Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 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 Singularities of Functions x 0 0.2 0.4 0.6 0.8 1 100 80 60 z 40 20 0 0 0.2 0.4 0.6 0.8 1 y The graph of f ( x ) = d ( x , A ) − γ has countably many components. Here, A ⊂ R 2 is the Sierpinski carpet. Then: f ∈ L 1 ([0 , 1] 2 ) ⇔ γ < 2 − log 8 / log 3 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Singularities of Functions Figure 11 : Fractal stalactites associated with the Sierpinski carpet, revisited.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Singularities of Functions The figure on the previous slide depicts another view of the graph of the same function y = d ( x , A ) − γ . The level set of this function tends to the Sierpinski carpet in the Hausdorff metric, when the level tends to + ∞ .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Singularities of Functions (a) Fractal stalagmites (b) Fractal stalactites Figure 12 : Soothing fractal cave of stalagmites and stalactites
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Singularities of Functions (a) Fractal stalagmites (b) Fractal stalactites Figure 13 : Soothing fractal cave of stalagmites and stalactites
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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: ∞ � ( ℓ j ) s , ζ L ( s ) = j =1 for s ∈ C with Re s large enough.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Zeta Functions of Fractal Strings Consider the open intervals I j = ( a j , a j − 1 ) for j ≥ 1, where � a j := ℓ k , and ℓ j := | I j | . k > 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 [LRˇ Z]. dim B A 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 Zeta Functions of Fractal Strings Theorem (L) The abscissa of convergence of ζ L is equal to dim B A : � � ∞ � ( ℓ j ) α < ∞ dim B A = inf α > 0 : . 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 ) → + ∞ .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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; LRˇ Z, 2013) The distance zeta function of A is defined by � d ( x , A ) s − N d x , ζ A ( s ) = A δ for s ∈ C with Re s sufficiently large.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 � d ( x , A ) s − N d x ζ A ( s ; A δ 1 ) − ζ A ( s ) = A δ,δ 1 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 � 1+ δ d ( x , A ) s − 1 d x , for δ ≥ ℓ 1 / 2 . ζ A ( s ) = − δ
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 � 1+ δ d ( x , A ) s − 1 d x , for δ ≥ ℓ 1 / 2 . ζ A ( s ) = − δ 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).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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; LRˇ Z, 2013).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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; 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 ζ A in U .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity Holomorphy Half-Plane of the Distance Zeta Function Let A be a nonempty bounded set in R N ; given δ a fixed positive � A δ d ( x , A ) s − N d x as before. number, let ζ A ( s ) = Theorem (LRˇ Z) 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 ) = + ∞ .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity Corollary (LRˇ Z) The abscissa of (absolute) convergence of ζ A is equal to dim B A , the (upper) Minkowski dimension of A : � � � d ( x , A ) α − N dx < ∞ D ( ζ A ) := inf α ∈ R : A δ = dim B A . Corollary (LRˇ Z) 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity Proof of Analyticity The proof of holomorphicity is based on the following result, which was stated earlier: Theorem (Harvey & Polking, 1970) Assume that A is a bounded set in R N and δ > 0 is given. Then � d ( x , A ) − γ d x < ∞ . γ < N − dim B A ⇒ A δ
