Multifractal Analysis. A selected survey Lars Olsen
Multifractal Analysis: The beginning 1974 Frontispiece of: Mandelbrot, “Intermittent turbulence in self-similar cas- cades: 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 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 ”
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 physi- cists Halsey et al: “ Fractal Measures and their Singularities ”
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 . log r r ց 0 Local dimensions are not new concepts: µ ( B ( x , r )) • Local dimensions are related to densities lim r ց 0 , and densities have a long history in rt (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 )
Definition. Multifractal spectrum. Let µ be a measure on a metric space X. The Hausdorff multifractal spectrum of µ is defined by ( ˛ ) log µ ( B ( x , r )) ˛ ˛ f H ,µ ( α ) = dim H x ∈ X ˛ lim = α , α ∈ R , ˛ r ց 0 log r The packing multifractal spectrum of µ is defined by ˛ ( ) ˛ log µ ( B ( x , r )) ˛ f P ,µ ( α ) = dim P x ∈ X ˛ lim = α , α ∈ R . ˛ r ց 0 log r
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 X µ ( Q ) q log Q is an r grid box with Q ∩ X � = ∅ τ µ ( q ) = lim inf , r ց 0 − log r X µ ( Q ) q log Q is an r grid box with Q ∩ X � = ∅ τ µ ( q ) = lim sup . − log r r ց 0 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 .
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 con- stant α µ such that ( ˛ ) ˛ log µ ( B ( x , r )) ˛ dim H x ∈ X ˛ lim = α µ = dim H X . ˛ r ց 0 log r 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.
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 P Q ∩ X � = ∅ µ ( Q ) q A version of the Ergodic Theorem. τ µ ( q ) = lim inf , − log r r ց 0 For many “natural” measures µ there is a constant α µ such that P Q ∩ X � = ∅ µ ( Q ) q τ µ ( q ) = lim sup . ˛ ( ) ˛ log µ ( B ( x , r )) − log r r ց 0 ˛ dim H x ∈ X ˛ lim = α µ = dim H X . ˛ r ց 0 log r Then for all α ≥ 0 , we have ( ˛ ) ˛ log µ ( B ( x , r )) = τ ∗ ˛ dim H x ∈ X ˛ lim = α µ ( α ) ˛ r ց 0 log r = 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 The Multifractal Formalism. A physics conjecture. ( ˛ ) log µ ( B ( x , r )) ˛ ˛ dim H x ∈ X ˛ lim = α > 0 . Let µ be a measure on a metric space X. ˛ r ց 0 log r For q ∈ R we define the lower and upper L q -dimensions of µ by A surprising relationship between global and local quantities. The L q -dimensions P Q ∩ X � = ∅ µ ( Q ) q τ µ ( q ) = lim inf , − log r r ց 0 τ µ ( q ) , τ µ ( q ) P Q ∩ X � = ∅ µ ( Q ) q τ µ ( q ) = lim sup . are global quantities; − log r r ց 0 the local dimension Then for all α ≥ 0 , we have log µ ( B ( x , r )) lim r ց 0 log r ( ˛ ) ˛ log µ ( B ( x , r )) = τ ∗ ˛ dim H x ∈ X ˛ lim = α µ ( α ) ˛ r ց 0 log r is a local quantity. = τ ∗ There are no reasons to expect any µ ( α ) . 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.
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 R d (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 We have daughter-intervals in the ratio 2 3 : 1 3 ` ´ ` ´ 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 + 2 3 Then ` ´ ` ´ µ = left part of µ + right part of µ = p 1 µ ◦ S − 1 + p 2 µ ◦ S − 1 1 2 A measure having this property is called self- similar. The precise definition is . . . ↓ µ | {z } | {z } left part of µ right part of µ
Example We have Subdivide the “mass” of any square between its 4 daughter-squares in the ratio 2 7 : 2 7 : 2 7 : 1 7 ` ´ ` ´ µ = bottom left part of µ + bottom right part of µ ` ´ ` ´ + top left part of µ + top right part of µ Let ( p 1 , p 2 , p 3 , p 4 ) = ( 1 7 , 2 7 , 2 7 , 2 7 ) Let S 1 ( x , y ) = 1 2 ( x , y ) , S 2 ( x , y ) = 1 2 ( x , y ) + ( 1 2 , 0 ) , S 3 ( x , y ) = 1 2 ( x , y ) + ( 0 , 1 2 ) , and S 4 ( 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 = p 1 µ ◦ S − 1 + p 2 µ ◦ S − 1 1 2 + p 3 µ ◦ S − 1 + p 4 µ ◦ S − 1 3 4 A measure having this property is called self- ↓ similar. The precise definition is . . . µ
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 = S i ( K ) , i X p i µ ◦ S − 1 µ = . 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, Cawley & Mauldin verified the Multifractal Formalism for self-similar measures satisfying the SSC. L Mejlbro, D Mauldin, F Topsøe, J P R Christensen
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 i r β ( q ) p q = 1 . i i For all q ∈ R , we have τ µ ( q ) = τ µ ( q ) = β ( q ) . h i log pi log pi For all α ∈ min i log ri , max i , we have log ri ( ˛ ) ˛ log µ ( B ( x , r )) = β ∗ ( α ) , ˛ dim H x ∈ K ˛ lim = α ˛ r ց 0 log r ˛ ( ) ˛ log µ ( B ( x , r )) = β ∗ ( α ) . ˛ dim P x ∈ K ˛ lim = α ˛ r ց 0 log r h i log pi log pi For all α �∈ min i log ri , max i , we have log ri ( ˛ ) ˛ log µ ( B ( x , r )) ˛ x ∈ K ˛ lim = α = ∅ . ˛ r ց 0 log r
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