A Tapestry of Complex Dimensions
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1 A Tapestry of Complex Dimensions John A. Rock May 7th,
2 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
3 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
4 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
5 /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
6 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
7 /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
8 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 ( ) s ( 2 j=1 ( j+2 3 m 2 ) s ) s ( s 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
9 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
10 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
11 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
12 References [1] M. Arbeiter and N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), [2] G. Brown, G. Michon, and J. Peyrière, On the multifractal analysis of measures, J. Statist. Phys. 66 (1992), [3] R. Cawley, R. D. Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), [4] D. L. Cohn, Measure Theory, Birkhäuser, Boston, [5] G. A. Edgar, R. D. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. London Math. Soc. 65 (1992), [6] R.S. Ellis, Large deviations for a general class of random vectors, Ann. Prob. 12 (1984), [7] K. Falconer, Fractal Geometry Mathematical foundations and applications, 2nd ed., John Wiley, Chichester,
13 [8] S. Jaffard, Multifractal formalism for functions, SIAM J. Math. Anal. 28 (1997), [9] S. Jaffard, Oscillation spaces: properties and applications to fractal and multifractal functions, J. Math. Phys. 38 (1998), [10] S. Jaffard, The multifractal nature of Lévy processes, Probab. Theory Related Fields 114 (1999), [11] S. Jaffard, Wavelet techniques in multifractal analysis, in: [13], pp [12] S. Jaffard and Y. Meyer, Wavelet methods for pointwise regularity and local oscilations of functions, Mem. Amer. Math. Soc., No. 587, 123 (1996), [13] M. L. Lapidus and M. van Frankenhuijsen (eds.), Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2, Amer. Math. Soc., Providence, RI, [14] M. L. Lapidus and J. A. Rock, Towards zeta functions and complex dimensions of mul- 13
14 tifractals, Complex Variables and Elliptic Equations, special issue dedicated to fractals, in press. (See also: Preprint, Institut des Hautes Etudes Scientfiques, IHES/M/08/34, 2008.) [15] M. L. Lapidus and J. A. Rock, Partition zeta functions and multifractal probability measures, preliminary version, [16] K. S. Lau and S. M. Ngai, L q spectrum of the Bernouilli convolution associated with the golden ration, Studia Math. 131 (1998), [17] J. Lévy Véhel, Introduction to the multifractal analysis of images, in: Fractal Images Encoding and Analysis (Y. Fisher, ed.), Springer- Verlag, Berlin, [18] J. Lévy Véhel and F. Mendivil, Multifractal strings and local fractal strings, preliminary version, [19] J. Lévy Véhel and R. Riedi, Fractional Brownian motion and data traffic modeling: The other end of the spectrum, in: Fractals in Engineering (J. Lévy Véhel, E. Lutton and C. Tricot, eds.), Springer-Verlag, Berlin,
15 [20] J. Lévy Véhel and S. Seuret, The 2-microlocal formalism, in: [13], pp [21] J. Lévy Véhel and R. Vojak, Multifractal analysis of Choquet capacities, Adv. in Appl. Math. 20 (1998), [22] B. B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of hight moments and dimension of the carrier, J. Fluid. Mech. 62 (1974), [23] B. B. Mandelbrot, Multifractals and 1/f Noise, Springer-Verlag, New York, [24] L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Math. Series, vol. 307, Longman Scientific and Technical, London, [25] L. Olsen, A multifractal formalism, Adv. Math. 116 (1996), [26] L. Olsen, Multifractal geometry, in: Fractal Geometry and Stochastics II (Greifswald/Koserow, 1998), Progress in Probability, vol. 46, Birkhäuser, Basel, 2000, pp
16 [27] G. Parisi and U. Frisch, Fully developed turbulence and intermittency inturbulence, and predictability in geophysical fluid dynamics and climate dynamics, in: International School of Enrico Fermi, Course 88 (M. Ghil, ed.), North-Holland, Amsterdam, 1985, pp [28] J. A. Rock, Zeta Functions, Complex Dimensions of Fractal Strings and Multifractal Analysis of Mass Distributions, Ph. D. Dissertation, University of California, Riverside,
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