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity Proof of Analyticity The proof of holomorphicity is based on the following result, which was stated earlier: Theorem (Harvey & Polking, 1970) Assume that A is a bounded set in R N and δ > 0 is given. Then � d ( x , A ) − γ d x < ∞ . γ < N − dim B A ⇒ A δ Remarks : If γ > N − dim B A , then the integral is equal to + ∞ . If D := dim B A exists and M D ∗ ( A ) > 0, then the converse also holds (ˇ Z., ISAAC Proc. 2009). The lower Minkowski content condition is essential (ˇ Z., RAE, 2005).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 (LRˇ Z) If M ∗ D ( A ) < ∞ , then s = D is a simple pole of ζ A and ∗ ( A ) ≤ res( ζ A ( · , A δ ) , D ) ≤ ( N − D ) M ∗ D ( A ) . ( N − D ) M D 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Residues of Distance Zeta Functions Theorem (LRˇ Z) 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 ) .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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) � δ ˜ t s − N − 1 | A t | d t , ζ A ( s ) = 0 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 ˜ ζ A .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 (LRˇ Z) 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 ) .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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, � 2 D � 1 − D 1 M D ∗ ( A ) = , 1 − D D � 1 � D − 1 M ∗ D ( A ) = 2(1 − a ) 2 − a , and � 1 � D 2 res(˜ ζ A , D ) = 2 − a . log 2
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Residues of Tube Zeta Functions Example (1 continued) For all a ∈ (0 , 1 / 2) , we have ∗ ( A ) < res(˜ ζ A ( s ) , D ) < M ∗ D ( A ) . M D 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Residues of Tube Zeta Functions Residues and Minkowski Contents for Generalized Cantor Sets —– refers to the residue of ˜ ζ A at D ( a ) = dim B A as a function of a ∈ (0 , 1 / 2), where A = C ( a ) . —– and —– refer to the lower and upper Minkowski contents of A = C ( a ) , respectively, as a function of a ∈ (0 , 1 / 2).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 j = j − a − ( j + 1) − a . L = ( ℓ j ) j ≥ 1 , We have: 1 D ( a ) = dim B A = 1 + a , 2 1 − D res(˜ ζ A , D ) = M D ( A ) = D (1 − D ) a D , res( ζ A , D ) = (1 − D ) M D ( A ) = 2 1 − D a D , D and res( ζ L , D ) = a D .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Residues of Tube Zeta Functions — refers to the Minkowski content M D ( A ) of the a -string, as a function of a > 0.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta ( 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):
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta ( 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): � { k − 1 /β } × (0 , k − α/β ) . Γ( α, β ) = k ∈ N
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta ( α, β )-chirps Distance Zeta Function of Geometric Chirps The distance zeta function of Γ( α, β ) can be computed as follows: ∞ � 1 k − α β − (1+ 1 β )( s − 1) , ζ Γ( α,β ) ( s ) ∼ s − 1 k =1 where, as before, we define 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta ( α, β )-chirps Distance Zeta Function of Geometric Chirps The distance zeta function of Γ( α, β ) can be computed as follows: ∞ � 1 k − α β − (1+ 1 β )( s − 1) , ζ Γ( α,β ) ( s ) ∼ s − 1 k =1 where, as before, we define 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 , 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 Meromorphic Extensions of Fractal Zeta Functions Minkowski Measurable Sets Theorem (LRˇ Z (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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Minkowski Measurable Sets Theorem (LRˇ Z (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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Example Let A ⊂ R be the bounded set associated to the a -string. Then A is Minkowski measurable and 3 | A t | = t 1 − D ( M + O ( t 2 D − ε )) as t → 0 + , where D = 1 / ( a + 1), M = 2 1 − D a D / D (1 − D ), and ε > 0 is arbitrary.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Example Let A ⊂ R be the bounded set associated to the a -string. Then A is Minkowski measurable and 3 | A t | = t 1 − D ( M + O ( t 2 D − ε )) as t → 0 + , where D = 1 / ( a + 1), M = 2 1 − D a D / D (1 − D ), and ε > 0 is arbitrary. Conclusion : ˜ ζ A and ζ A have a (unique) meromorphic extension to { Re s > D − 3 2 D = − 1 2 D } . In fact, D mer (˜ ζ A ) = −∞ (This can be deduced from [L-vF, 2000].)
