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Multifractal Analysis. A selected survey

Lars Olsen

Multifractal Analysis: The beginning 1974

Multifractal analysis refers to a particular way

• f analysing the local structure of measures.

The idea of multifractals originates from 1974 in a paper by Mandelbrot analyzing the dissi- pation of energy in a turbulent fluid: “Intermittent turbulence in self-similar cas- cades: divergence of high moments and di- mension of the carrier ” Frontispiece of: Mandelbrot, “Intermittent turbulence in self-similar cas- cades: divergence of high moments and dimension of the carrier”, 1974

Multifractal Analysis: Revisited by physicists 1986

Mandelbrot’s multifractal ideas were revisited in a broader context while being expanded and clarified in 1986 in a paper by theoretical physi- cists Halsey et al: “Fractal Measures and their Singularities ” Frontispiece of: Halsey et al, “Fractal Measures and their Singularities”, 1986

In order to explain the ideas behind multifractal analysis we require two concepts: Local dimensions and the multifractal spectrum; Lq-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 dimloc(x; µ) = lim

rց0

log µ(B(x, r)) log r . Local dimensions are not new concepts:

• Local dimensions are related to densities limrց0

µ(B(x,r)) rt

, 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)) ∼ rdimloc (x;µ)

• Definition. Multifractal spectrum.

Let µ be a measure on a metric space X. The Hausdorff multifractal spectrum of µ is defined by fH,µ(α) = dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) , α ∈ R , The packing multifractal spectrum of µ is defined by fP,µ(α) = dimP ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) , α ∈ R .

Lq-dimensions

• Definition. Lq-dimensions.

Let µ be a measure on a metric space X. For q ∈ R we define the lower and upper Lq-dimensions of µ by τµ(q) = lim inf

rց0

log X

Q is an r grid box with Q ∩ X = ∅

µ(Q)q − log r , τµ(q) = lim sup

rց0

log X

Q is an r grid box with Q ∩ X = ∅

µ(Q)q − log r . Lq-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 Lq-dimensions was introduced by theoretical physicists in the 1980’s (Halsey et

al, Proccacia, Grassberger and others). Lq-dimensions extend the usual fractal dimensions: τµ(0) = the lower box dimension of X , τµ(0) = the upper box dimension of X .

So . . . what did Halsey et al say in their 1986 paper?

The Ergodic Theorem shows the following: for many “natural” measures µ there is a con- stant αµ such that dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = αµ ) = dimH X . In 1986 theoretical physicists Halsey et al’s paper “Fractal Measures and their Singulari- ties ” suggested to following remarkable result, known as the Multifractal Formalism, revealing an enormous complexity not foreseen by the Ergodic Theorem. Frontispiece of: Halsey et al, “Fractal Measures and their Singularities”, 1986

A version of the Ergodic Theorem. For many “natural” measures µ there is a constant αµ such that dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = αµ ) = dimH X. 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 Lq-dimensions of µ by τµ(q) = lim inf

rց0

P

Q ∩ X = ∅ µ(Q)q

− log r , τµ(q) = lim sup

rց0

P

Q ∩ X = ∅ µ(Q)q

− log r . Then for all α ≥ 0, we have dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) = τ∗

µ(α)

= τ∗

µ(α).

The Multifractal Formalism is remarkable: Revealing an enormous complexity not foreseen by the Ergodic Theorem. There is an uncountable number of α such that dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) > 0 . A surprising relationship between global and local quantities. The Lq-dimensions τµ(q) , τµ(q) are global quantities; the local dimension lim

rց0

log µ(B(x, r)) log r is a local quantity. There are no reasons to expect any relationship between the Lq-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. 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 Lq-dimensions of µ by τµ(q) = lim inf

rց0

P

Q ∩ X = ∅ µ(Q)q

− log r , τµ(q) = lim sup

rց0

P

Q ∩ X = ∅ µ(Q)q

− log r . Then for all α ≥ 0, we have dimH ( x ∈ X ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) = τ∗

µ(α)

= τ∗

µ(α).

Multifractal Analysis: Explored by mathematicians 1989–1992

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 Rd (Cawley & Mauldin).

William Blake (28 November 1757 - 12 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

↓ µ | {z }

left part of µ

| {z }

right part of µ

We have µ = ` left part of µ ´ + ` right part of µ ´ Let (p1, p2) = ( 2

3 , 1 3 )

Let S1(x) = 1

3 x and S2(x) = 1 3 x + 2 3

Then µ = ` left part of µ ´ + ` right part of µ ´ = p1 µ ◦ S−1

1

+ p2 µ ◦ S−1

2

A measure having this property is called self-

• similar. The precise definition is . . .

