD(A, Ω, h) = (−1)m+1m! lim

s→D

ψ(s) (s − D)m , (5) where h(t) := 1 (− log t)m+1 .

Examples of sets that have zero-log complex dimensions?

Not easy to find nice examples where we can calculate everything explicitly

Examples of sets that have zero-log complex dimensions?

Not easy to find nice examples where we can calculate everything explicitly They will arise naturally as orbits of some discrete dynamical systems

Examples of sets that have zero-log complex dimensions?

Not easy to find nice examples where we can calculate everything explicitly They will arise naturally as orbits of some discrete dynamical systems Theorem (Mardeˇ si´ c, Resman, ˇ Zupanovi´ c) Let f ∈ Diffr(0, a) be continuous on [0, a) , positive on (0, a) and let f (0) = f ′(0) = 0. Assume 1 < x(log(f ))′(x). Put g = id − f and let Sg(x0) = {xn|n ∈ N} be an orbit of g, x0 < a. |At(Sg(x0))| ≍ g−1(t)

Examples of sets that have zero-log complex dimensions?

Not easy to find nice examples where we can calculate everything explicitly They will arise naturally as orbits of some discrete dynamical systems Theorem (Mardeˇ si´ c, Resman, ˇ Zupanovi´ c) Let f ∈ Diffr(0, a) be continuous on [0, a) , positive on (0, a) and let f (0) = f ′(0) = 0. Assume 1 < x(log(f ))′(x). Put g = id − f and let Sg(x0) = {xn|n ∈ N} be an orbit of g, x0 < a. |At(Sg(x0))| ≍ g−1(t) Let g(x) = xk log◦m(x−1), then g−1(t) ≍ t1/k log◦m(t−1)1/k

Examples of sets that have zero-log complex dimensions?

Not easy to find nice examples where we can calculate everything explicitly They will arise naturally as orbits of some discrete dynamical systems Theorem (Mardeˇ si´ c, Resman, ˇ Zupanovi´ c) Let f ∈ Diffr(0, a) be continuous on [0, a) , positive on (0, a) and let f (0) = f ′(0) = 0. Assume 1 < x(log(f ))′(x). Put g = id − f and let Sg(x0) = {xn|n ∈ N} be an orbit of g, x0 < a. |At(Sg(x0))| ≍ g−1(t) Let g(x) = xk log◦m(x−1), then g−1(t) ≍ t1/k log◦m(t−1)1/k we also have |At(Sg(x0))| ≍ t(− log t) for appropriate differentiable f

The 1/2-square fractal

Figure: Here, G := G1 ∪ G2 is the single generator of the corresponding self-similar spray or RFD (A, Ω), where Ω = (0, 1)2.

Fractal tube formula for the 1/2-square fractal

ζA(s) = 2−s s(s − 1)(2s − 2) + 4 s − 1 + 2π s , (6) D(ζA) = 1, P(ζA) := P(ζA, C) = {0} ∪ ( 1 + 2π log 2iZ ) . (7)

Fractal tube formula for the 1/2-square fractal

ζA(s) = 2−s s(s − 1)(2s − 2) + 4 s − 1 + 2π s , (6) D(ζA) = 1, P(ζA) := P(ζA, C) = {0} ∪ ( 1 + 2π log 2iZ ) . (7) |At| = ∑

ω∈P(ζA)

res ( t2−s 2 − s ζA(s), ω ) = 1 4 log 2t log t−1 + t G ( log2(4t)−1) + 1 + 2π 2 t2, (8) valid for all t ∈ (0, 1/2), where G is a nonconstant 1-periodic function on R bounded away from zero and ∞. The 1/2-square fractal is critically fractal in dimension 1.

Further research directions

Riemann surfaces generated by relative fractal drums Extending the notion of complex dimensions to include complicated “mixed” singularities/branching points and connecting them with various gauge functions Obtaining corresponding tube formulas and gauge-Minkowski measurability criteria Applying the theory to problems from dynamical systems

References

• C. Q. He and M. L. Lapidus, Generalized Minkowski content,

spectrum of fractal drums, fractal strings and the Riemann zeta-function, Mem. Amer. Math. Soc. No. 608, 127 (1997), 1–97.

• M. L. Lapidus and M. van Frankenhuijsen, Fractality, Complex

Dimensions, and Zeta Functions: Geometry and Spectra of Fractal Strings, second revised and enlarged edition (of the 2006 edn.), Springer Monographs in Mathematics, Springer, New York, 2013.

• M. L. Lapidus, G. Radunovi´

c, Fractal zeta functions and Logarithmic gauge Minkowski measurability, in preparation, 2017.

• M. L. Lapidus, G. Radunovi´

c and D. ˇ Zubrini´ c, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2017, to appear, approx. 680 pages.

References

• P. Mardeˇ

si´ c, M. Resman and V. ˇ Zupanovi´ c, Multiplicity of fixed points and growth of ϵ-neighborhoods of orbits, Journal of Diff.

• Equ. 253 (2012) 2493–2514.