Multifractal Analysis. A selected survey. Lars Olsen

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1 Multifractal Analysis. A selected survey Lars Olsen

$Multifractal Analysis: The beginning 1974 Frontispiece of: Mandelbrot, Intermittent turbulence in self-similar cascades:$ $The idea of multifractals originates from 1974 in a paper by Mandelbrot analyzing the dissipation of energy in a turbulent$

2 Multifractal Analysis: The beginning 1974 Frontispiece of: Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, 1974 Multifractal analysis refers to a particular way of analysing the local structure of measures. The idea of multifractals originates from 1974 in a paper by Mandelbrot analyzing the dissipation of energy in a turbulent fluid: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier

$Multifractal Analysis: Revisited$ of: Halsey et al, Fractal

$1986 Mandelbrot s multifractal$ context while being expanded and

3 Multifractal Analysis: Revisited by physicists 1986 Frontispiece of: Halsey et al, Fractal Measures and their Singularities, 1986 Mandelbrot s multifractal ideas were revisited in a broader context while being expanded and clarified in 1986 in a paper by theoretical physicists Halsey et al: Fractal Measures and their Singularities

4 In order to explain the ideas behind multifractal analysis we require two concepts: Local dimensions and the multifractal spectrum; L q -dimensions. Local dimensions and the multifractal spectrum Definition. Local dimension. Let µ be a measure on a metric space X. The local dimension of µ at x X is defined by log µ(b(x, r dim loc (x; µ = lim. Local dimensions are not new concepts: µ(b(x,r Local dimensions are related to densities lim r t, and densities have a long history in (geometric measure theory starting in the 1920 s; Local dimensions are related to the mass distribution principle starting in the 1930 s (Frostman, Billingsley and others; The local dimension of µ at x X measures the dimensional behaviour of µ in a neighbourhood of x: µ(b(x, r r dim loc (x;µ

5 Definition. Multifractal spectrum. Let µ be a measure on a metric space X. The Hausdorff multifractal spectrum of µ is defined by f H,µ (α = dim H (x X lim log µ(b(x, r = α, α R, The packing multifractal spectrum of µ is defined by f P,µ (α = dim P (x X lim log µ(b(x, r = α, α R.

6 L q -dimensions Definition. L q -dimensions. Let µ be a measure on a metric space X. For q R we define the lower and upper L q -dimensions of µ by τ µ (q = lim inf X log µ(q q Q is an r grid box with Q X, log r τ µ(q = lim sup log X Q is an r grid box with Q X log r µ(q q. L q -dimensions are not new concepts: Moments of measures have a long history in probability theory; Related dimensions were introduced in information theory in the 1950 s (Renyi and others; The modern definition of L q -dimensions was introduced by theoretical physicists in the 1980 s (Halsey et al, Proccacia, Grassberger and others. L q -dimensions extend the usual fractal dimensions: τ µ (0 = the lower box dimension of X, τ µ(0 = the upper box dimension of X.

7 So... what did Halsey et al say in their 1986 paper? Frontispiece of: Halsey et al, Fractal Measures and their Singularities, 1986 The Ergodic Theorem shows the following: for many natural measures µ there is a constant α µ such that dim H (x X lim log µ(b(x, r = α µ = dim H X. In 1986 theoretical physicists Halsey et al s paper Fractal Measures and their Singularities suggested to following remarkable result, known as the Multifractal Formalism, revealing an enormous complexity not foreseen by the Ergodic Theorem.

8 The Multifractal Formalism. A physics conjecture. Let µ be a measure on a metric space X. For q R we define the lower and upper L q -dimensions of µ by A version of the Ergodic Theorem. For many natural measures µ there is a constant α µ such that dim H (x X lim log µ(b(x, r = α µ = dim H X. τ µ (q = lim inf τ µ(q = lim sup Then for all α 0, we have PQ X µ(qq, log r PQ X µ(qq. log r dim H (x X lim log µ(b(x, r = α = τ µ (α = τ µ (α.

