A Tapestry of Complex Dimensions John A. Rock May 7th, 2009 1
f (α) dim M(supp( β)) (a) (b) α Figure 1: (a) Construction of the binomial measure β. spectrum f(α) of the measure β. (b) The multifractal 1 Multifractal Measures and Their Spectra Multifractals are used to model natural phenomena which have very irregular structure. The distribution of stars in a galaxy, the distribution of minerals in a mine, and the formation of lightning are considered to be multifractal and are mathematically modeled by measures. GOAL: In this talk, we ll find most members of the family of multifractal zeta functions for a simple measure σ and compute the corresponding multifractal spectrum and complex dimensions. 2
2 Measures and Regularity The measure we consider acts on closed subintervals U of the unit interval [0, 1]. Multifractal zeta functions are parameterized regularity, which connects the size of a interval with its mass. Definition 2.1. The regularity A(U) of an interval U with respect to the measure β is A(U) = log β(u) log U, where U is the length of U. Regularity A(U) is also known as the coarse Hölder exponent α which satisfies U α = β(u). 3
3 Multifractal Zeta Functions Collecting the lengths of the intervals K n p (α) according to their regularity α allows us to define the multifractal zeta functions. Definition 3.1. The multifractal zeta function of a measure µ, sequence os scales N and with associated regularity value α is ζ µ N (α, s) = n=1 for Re(s) large enough. k n (α) p=1 K n p (α) s, What are these K n p (α), exactly? Ask me later. 4
1100 0000000 1111111 0000000000000000000 1111111111111111111 1100 0000000 1111111 0000000000000000000 1111111111111111111 1100 0000000 1111111 0000000000000000000 1111111111111111111...1/27 1/9 1/3 σ 4 Simple σ Figure 2: Approximation of the measure σ. Let N = {3 n } n=1 and σ = 3 j δ 3 j. j=1 [0,1] goal: Find all the regularity values attained by σ with intervals U that have length in N. 5
The positive values of σ(u) are obtained in one of the two following ways: Case 1: U contains exactly one point-mass of size 3 j where j N. Case 2: U contains two or more point-masses, necessarily including the point-mass 3 N. If any other smaller point-mass 3 p is also contained in U, so are all point-masses 3 j between 3 p and 3 N (i.e., N j p). That is, U contains any finite or infinite sequence of point-masses {3 j } p j=n, where p > N. Lemma 4.1. For the measure σ and sequence of scales N = {3 n } n=1, the possible finite regularity values of U where U = 3 N for some fixed N N are: α(m 1, m 2 ) = log 3 m 1n log 3 m 2n = m 1 m 2, α N (p) = log ((3p N+1 1)/2) N log 3 + p + 1 N, where m 1 < m 2 and (m 1, m 2 ) = 1 for m 1, m 2 N, and p N { } and p > N. These regularity values are all distinct from one another. 6
1100 0000000 1111111 0000000000000000000 1111111111111111111 1100 0000000 1111111 0000000000000000000 1111111111111111111 1100 0000000 1111111 0000000000000000000 1111111111111111111...1/27 1/9 1/3 [0,1] σ 2 K (1/2) 4 K (1/2) 6 K (1/2) Figure 3: Approximation of σ and the construction of ζn σ (1/2, s). The solid black bars represent the K n p (α) with α = 1/2 that generate the terms of the multifractal zeta function. For α(m 1, m 2 ) = m 1 /m 2, only stages at multiples of m 2 have intervals with the correct regularity, hence the other stages are skipped. 7
The breakdown for all possible regularity values associated with the measure σ and sequence N provided by Lemma 4.1 allows for the complete breakdown of all the possible multifractal zeta functions of σ with N. Theorem 4.2. For the measure σ and sequence N = {3 n } n=1, the nontrivial multifractal zeta functions have the following forms*: ) ( ) s ( ) s ( 2 2 = ζ σ N ( m1 m 2, s = ζ σ N (1, s) = = j=1 3 m 2j ( ) s 5 + 9 ( ) s 5 + 9 ( 2 j=1 ( 2 27 3 j+2 3 m 2 ) s ) s ( 1 1 3 s 1 1 3 m 2s ), where m 1 < m 2 and (m 1, m 2 ) = 1 for all n, m 1, and m 2 N. For all other regularity values, the corresponding multifractal zeta functions are entire. ), *This list is incomplete, technically. 8
5 Complex Dimensions Remark 5.1. Similar to what is done with geometric zeta functions, under appropriate conditions it is assumed that, as a function of s C, ζ µ N (α, s) admits a meromorphic continuation to an open neighborhood of a window W. We may then consider the poles of these zeta functions as complex dimensions. Definition 5.2. For a measure µ, sequence N which tends to zero and regularity value α, the set of complex dimensions with parameter α is given by D µ N (α, W ) = {ω W ζµ N (α, s) has a pole at ω}. 9
The formulas for the multifractal zeta functions provided by Theorem 4.2 immediately yield the following collections of complex dimensions with all regularity values α. Corollary 5.3. Under the assumptions of Theorem 4.2, the complex dimensions with parameter α of the measure σ and sequence N = {3 n } n=1 are the poles of the multifractal zeta function ζn σ (α, s). For the nontrivial values of α described in Theorem 4.2, { DN σ (m 1 /m 2, W ) = ω W ω is a pole of ζn σ { } 2πiz =, m 2 log 3 z Z ( )} m1, s m 2 DN σ (1, W ) = {ω W ω is a pole of ζn σ (1, s)} { } 2πiz =, log 3 for appropriate windows W. z Z 10
Remark 5.4. All of the poles above have real part zero. Consider the space R C, where to R we associate the collection of finite regularity values α and to C we associate the corresponding complex dimensions with parameter α. For the measure σ and sequence N, the full family of complex dimensions of all α is a dense subset of the strip in R C given by [0, 1] {s C Re(s) = 0}. Specifically, we get the set { (α, ω) α [0, 1] Q, Re(ω) = 0, Im(ω) = 2πiz k log 3 for k N This is the tapestry of complex dimensions corresponding to the measure σ. 11
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