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (LRˇ Z (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 : ∗ ( A ) = min G , M ∗ D ( A ) = max G , and D (˜ dim B A = D , M D ζ A ) = D .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (LRˇ Z (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 : ∗ ( A ) = min G , M ∗ D ( A ) = max G , and D (˜ dim B A = D , M 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 � � s k = D + 2 π G 0 ( k P (˜ T i k : ˆ ζ A ) = T ) � = 0 , k ∈ Z ;
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (. . . continued) � T they are all simple. Here, ˆ 0 e − 2 π i s · τ G ( τ ) d τ. G 0 ( s ) := For all s k ∈ P (˜ ζ A ) , res(˜ T ˆ ζ A , s k ) = 1 G 0 ( k T ) . We have � T ζ A , s k ) | ≤ 1 | res(˜ k →±∞ res(˜ G ( τ ) d τ, lim ζ A , s k ) = 0 . T 0
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Fractal Zeta Functions Minkowski Nonmeasurable Sets Theorem (. . . continued) � T they are all simple. Here, ˆ 0 e − 2 π i s · τ G ( τ ) d τ. G 0 ( s ) := For all s k ∈ P (˜ ζ A ) , res(˜ T ˆ ζ A , s k ) = 1 G 0 ( k T ) . We have � T ζ A , s k ) | ≤ 1 | res(˜ k →±∞ res(˜ G ( τ ) d τ, lim ζ A , s k ) = 0 . T 0 Moreover, � T ζ A , D ) = 1 res(˜ G ( τ ) d τ T 0 and ∗ ( A ) < res(˜ M D ζ A , 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 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 � T M = 1 res(˜ ζ A , D ) = � G ( τ ) d τ. T 0 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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: � log 3 } � � 3 � −{ τ − log 2 2 { τ − log 2 log 3 } + G ( τ ) = 2 1 − D , 2 where { x } := x − ⌊ x ⌋ is the fractional part of x and ⌊ x ⌋ is the integer part of x .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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: � log 3 } � � 3 � −{ τ − log 2 2 { τ − log 2 log 3 } + G ( τ ) = 2 1 − D , 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 3 2 + 2 π log 3 i Z .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 [LRˇ Z] for details.)
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity of Relative Zeta Functions Definition (Relative distance zeta function of ( A , Ω) , LRˇ Z) � d ( x , A ) s − N d x , ζ A ( s , Ω) = Ω for s ∈ C with Re s sufficiently large.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Analyticity of Relative Zeta Functions Definition (Relative distance zeta function of ( A , Ω) , LRˇ Z) � d ( x , A ) s − N d x , ζ A ( s , Ω) = Ω for s ∈ C with Re s sufficiently large. Theorem (LRˇ Z) 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) → + ∞ .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Definition (Relative tube zeta function of ( A , Ω) , LRˇ Z) � δ ˜ t s − N − 1 | A t ∩ Ω | dt , ζ A ( s , Ω) = 0 for s ∈ C with Re s sufficiently large. (Here, δ > 0 is fixed.)
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Definition (Relative tube zeta function of ( A , Ω) , LRˇ Z) � δ ˜ t s − N − 1 | A t ∩ Ω | dt , ζ A ( s , Ω) = 0 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) .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Relative Zeta Functions Theorem (LRˇ Z, logarithmic gauge functions) (1. Minkowski measurable case) Let ( A , Ω) be a relative fractal drum in R N such that | A t ∩ Ω | = t N − D (log t − 1 ) m ( M + O ( t α )) as t → 0 + , where m ∈ N 0 . Then dim B ( A , Ω) = D (˜ ζ A ( · , Ω)) = D and ˜ ζ A ( · , Ω) has a unique meromorphic extension to { Re s > D − α } .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Relative Zeta Functions Theorem (LRˇ Z, logarithmic gauge functions) (1. Minkowski measurable case) Let ( A , Ω) be a relative fractal drum in R N such that | A t ∩ Ω | = t N − D (log t − 1 ) m ( M + O ( t α )) as t → 0 + , where m ∈ N 0 . Then dim B ( A , Ω) = D (˜ ζ A ( · , Ω)) = D and ˜ ζ A ( · , Ω) has a unique meromorphic extension to { Re s > D − α } . Moreover, s = D is the unique pole in this half-plane; it is of order m + 1 , and c − m − 1 = m ! M , c − m = · · · = c − 1 = 0 , ζ A ( s , Ω) = � ∞ where ˜ j = −∞ c j ( s − D ) j is the Laurent series of ˜ ζ A ( · , Ω) near s = D.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Relative Zeta Functions Theorem (LRˇ Z, logarithmic gauge functions) (2. Minkowski nonmeasurable case) Let ( A , Ω) be a relative fractal drum in R N such that there exist D ≥ 0 , a nonconstant periodic function G : R → R with minimal period T > 0 , m ∈ N 0 , and α > 0 satisfying | A t ∩ Ω | = t N − D (log t − 1 ) m � � G (log t − 1 ) + O ( t α ) as t → 0 + .