Projections of Fractals - Old and New Kenneth Falconer University - PowerPoint PPT Presentation

Projections of Fractals - Old and New Kenneth Falconer University of St Andrews, Scotland, UK Kenneth Falconer Projections of Fractals - Old and New

Marstrand’s projection theorems Theorem (Marstrand 1954) Let E ⊂ R 2 be a Borel set. For all θ ∈ [0 , π ) (i) dim H proj θ E ≤ min { dim H E , 1 } . For almost all θ ∈ [0 , π ), (ii) dim H proj θ E = min { dim H E , 1 } , (iii) L (proj θ E ) > 0 if dim H E > 1. [proj θ denotes orthogonal projection onto the line L θ , dim H is Hausdorff dimension, L is Lebsegue measure or length on L θ .] Kenneth Falconer Projections of Fractals - Old and New

Energy characterisation of Hausdorff dimension That dim H proj θ E ≤ min { dim H E , 1 } for all θ follows since projection is a Lipschitz map which cannot increase dimension. Marstrand’s lower bound proof was geometric and intricate. Kaufman’s (1968) potential theoretic proof has become the standard approach for such problems. This depends on the following energy characterisation of Hausdorff dimension. Let M ( E ) be the set of probability measures on E � � d µ ( x ) d µ ( y ) 1 � � dim H = sup s : C ( s ) ( E ) ≡ inf < ∞ | x − y | s µ ∈M ( E ) s : C ( s ) ( E ) > 0 � � = sup . Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections The box-counting dimension of a non-empty and compact E ⊂ R 2 is log N r ( E ) dim B E = lim − log r r → 0 where N r ( E ) is the least number of sets of diameter r covering E . [Taking lower/upper limits gives the lower/upper box dimensions.] Is there a Marstrand-type theorem for box-dimensions of projections? For E ⊂ R 2 dim B E 2 dim B E ≤ dim B proj θ E ≤ min { dim B E , 1 } (almost all θ ∈ [0 , π )) 1 + 1 and examples show that these bounds are best possible. Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections The box-counting dimension of a non-empty and compact E ⊂ R 2 is log N r ( E ) dim B E = lim − log r r → 0 where N r ( E ) is the least number of sets of diameter r covering E . [Taking lower/upper limits gives the lower/upper box dimensions.] Is there a Marstrand-type theorem for box-dimensions of projections? For E ⊂ R 2 dim B E 2 dim B E ≤ dim B proj θ E ≤ min { dim B E , 1 } (almost all θ ∈ [0 , π )) 1 + 1 and examples show that these bounds are best possible. Even so, dim B proj θ E and dim B proj θ E must be constant for almost all θ ; for a messy argument and indirect value see (F & Howroyd, 1996, 2001). Using capacities things become much simpler. Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Define potential kernels φ s r ( x ) by � r s � � � ( x ∈ R 2 or R n ) φ s r ( x ) = min 1 , | x | r ( E ) of a compact E ⊂ R n w.r.t. φ s The capacity C s r is 1 � � φ s r ( E ) = inf r ( x − y ) d µ ( x ) d µ ( y ) , C s µ ∈M ( E ) where M ( E ) are the probability measures on E . The infimum is attained by some equilibrium measure µ 0 ∈ M ( E ), and moreover � 1 φ s r ( x − y ) d µ 0 ( y ) ≥ ( x ∈ E ) , C s r ( E ) with equality for µ 0 -almost all x ∈ E . Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Then for E ⊂ R n � c 2 log(1 / r ) C s r ( E ) if s = n c 1 C s r ( E ) ≤ N r ( E ) ≤ (1) , c 2 C s r ( E ) if s > n ( c 1 , c 2 depend on n , s , diam E ). In particular for E ⊂ R n log C n r ( E ) log N r ( E ) lim = lim = dim B E − log r − log r r → 0 r → 0 (we can replace dim B and lim by either dim B and lim, or by dim B and lim). Note: Inequalities (1) fail if 0 < s < n . Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Theorem Let E ⊂ R 2 be non-empty compact. For all θ ∈ [0 , π ) log C 1 r ( E ) (i) dim B proj θ E ≤ lim − log r r → 0 For almost all θ ∈ [0 , π ), log C 1 r ( E ) (ii) dim B proj θ E = lim − log r r → 0 [We can replace dim B and lim by either dim B and lim, or by dim B and lim.] Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Theorem Let E ⊂ R 2 be non-empty compact. For all θ ∈ [0 , π ) log C 1 r ( E ) ≡ dim 1 (i) dim B proj θ E ≤ lim B E . − log r r → 0 For almost all θ ∈ [0 , π ), log C 1 r ( E ) ≡ dim 1 (ii) dim B proj θ E = lim B E . − log r r → 0 [We can replace dim B and lim by either dim B and lim, or by dim B and lim.] We call log C s r ( E ) ( E ⊂ R 2 or R n ) , dim s B E := lim − log r r → 0 � r s � using capacity with respect to the kernel φ s � � r ( x ) = min 1 , , | x | the s -box-dimension profile of E , which should be thought of as the ’box-dimension of E when regarded from an s -dimensional viewpoint’. Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F , and I r ( F ) the intervals [ ir , ( i + 1) r ) , ( i ∈ Z ) that intersect F . � 2 ≤ N r ( F ) ν ( I ) 2 ≤ � � � 1 = ν ( I ) I ∈I r ( F ) I ∈I r ( F ) � N r ( F ) ( ν × ν ) { ( w , z ) ∈ I × I } ≤ N r ( F )( ν × ν ) { ( w , z ) : | w − z | ≤ r } . (1) I ∈I r ( F ) Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F , and I r ( F ) the intervals [ ir , ( i + 1) r ) , ( i ∈ Z ) that intersect F . � 2 ≤ N r ( F ) ν ( I ) 2 ≤ � � � 1 = ν ( I ) I ∈I r ( F ) I ∈I r ( F ) � N r ( F ) ( ν × ν ) { ( w , z ) ∈ I × I } ≤ N r ( F )( ν × ν ) { ( w , z ) : | w − z | ≤ r } . (1) I ∈I r ( F ) Let µ be an equilibrium measure on E ⊂ R 2 , with projections µ θ onto L θ . � � ( µ θ × µ θ ) { ( w , z ): | w − z | ≤ r } d θ = ( µ × µ ) { ( x , y ): | proj θ x − proj θ y | ≤ r } d θ �� �� c φ 1 = L{ θ : | proj θ ( x − y ) | ≤ r } d µ ( x ) d µ ( y ) ≤ c r ( x − y ) d µ ( x ) d µ ( y ) = r ( E ) . C 1 Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F , and I r ( F ) the intervals [ ir , ( i + 1) r ) , ( i ∈ Z ) that intersect F . � 2 ≤ N r ( F ) ν ( I ) 2 ≤ � � � 1 = ν ( I ) I ∈I r ( F ) I ∈I r ( F ) � N r ( F ) ( ν × ν ) { ( w , z ) ∈ I × I } ≤ N r ( F )( ν × ν ) { ( w , z ) : | w − z | ≤ r } . (1) I ∈I r ( F ) Let µ be an equilibrium measure on E ⊂ R 2 , with projections µ θ onto L θ . � � ( µ θ × µ θ ) { ( w , z ): | w − z | ≤ r } d θ = ( µ × µ ) { ( x , y ): | proj θ x − proj θ y | ≤ r } d θ �� �� c φ 1 = L{ θ : | proj θ ( x − y ) | ≤ r } d µ ( x ) d µ ( y ) ≤ c r ( x − y ) d µ ( x ) d µ ( y ) = r ( E ) . C 1 2 − k ( E ) − 1 < ∞ then there are M θ < ∞ for a.a. θ such that k 2 sk C 1 If � 2 sk ( µ θ × µ θ ) { ( w , z ): | w − z | ≤ 2 − k } ≤ M θ ( k ∈ N ) , so from (1), 1 ≤ N 2 − k (proj θ E ) 2 − sk M θ . 1 Hence if dim B ( E ) > s then dim B (proj θ E ) ≥ s for almost all θ . Kenneth Falconer Projections of Fractals - Old and New