Example

Subdivide the “mass” of any square between its 4 daughter-squares in the ratio 2

7 : 2 7 : 2 7 : 1 7

↓ µ 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) = ( 1

7 , 2 7 , 2 7 , 2 7 )

Let S1(x, y) = 1

2 (x, y),

S2(x, y) = 1

2 (x, y) + ( 1 2 , 0),

S3(x, y) = 1

2 (x, y) + (0, 1 2 ), and

S4(x, y) = 1

2 (x, y) + ( 1 2 , 1 2 )

Then µ = ` bottom left part of µ ´ + ` bottom right part of µ ´ + ` top left part of µ ´ + ` top right part of µ ´ = p1 µ ◦ S−1

1

+ p2 µ ◦ S−1

2

+ p3 µ ◦ S−1

3

+ p4 µ ◦ S−1

4

A measure having this property is called self-

• similar. The precise definition is . . .

• Definition. Self-similar set and self-similar measure. Hutchinson (1981).

Let (S1, . . . , SN) be a list of similarities Si : Rd → Rd . Write ri for the contraction ratio of Si Let (p1, . . . , pN) be a probability vector. Let K and µ be the self-similar set and the self-similar measure associated with (Si , pi )N

i=1, i.e.

K = [

i

Si (K) , µ = X

i

pi µ ◦ S−1

i

. Usually people assume various separation conditions.

• Definition. Open Set Condition (OSC).

The (S1, . . . , SN) satisfies the OSC, if there is a non-empty and bounded open set such that Si (U) ⊆ U for all i and Si (U) ∩ Sj (U) = ∅ for all i and j with i = j.

• Definition. Strong Separation Condition (SSC).

The (S1, . . . , SN) satisfies the OSC, if Si (K) ∩ Sj (K) = ∅ for all i and j with i = j.

Multifractal Analysis of Self-Similar Measures)

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 Frontispiece of: Cawley & Mauldin, “Multifractal Decomposition of Moran Fractals”, 1992

• Theorem. Cawley & Mauldin (1992).

Let K and µ be the self-similar set and measure associated with the list (Si , pi )N

i=1. Assume that the SSC is

satisfied. Define β : R → R by X

i

pq

i rβ(q) i

= 1 . For all q ∈ R, we have τµ(q) = τµ(q) = β(q) . For all α ∈ h mini

log pi log ri , maxi log pi log ri

i , we have dimH ( x ∈ K ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) = β∗(α) , dimP ( x ∈ K ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) = β∗(α) . For all α ∈ h mini

log pi log ri , maxi log pi log ri

i , we have ( x ∈ K ˛ ˛ ˛ ˛ ˛ lim

rց0

log µ(B(x, r)) log r = α ) = ∅ .

Multifractal Analysis: Studied by mathematicians after 1992 . . .

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

• f 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 ”∗

Multifractal Analysis of Divergence Points of Self-Similar Measures

Of course, the local dimension limrց0

log µ(B(x,r)) log r

may not exist! A point x for which the local dimension limrց0

log µ(B(x,r)) log r

does not exist is called a divergence point. How many divergence points are there? 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 (Si , pi )N

i=1. Assume that the SSC is

• satisfied. Then

dimH ( x ∈ K ˛ ˛ ˛ ˛ ˛ the limit lim

rց0

log µ(B(x, r)) log r does not exist ) = dimH K .

A detailed analysis of the set of divergence points. For function ϕ : (0, ∞) → R, write accrց0ϕ(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 FH,µ(C) = ( x ∈ X ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) , C ⊆ R . The packing multifractal divergence spectrum of µ is defined by FP,µ(C) = ( x ∈ X ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) , C ⊆ R . The divergence spectra extend the usual spectra: FH,µ ` {α} ´ = fH,µ(α) FP,µ ` {α} ´ = fP,µ(α)

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 (Si , pi )N

i=1. Assume that the SSC is

satisfied. Define β : R → R by X

i

pq

i rβ(q) i

= 1 . For all q ∈ R, we have τµ(q) = τµ(q) = β(q) . If C ⊆ h mini

log pi log ri , maxi log pi log ri

i and C is a closed interval, we have dimH ( x ∈ K ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) = inf

α∈C β∗(α) ,

dimP ( x ∈ K ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) = sup

α∈C

β∗(α) , If C ⊆ h mini

log pi log ri , maxi log pi log ri

i

• r C is not a closed interval, we have

( x ∈ K ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) = ∅ .

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

• r (less grandiose)

A naive question

Question. Let K and µ be the self-similar set and measure associated with the list (Si , pi )N

i=1 and assume that µ is not equal

to the normalized Hausdorff t-dimensionsal measure restricted to K where t = dimH K. Assume that the SSC is satisfied. Let C ⊆ h mini

log pi log ri , maxi log pi log ri

i be a closed interval. Is the multifractal set ( x ∈ K ˛ ˛ ˛ ˛ ˛ accrց0 log µ(B(x, r)) log r = C ) a Π0

3-complete set?

[A set M ⊆ K is called Π0

3-complete if for any E ∈ Π0 3, there is a continuous map f : E → K such that

E = f −1(M).]

We have come full circle

. . . and The End: multifractal ge-

• metry

ergodic theory/descriptive set theory The Beginning: multifractal geom- etry

Thank you