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Meromorphic Extensions of Relative Zeta Functions Theorem (LRˇ Z, logarithmic gauge functions) (2. Minkowski nonmeasurable case) Let ( A , Ω) be a relative fractal drum in R N such that there exist D ≥ 0 , a nonconstant periodic function G : R → R with minimal period T > 0 , m ∈ N 0 , and α > 0 satisfying | A t ∩ Ω | = t N − D (log t − 1 ) m � � G (log t − 1 ) + O ( t α ) as t → 0 + . Then dim B ( A , Ω) = D (˜ ζ A ( · , Ω)) = D , and ˜ ζ A ( · , Ω) has a unique meromorphic extension ( at least ) to { Re s > D − α } .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Theorem (... continued) All of the ( visible ) poles of ˜ ζ A ( · , Ω) are of order m + 1 , and � � s k = D + 2 π G 0 ( k P (˜ T ik ∈ C : ˆ ζ A ( · , Ω)) = T ) � = 0 , k ∈ Z , where ( as before ) � T ˆ e − 2 π i s · τ G ( τ ) d τ. G 0 ( s ) := 0 Also, s 0 = D ∈ P (˜ ζ A ( · , Ω)) . Remark : Provided D < N , the exact same theorem holds for the relative distance zeta function ζ A ( · , Ω) (instead of ˜ ζ A ( · , Ω)).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Figure 14 : The complex dimensions of a relative fractal drum.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 2 − 1 + 4 π (log 2) i Z 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 } .
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Quasiperiodic Sets and Hyperfractals Definition A set A ⊂ R N is quasiperiodic if | A t | = t N − D ( G (log 1 / t ) + O ( t α )) as t → 0 + , for some D ≥ 0, α > 0, and where G ( τ ) is a quasiperiodic function (see the definition below). If G = G 1 + G 2 and for j = 1 , 2, the functions G j are T j -periodic with T 1 / T 2 irrational , then A is quasiperiodic (and not periodic).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta The proper definition of quasiperiodic function needed in the present context will be given shortly. It is quite different from the classic definition of quasiperiodicity (used for example, in the theory of quasicrystals), which we now recall. Definition A quasiperiodic function g ( in the classical sense ) is the restriction to an affine subspace E of R m of a multiply periodic function on R m with period lattice Λ ⊂ R m . If E is in “general position” and m ≥ 2, then g is not periodic (with a single minimal period).
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta Example Let C ( m , a ) be a suitably constructed two-parameter family of generalized Cantor sets. (See [LRˇ Z].) Let A = C (1 / 3 , 2) ∪ C ( b , 3) ⊂ [0 , 1] ∪ [2 , 3] , where b is chosen so that D = log 3 2 = log 1 / b 3 ( b = 3 − log 2 3 ) . Then T 1 = log 3, T 2 = log(1 / b ), and T 1 / T 2 = log 3 2 is irrational; it is even transcendental, by the Gel’fond–Schneider theorem (1934). Hence, A is a quasiperiodic set.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 [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 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.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 to the bibliography.
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta 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 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
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta References M. L. Lapidus, M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of 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]) c, D. ˇ M. L. Lapidus, G. Radunovi´ Zubrini´ c, Fractal Zeta Functions and Fractal Drums : Higher-Dimensional Theory of Complex Dimensions , research monograph, Springer, 2016, approx. 450 pages. ([LRˇ Z])
Outline Definitions and Motivations Fractal strings Distance and Tube Zeta Functions Relative Distance and Tube Zeta References c and D. ˘ M. L. Lapidus, G. Radunovi´ Zubrini´ c, Distance and tube zeta functions of fractals and arbitrary compact sets, preprint, 2015. c and D. ˘ M. L. Lapidus, G. Radunovi´ Zubrini´ c, Complex dimensions of fractals and meromorphic extensions of fractal zeta functions, preprint, 2015. c and D. ˘ M. L. Lapidus, G. Radunovi´ Zubrini´ c, Zeta functions and complex dimensions of relative fractal drums: Theory, examples and applications, preprint, 2015.
Recommend
More recommend
