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

Log-Minkowski measurability and complex dimensions Goran Radunovi´ c University of California, Riverside 14 th 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 ⊂ R N , Ω ⊂ R N , Lebesgue measurable, i.e., | Ω | < ∞ δ -neighbourhood of A : A δ = { x ∈ R N : d ( x , A ) < δ } upper r -dimensional Minkowski content of ( A , Ω): | A δ ∩ Ω | M r ( A , Ω) := lim sup δ N − r δ → 0 + upper Minkowski dimension of ( A , Ω): dim B ( A , Ω) = inf { r ∈ R : M r ( A , Ω) = 0 } lower Minkowski content and dimension defined via lim inf

Minkowski measurability dim B ( A , Ω) = dim B ( A , Ω) ⇒ ∃ dim B ( A , Ω) if ∃ D ∈ R such that 0 < M D ( A , Ω) = M D ( A , Ω) < ∞ , we say ( A , Ω) is Minkowski measurable ; in that case D = dim B ( 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 R N , s ∈ C and fix δ > 0 the distance zeta function of ( A , Ω): ∫ d ( x , A ) s − N dx ζ A , Ω ( s ; δ ) := A δ ∩ Ω dependence on δ is not essential

The relative distance zeta function ( A , Ω) RFD in R N , s ∈ C and fix δ > 0 the distance zeta function of ( A , Ω): ∫ d ( x , A ) s − N dx ζ A , Ω ( s ; δ ) := A δ ∩ Ω dependence on δ is not essential the complex dimensions of ( A , Ω) are defined as the poles of ζ A , Ω

The relative distance zeta function ( A , Ω) RFD in R N , s ∈ C and fix δ > 0 the distance zeta function of ( A , Ω): ∫ d ( x , A ) s − N dx ζ A , Ω ( s ; δ ) := A δ ∩ Ω dependence on δ is not essential the complex dimensions of ( A , Ω) are defined as the poles of ζ A , Ω take Ω to be an open neighborhood of A in order to recover the classical ζ A

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu] Theorem • ( A , Ω) RFD in R N : ( a ) ζ A , Ω ( s ) is holomorphic on { Re s > dim B ( A , Ω) }

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu] Theorem • ( A , Ω) RFD in R N : ( a ) ζ A , Ω ( s ) is holomorphic on { Re s > dim B ( A , Ω) } ( b ) R ∋ s < dim B ( A , Ω) ⇒ the integral defining ζ A , Ω ( s ) diverges

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu] Theorem • ( A , Ω) RFD in R N : ( a ) ζ A , Ω ( s ) is holomorphic on { Re s > dim B ( A , Ω) } ( b ) R ∋ s < dim B ( A , Ω) ⇒ the integral defining ζ A , Ω ( s ) diverges ( c ) if ∃ D = dim B ( A , Ω) < N and M D ( A , Ω) > 0, then ζ A , Ω ( s ) → + ∞ when R ∋ s → D +

Holomorphicity theorem for the relative distance zeta function [LapRaˇ Zu] Theorem • ( A , Ω) RFD in R N : ( a ) ζ A , Ω ( s ) is holomorphic on { Re s > dim B ( A , Ω) } ( b ) R ∋ s < dim B ( A , Ω) ⇒ the integral defining ζ A , Ω ( s ) diverges ( c ) if ∃ D = dim B ( A , Ω) < N and M D ( A , Ω) > 0, then ζ A , Ω ( s ) → + ∞ when R ∋ s → D + • we call { Re s = dim B ( A , Ω) } the critical line