9 The Multifractal Formalism is remarkable: Revealing an enormous complexity not foreseen by the Ergodic Theorem. There is an uncountable number of α such that dim H (x X lim log µ(b(x, r log r = α > 0. A surprising relationship between global and local quantities. The L q -dimensions τ µ (q, τ µ(q The Multifractal Formalism. A physics conjecture. Let µ be a measure on a metric space X. For q R we define the lower and upper L q -dimensions of µ by τ µ (q = lim inf PQ X µ(qq, log r are global quantities; the local dimension log µ(b(x, r lim is a local quantity. There are no reasons to expect any relationship between the L q -dimensions and the local dimensions. Clearly false. The Multifractal Formalism is also remarkable because it is clearly false: it is easy to find measures that do not satisfy the Multifractal Formalism; it is difficult to find interesting measures that satisfies the Multifractal Formalism. τ µ(q = lim sup Then for all α 0, we have PQ X µ(qq. log r dim H (x X lim log µ(b(x, r = α = τ µ (α = τ µ (α.

10 Multifractal Analysis: Explored by mathematicians The Multifractal Formalism was quickly seized by the mathematical community. Mathematical objectives: investigate the validity of the Multifractal Formalism; provide rigorous foundations for the heuristic arguments in physics. By 1992 two papers had appeared verifying the Multifractal Formalism for two types of measures exhibiting some degree of self-similarity: Gibbs states on hyperbolic cookie-cutters in R (Rand; Moran self-similar measures in R d (Cawley & Mauldin.

William Blake (28 November 1757-12 August 1827 The true method of knowledge is by

11 William Blake (28 November August 1827 The true method of knowledge is by example. Let us follow Blake s advice and consider an example, namely, self-similar measures.

Self-similar measures Example Subdivide the mass of any interval between its 2 daughter-intervals in the ratio 2 3 : 1 3 We have µ = `left part of µ + `right part of µ Let (p 1, p 2 = ( 2 3, 1 3 Let

12 Self-similar measures Example Subdivide the mass of any interval between its 2 daughter-intervals in the ratio 2 3 : 1 3 We have µ = `left part of µ + `right part of µ Let (p 1, p 2 = ( 2 3, 1 3 Let S 1 (x = 1 3 x and S 2(x = 1 3 x Then µ = `left part of µ + `right part of µ = p 1 µ S 1 + p 1 2 µ S 1 2 A measure having this property is called selfsimilar. The precise definition is... µ {z } left part of µ {z } right part of µ

12, 0, S3 (x, y = 12 (x, y + (0, 12, and S4 (x, y = 12 (x, y + ( 12, 12 Then ` ` µ = bottom left part of µ + bottom right part of µ ` ` + top left part

13 Example Subdivide the mass of any square between its 4 daughter-squares in the ratio 72 : 72 : 27 : 17 We have ` ` µ = bottom left part of µ + bottom right part of µ ` ` + top left part of µ + top right part of µ Let (p1, p2, p3, p4 = ( 17, 27, 27, 27 Let S1 (x, y = 12 (x, y, S2 (x, y = 21 (x, y + ( 12, 0, S3 (x, y = 12 (x, y + (0, 12, and S4 (x, y = 12 (x, y + ( 12, 12 Then ` ` µ = bottom left part of µ + bottom right part of µ ` ` + top left part of µ + top right part of µ 1 = p1 µ S1 + p3 µ µ 1 + p2 µ S2 1 S3 1 + p4 µ S4 A measure having this property is called selfsimilar. The precise definition is...

14 Definition. Self-similar set and self-similar measure. Hutchinson (1981. Let (S 1,..., S N be a list of similarities S i : R d R d. Write r i for the contraction ratio of S i Let (p 1,..., p N be a probability vector. Let K and µ be the self-similar set and the self-similar measure associated with (S i, p i N i=1, i.e. K = µ = [ S i (K, i X p i µ S 1. i i Usually people assume various separation conditions. Definition. Open Set Condition (OSC. The (S 1,..., S N satisfies the OSC, if there is a non-empty and bounded open set such that S i (U U for all i and S i (U S j (U = for all i and j with i j. Definition. Strong Separation Condition (SSC. The (S 1,..., S N satisfies the OSC, if S i (K S j (K = for all i and j with i j.

$Multifractal Analysis of Self-Similar Measures Frontispiece of: Cawley & Mauldin, Multifractal Decomposition of Moran Fractals, 1992 In 1992,$

15 Multifractal Analysis of Self-Similar Measures Frontispiece of: Cawley & Mauldin, Multifractal Decomposition of Moran Fractals, 1992 In 1992, Cawley & Mauldin verified the Multifractal Formalism for self-similar measures satisfying the SSC. L Mejlbro, D Mauldin, F Topsøe, J P R Christensen

16 Theorem. Cawley & Mauldin (1992. Let K and µ be the self-similar set and measure associated with the list (S i, p i N i=1. Assume that the SSC is satisfied. Define β : R R by X p q r β(q = 1. i i i For all q R, we have h i log p For all α min i log p i, max i log r i, we have i log r i τ µ (q = τ µ(q = β(q. dim H (x K lim log µ(b(x, r = α = β (α, dim P (x K lim log µ(b(x, r = α = β (α. h i log p For all α min i log p i, max i log r i, we have i log r i ( x K lim log µ(b(x, r = α =.