Exceptional directions Marstrand’s theorem tells nothing about which particular directions may have projections with dimension or measure smaller than usual, i.e. when dim H proj θ E < min { dim H E , 1 } , or if dim H E > 1 when L (proj θ E ) = 0. The set shown has dimension log 4 / log(5 / 2) = 1 . 51, but with some projections of dimension < 1. Kenneth Falconer Projections of Fractals - Old and New

Exceptional directions The set of exceptional directions can’t be ‘too big’: Theorem (Kaufman, 1968) If E ⊂ R 2 and dim H E ≤ 1, dim H { θ : dim H proj θ E < dim H E } ≤ dim H E . – follows from an energy estimate Theorem (F, 1982) If E ⊆ R 2 and dim H E > 1, dim H { θ : L (proj θ E ) = 0 } ≤ 2 − dim H E . – proof uses Fourier transforms. Theorem (F, Howroyd, 1997) For compact E ⊂ R 2 and let 0 ≤ s ≤ 1. Then dim H { θ ∈ [0 , π ) such that dim B proj θ E < dim s B E } ≤ s . Kenneth Falconer Projections of Fractals - Old and New

General problem Theorem (Marstrand 1954) Let E ⊂ R 2 be a Borel set. For all θ ∈ [0 , π ) (i) dim H proj θ E ≤ min { dim H E , 1 } . For almost all θ ∈ [0 , π ), (ii) dim H proj θ E = min { dim H E , 1 } , (iii) L (proj θ E ) > 0 if dim H E > 1. General problem: find sets or classes of sets for which there are no exceptional directions for Marstrand’s theorem, i.e. where (ii) or (iii) hold for all θ , or at least where the exceptional directions can be identified. We will consider self-similar and self-affine sets and their random counterparts. Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets Given an iterated function system of contracting similarities f 1 , . . . , f m : R 2 → R 2 there exists a unique non-empty compact E ⊂ R 2 called a self-similar set such that m � E = f i ( E ) . ( ∗ ) i =1 We assume that the union ( ∗ ) is disjoint or perhaps ‘nearly disjoint’ (i.e. OSC), so dim H E = dim B E = s , where s is the similarity dimension, given by � m i =1 r s i = 1. Kenneth Falconer Projections of Fractals - Old and New