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

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

(Generalized) complex dimensions of an RFD Definition Let W be a connected open set s.t. { Re s > dim B ( A , Ω) } ⊂ W and ζ A , Ω is holomorphic on W . The set of visible complex dimensions of ( A , Ω) ( with respect to W ) is the set of singularities P ( ζ A , Ω , W ) ⊂ ∂ W of ζ A , Ω . principal complex dimensions : dim PC ( A , Ω) := { ω ∈ P ( ζ A , Ω , W ) : Re ω = dim B ( 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 > dim B ( A , Ω) } ⊂ W and ζ A , Ω is holomorphic on W . The set of visible complex dimensions of ( A , Ω) ( with respect to W ) is the set of singularities P ( ζ A , Ω , W ) ⊂ ∂ W of ζ A , Ω . principal complex dimensions : dim PC ( A , Ω) := { ω ∈ P ( ζ A , Ω , W ) : Re ω = dim B ( 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 > dim B ( A , Ω) } ⊂ W and ζ A , Ω is holomorphic on W . The set of visible complex dimensions of ( A , Ω) ( with respect to W ) is the set of singularities P ( ζ A , Ω , W ) ⊂ ∂ W of ζ A , Ω . principal complex dimensions : dim PC ( A , Ω) := { ω ∈ P ( ζ A , Ω , W ) : Re ω = dim B ( 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 �→ | A t ∩ Ω | as t → 0 + in terms of ζ A , Ω .

Fractal tube formulas for relative fractal drums An asymptotic formula for the tube function t �→ | A t ∩ Ω | as t → 0 + in terms of ζ A , Ω . Theorem (Simplified pointwise formula with error term) • α < dim B ( A , Ω) < N ; ζ A , Ω satisfies suitable rational growth conditions ( d -languidity ) on the half-plane W := { Re s > α } , then: ( t N − s ) ∑ + O ( t N − α ) . | A t ∩ Ω | = res N − s ζ A , Ω ( s ) , ω ω ∈P ( ζ A , Ω , W )

Fractal tube formulas for relative fractal drums An asymptotic formula for the tube function t �→ | A t ∩ Ω | as t → 0 + in terms of ζ A , Ω . Theorem (Simplified pointwise formula with error term) • α < dim B ( A , Ω) < N ; ζ A , Ω satisfies suitable rational growth conditions ( d -languidity ) on the half-plane W := { Re s > α } , then: ( t N − s ) ∑ + O ( t N − α ) . | A t ∩ Ω | = res N − s ζ A , Ω ( s ) , ω ω ∈P ( ζ A , Ω , W ) if we allow polynomial growth of ζ A , 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 := dim B ( A , Ω) and D < N • ζ A , 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 ζ A , Ω located on the critical line { Re s = D } and it is simple. M D ( A , Ω) = res( ζ A , 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 R N +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 r 1 = r 2 = r 3 = 1 / 2 ( A , Ω) = ( ∂ G , G ) ⊔ ⊔ 3 j =1 ( r j A , r j Ω)

i i i Fractal tube formula for The Sierpi´ nski gasket √ 3) 1 − s 2 − s s + 3 δ s − 1 s ( s − 1)(2 s − 3) + 2 πδ s 6( ζ A ( s ; δ ) = s − 1

i i Fractal tube formula for The Sierpi´ nski gasket √ 3) 1 − s 2 − s s + 3 δ s − 1 s ( s − 1)(2 s − 3) + 2 πδ s 6( ζ A ( s ; δ ) = s − 1 ( log 2 3 + 2 π ) P ( ζ A ) = { 0 , 1 } ∪ log 2 i Z

i Fractal tube formula for The Sierpi´ nski gasket √ 3) 1 − s 2 − s s + 3 δ s − 1 s ( s − 1)(2 s − 3) + 2 πδ s 6( ζ A ( s ; δ ) = s − 1 ( log 2 3 + 2 π ) P ( ζ A ) = { 0 , 1 } ∪ log 2 i Z By letting ω k := log 2 3 + p k i and p := 2 π/ log 2 we have that

i Fractal tube formula for The Sierpi´ nski gasket √ 3) 1 − s 2 − s s + 3 δ s − 1 s ( s − 1)(2 s − 3) + 2 πδ s 6( ζ A ( s ; δ ) = s − 1 ( log 2 3 + 2 π ) P ( ζ A ) = { 0 , 1 } ∪ log 2 i Z By letting ω k := log 2 3 + p k i and p := 2 π/ log 2 we have that ( t 2 − s ) ∑ | A t | = res 2 − s ζ A ( s ; δ ) , ω ω ∈P ( ζ A )