A Selective Survey of Self-Similar Sets and Suchlike Structures Kenneth Falconer University of St Andrews, Scotland Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Kenneth Falconer, Dan Mauldin & Toby O’Neil – 1997 Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Bagpipes! Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Carn Liath Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Iterated function systems A family S 1 , . . . , S m of contractions on D ⊆ R N , i.e. x , y ∈ R N , | S i ( x ) − S i ( y ) | ≤ c i | x − y | c i < 1 is called an iterated function system (IFS). Given an IFS there exists a unique, non-empty compact set E satisfying m � E = S i ( E ) , i =1 called the attractor of the IFS. [Hutchinson (1981)] If the S i are similarities E is called a self-similar set. Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Some self-similar sets Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Dimension Let E ⊆ R N . For δ > 0 we let N δ ( E ) be the least number of sets of diameter at most δ that can cover E . We define the box counting dimension or box dimension of E by log N δ ( E ) dim B E = lim − log δ δ → 0 (assuming this exists, otherwise lower and upper limits define the lower and upper box dimensions). Roughly speaking N δ ( E ) ∼ δ − dim B E as δ → 0 . We will also mention Hausdorff dimension dim H E , defined via Hausdorff measures H s . Finding the dimensions of fractal attractors has attracted much interest. Let’s recall some classes of attractor where formulae are known. Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Self-similar sets If the S i are similarities of ratio r i , i.e. | S i ( x ) − S i ( y ) | = r i | x − y | , the self-similar set E , satisfying E = � m i =1 S i ( E ), has m � r s dim H E = dim B E = s where i = 1 , i =1 provided the open set condition holds, that is there exists an non-empty bounded open set O such that ∪ m i =1 S i ( O ) ⊂ O . [Moran (1946), Hutchinson (1981)] Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Statistically self-similar sets We can randomise these constructions in a natural way: Let ( S 1 , . . . , S m ) be a random m -tuple of contracting similarities. For each i = ( i 1 , i 2 , . . . , i k ) ∈ { 1 , 2 , . . . , m } k ; k = 0 , 1 , 2 , . . . let ( S i , 1 , . . . , S i , m ) be an independent realisation of ( S 1 , . . . , S m ). Starting with some compact set A = A ∅ , define a hierarchy of sets A i = A i 1 ,..., i k with A i , 1 ≡ S i , 1 ( A i ) , · · · , A i , m ≡ S i , m ( A i ) (essentially) disjoint subsets of A i . We get a statistically self-similar set ∞ � � E = A i 1 ,..., i k . k =0 i 1 ,..., i k Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
A random von Koch curve Suppose that the ( S 1 , . . . , S m ) have (random) scaling ratios ( R 1 , . . . , R m ). Then with probability 1, m � � � R s dim H E = dim B E = s where = 1 . E i i =1 [Mauldin, Graf, Williams ( ∼ 1986)] + exact dimension Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Graph-directed sets Let G be a directed graph on m vertices (multiple edges allowed). For each directed edge ( i , j ) ∈ G let S i , j : R N → R N be a contracting similarity of ratio r i , j . This defines m graph-directed self-similar sets E 1 , . . . , E m such that � E i = S i , j ( E j ) ( i = 1 , 2 , . . . , m ) . ( i , j ) ∈ G Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Under reasonable conditions ( G strongly connnected, open set condition), ρ [ a s for i = 1 , . . . , m , dim H E i = dim B E i = s where i , j ] = 1 , where ρ [ b i , j ] is the spectral radius of the matrix with entries b i , j . [Mauldin & Williams (1988)] Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Self-conformal sets Now let S 1 , . . . , S m be conformal contractions on D ⊆ R N . The attractor E is called a self-conformal set. Examples include sets of continued fractions, e.g. the attractor of S 1 ( x ) = 1 + 1 / x ; S 2 ( x ) = 2 + 1 / x is 1 1 1 � � x = a 0 + a 3 + · · · : a i ∈ { 1 , 2 } , a 1 + a 2 + and certain Julia sets, e.g. repellers of the complex mapping f ( z ) = z 2 + c for suitable c ∈ C . Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Dimension calculations go back to Bowen’s formula which uses the thermodynamic formalism to express the dimension as a zero of a pressure functional P . For a function f on a compact domain D and a H¨ older function ψ : D → R , the pressure is given by P ( ψ ) = 1 � ψ ( x ) + ψ ( fx ) + · · · + ψ ( f k − 1 x ) � � k log lim exp . k →∞ x ∈ Fixf k Let f be defined on each S i ( D ) by the inverses of the S i (suitable separation conditions) and let ψ ( x ) = − s log | f ′ ( x ) | . Then dim H E = dim B E = s where P ( − s log | f ′ | ) = 0 , with 0 < H s ( E ) < ∞ . [Bowen (1979)] This can be interpreted as a limiting case of the Moran-Hutchinson formula: For some fixed x ∈ D let � Φ s | ( S i 1 ◦ · · · ◦ S i k ) ′ ( x ) | s , k ≡ i 1 ... i k Note that | ( S i 1 ◦ · · · ◦ S i k ) ′ ( x ) | is essentially the diameter of S i 1 ◦ · · · ◦ S i k ( D ). Using bounded distortion and submultiplicativity, k ) 1 / k ≡ Φ s k →∞ (Φ s lim exists and dim H E = dim B E = s where Φ s = 1 . Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Infinite self-conformal systems Now let S 1 , S 2 , . . . be a sequence of conformal contractions on some compact subset D of R N . The limit set ∞ � � E = S i 1 ◦ · · · ◦ S i k ( A ) , k =0 i 1 ,..., i k need not be compact, and there may be no sets or several sets satisfying Apollonian gasket - limit set of an F = � ∞ i =1 S i ( F ). infinite conf. IFS, dim ∼ 1 . 3057 One can extend the definition of ‘pressure’ P ( s ) to infinite systems. Then dim H E = inf { s : P ( s ) < 0 } . In general, dim H E � = dim P E . However, if P ( s ) = 0 for some s then 0 < H s ( E ) , P s ( E ) < ∞ . dim H E = dim B E = dim P E = s with [Mauldin, Urbanski ∼ 1998, ...] Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Self-affine sets Now let the S i be affine contractions, i.e. S i ( x , y ) = T i ( x , y ) + ( c i , d i ) where T i is a linear mapping and ( c i , d i ) is a translation. Some self-affine sets - J Miao Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Some more self-affine sets Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
A problem ... If λ > 0 then dim H E ≥ 1 Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
A problem ... If λ = 0 then dim H E = log 2 log 3 < 1 Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Dimensions of self-affine carpets A carpet on R 2 is the attractor of an IFS of the form S i ( x , y ) = ( a i x + c i , b i y + d i ) i = 1 , . . . , m (i.e. contractions that map a square to rectangles). Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
A generic formula for the dimension of carpets E defined by S i ( x , y ) = ( a i x + c i , b i y + d i ) i = 1 , . . . , m (with | a i | , | b i | < 1 2 ) is given by dim H E = dim B E = m m � b i � max { s 1 − 1 , 0 } � a i � max { s 2 − 1 , 0 } � � � a s 1 � b s 2 max s 1 , s 2 : = 1 , = 1 , i i a i b i i =1 i =1 valid for Lebesgue a.a. ( c 1 , d 1 , . . . , c m , d m ) ∈ R 2 m . [F, Solomyak] Note that the formula depends on whether the contractions in the x -direction or in the y -direction dominate. [In fact this generic formula also holds where S i ( x , y ) = ( a i x + e i y + c i , b i y + d i ) i = 1 , . . . , m ] [F, Miao] Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Exceptional set of translations When does the generic formula give the right answer? What can we say about the set of translations ( c 1 , d 1 , . . . , c m , d m ) for which the dimension of the attractor is smaller than the affinity dimension? For t ≤ s = the almost sure value of the dimension, write E ( t ) = { ( c 1 , d 1 , . . . , c m , d m ) ∈ R 2 m : dim H E ω < t } . Then: ( a ) dim H E ( t ) ≤ 2 m − c ( s − t ) where c > 0 is a constant ( b ) dim F E ( t ) ≤ 2 t , where dim F denotes Fourier dimension, i.e. µ ( x ) = O ( | x | − t / 2 ) as | x | → ∞} dim F A = sup { t : ∃ µ on A s.t. ˆ [F & Miao 2008] Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
Carpets where dim E is not the generic value Bedford McMullen self-affine carpets Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures
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