17 Multifractal Analysis: Studied by mathematicians after Multifractal analysis of other types of measures (Self-conformal measures, self-affine measures,... Attempts to construct general axiomatic multifractal formalisms Multifractal analysis of measures from different viewpoints (Divergence points, multifractal properties of typical (in the sense of Baire measures, multifractal properties of prevalent (in the sense of Christensen and Hunt, Sauer & York measures Multifractal analysis in dynamical systems: (Multifractal analysis of local entropies, multifractal analysis of Lyapunov exponents,... Multifractal analysis in ergodic theory Multifractal analysis in number theory Non-commutative multifractal geometry. Despite the substantial developments in the past 20 years, Cawley & Mauldin s result remains enigmatic, influential and representative: all other multifractal results have the form the multifractal spectrum = a natural auxiliary function

18 Multifractal Analysis of Divergence Points of Self-Similar Measures Of course, the local dimension lim log µ(b(x,r log r may not exist! A point x for which the local dimension lim log µ(b(x,r log r How many divergence points are there? does not exist is called a divergence point. Well... for self-similar measures there are many divergence points! More precisely... Theorem. Chen & Xiong (1999, Barreire & Schmeling (2000, Olsen (2002. Let K and µ be the self-similar set and measure associated with the list (S i, p i N i=1. Assume that the SSC is satisfied. Then dim H (x K the limit lim log µ(b(x, r does not exist = dim H K.

19 A detailed analysis of the set of divergence points. For function ϕ : (0, R, write acc ϕ(r = the set of accumulation points of ϕ as r tends to 0 Definition. Multifractal divergence spectrum. Let µ be a measure on a metric space X. The Hausdorff multifractal divergence spectrum of µ is defined by ( F H,µ (C = x X acc log µ(b(x, r = C, C R. log r The packing multifractal divergence spectrum of µ is defined by ( F P,µ (C = x X acc log µ(b(x, r = C, C R. log r The divergence spectra extend the usual spectra: F H,µ` {α} = fh,µ (α F P,µ` {α} = fp,µ (α

20 The next results shows that the set of divergence points has an enormous complexity not foreseen by Cawley & Mauldin s Multifractal Formalism Theorem for self-similar measures. Theorem. Olsen & Winter (2004, Olsen (2009. Let K and µ be the self-similar set and measure associated with the list (S i, p i N i=1. Assume that the SSC is satisfied. Define β : R R by X p q r β(q = 1. i i i For all q R, we have τ µ (q = τ µ(q = β(q. h i log p If C min i log p i, max i log r i and C is a closed interval, we have i log r i dim H (x K acc log µ(b(x, r = C log r dim P (x K acc log µ(b(x, r = C log r h i log p If C min i log p i, max i log r i or C is not a closed interval, we have i log r i ( x K acc log µ(b(x, r = C =. log r = inf α C β (α, = sup β (α, α C

21 Multifractal Analysis of Divergence Points of Self-Similar Measures and Descriptive Set Theory... an attempt to honor the breath of this meeting: Multifractals and descriptive set theory or (less grandiose A naive question Question. Let K and µ be the self-similar set and measure associated with the list (S i, p i N i=1 and assume that µ is not equal to the normalized Hausdorff t-dimensionsal measure restricted to K where t = dim H K. Assume that the SSC is satisfied. h i log p Let C min i log p i, max i log r i be a closed interval. i log r i ( Is the multifractal set x K acc log µ(b(x, r = C a Π 0 3-complete set? log r [A set M K is called Π 0 3 -complete if for any E Π0 3, there is a continuous map f : E K such that E = f 1 (M.]

22 We have come full circle... and The End: multifractal geometry The Beginning: multifractal geometry ergodic theory/descriptive set theory

23 Thank you

A non-uniform Bowen s equation and connections to multifractal analysis

$A non-uniform Bowen s equation and connections to multifractal analysis$ A non-uniform Bowen s equation and connections to multifractal analysis Vaughn Climenhaga Penn State November 1, 2009 1 Introduction and classical results Hausdorff dimension via local dimensional characteristics