Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures The Weyl-Berry Conjecture Michael Berry (1979): For non-smooth boundaries, change N ( λ ) = c d λ d / 2 vol d (Ω) + c d − 1 λ ( d − 1 ) / 2 vol d - 1 ( ∂ Ω) + o ( λ ( d − 1 ) / 2 ) to N ( λ ) = c d λ d / 2 vol d (Ω) + c D λ D / 2 H D ( ∂ Ω) + o ( λ ( D − 1 ) / 2 ) where D is the Hausdorff dimension of the boundary. Brossard and Carmona (1986): “No. But maybe Minkowski dimension instead?” Lapidus and Pomerance (1993): “Yes, for subsets of R .” N ( λ ) = λ vol 1 (Ω) + c D M D ( ∂ Ω) λ D + o ( λ D ) , as λ → ∞ , where D is Minkowski dimension and M D is Minkowski content. Kigami and Lapidus (1993): similar results for ∆ on fractals.
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals.
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals. Can one hear D (the dimension of ∂ Ω )? Lapidus and Pomerance (1993): “Yes.”
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals. Can one hear D (the dimension of ∂ Ω )? Lapidus and Pomerance (1993): “Yes.” Can one hear if ∂ Ω is Minkowski measurable? Lapidus and Maier (1995): “Iff ζ ( s ) has no zero on Re s = D .” � ∞ n − s is the Riemann zeta function. Here ζ ( s ) = n = 1
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let A , B ⊆ R d be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B . A W B
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let A , B ⊆ R d be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B . Can one hear D (the dimension of ∂ Ω )? Lapidus and van Frankenhuijsen (2000): “Yes.” Can one hear if ∂ Ω is Minkowski measurable? Now measurability can fail for more than one reason. Arithmetic properties of scaling ratios. Minkowski measurability of ∂ B .
Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures Can you hear the shape of a fractal drum? Let A , B ⊆ R d be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B . Can one hear D (the dimension of ∂ Ω )? Lapidus and van Frankenhuijsen (2000): “Yes.” Can one hear if ∂ Ω is Minkowski measurable? Now measurability can fail for more than one reason. Arithmetic properties of scaling ratios. Minkowski measurability of ∂ B . So there is some interest in having a robust collection of examples of sets which are (or are not) Minkowski measurable.
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Minkowski dimension is box-counting dimension ε = 150 km ε = 75 km ε = 37 . 5 km N ( ε ) = 22 N ( ε ) = 53 N ( ε ) = 130 N ( ε ) = # { boxes of size ε required to cover the set (coastline) } .
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Minkowski dimension is box-counting dimension ε = 150 km ε = 75 km ε = 37 . 5 km N ( ε ) = 22 N ( ε ) = 53 N ( ε ) = 130 N ( ε ) = # { boxes of size ε required to cover the set (coastline) } . Idea: if C is the measure of the set, then ( number of boxes ) · ( box size ) = N ( ε ) · ( ε ) D ≈ C
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Minkowski dimension is box-counting dimension ε = 150 km ε = 75 km ε = 37 . 5 km N ( ε ) = 22 N ( ε ) = 53 N ( ε ) = 130 N ( ε ) = # { boxes of size ε required to cover the set (coastline) } . Idea: if C is the measure of the set, then ( number of boxes ) · ( box size ) = N ( ε ) · ( ε ) D ≈ C � 1 � D = ⇒ N ( ε ) ≈ C ε
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting How to find D : box-counting N ( ε ) ≈ C ( 1 ε ) D
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting How to find D : box-counting � ε ) D � N ( ε ) ≈ C ( 1 ε ) D C ( 1 = ⇒ log N ( ε ) ≈ log log N ( ε ) ≈ log C + D log ( 1 ε ) log N ( ε ) log C ε ) ≈ ε ) + D log ( 1 log ( 1 log N ( ε ) D = lim log ( 1 ε ) ε → 0
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 . A ⊆ � J N ext ( ε ) = min { J . j = 1 B ( x j , ε ) , where ( x j ) j ∈ J ⊆ R d } . . A ⊆ � J N int ( ε ) = min { J . . j = 1 B ( x j , ε ) , where ( x j ) j ∈ J ⊆ A } . B ( x j , ε ) ∩ B ( x k , ε ) = ∅ , j � = k , where ( x j ) j ∈ J ⊆ A } N pack ( ε ) = max { J . . . | x j − x k | ≥ ε, where ( x j ) j ∈ J ⊆ A } N net ( ε ) = max { J . .
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 . A ⊆ � J N ext ( ε ) = min { J . j = 1 B ( x j , ε ) , where ( x j ) j ∈ J ⊆ R d } . . A ⊆ � J N int ( ε ) = min { J . . j = 1 B ( x j , ε ) , where ( x j ) j ∈ J ⊆ A } . B ( x j , ε ) ∩ B ( x k , ε ) = ∅ , j � = k , where ( x j ) j ∈ J ⊆ A } N pack ( ε ) = max { J . . . | x j − x k | ≥ ε, where ( x j ) j ∈ J ⊆ A } N net ( ε ) = max { J . . N net ( 2 ε ) = N pack ( ε ) ≤ 2 n N ext ( ε ) N ext ( ε ) ≤ N int ( ε ) ≤ N net ( ε ) . and
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 log V A ( ε ) Minkowski’s formulation: D = n − lim . log ε ε → 0 Here, V A ( ε ) = vol d ( A − ε ) is an inner tube formula , and A − ε is the inner ε -parallel set of A : . d ( x , A ) < ε } . A − ε = { x ∈ A . . A A
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 log V A ( ε ) Minkowski’s formulation: D = n − lim . log ε ε → 0 Here, V A ( ε ) = vol d ( A − ε ) is an inner tube formula , and A − ε is the inner ε -parallel set of A : . d ( x , A ) < ε } . A − ε = { x ∈ A . . A -e
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 log V A ( ε ) Minkowski’s formulation: D = n − lim . log ε ε → 0 Here, V A ( ε ) = vol d ( A − ε ) is an inner tube formula , and A − ε is the inner ε -parallel set of A : . d ( x , A ) < ε } . A − ε = { x ∈ A . . A -e So you can find D if you can find V A ( ε )
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting Alternative formulations log N ( ε ) Box-counting formulation: D = lim ε ) . log ( 1 ε → 0 log V A ( ε ) Minkowski’s formulation: D = n − lim . log ε ε → 0 Here, V A ( ε ) = vol d ( A − ε ) is an inner tube formula , and A − ε is the inner ε -parallel set of A : . d ( x , A ) < ε } . A − ε = { x ∈ A . . P ROBLEM : The limits may not exist.
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski measurability Definition. The set A ⊆ R d is Minkowski measurable in dimension α iff ε → 0 + V A ( ε ) ε − ( d − α ) M α ( A ) = lim exists with 0 < M α ( A ) < ∞ . In this case, M α ( A ) is the α -dimensional Minkowski content of A .
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski measurability Definition. The set A ⊆ R d is Minkowski measurable in dimension α iff ε → 0 + V A ( ε ) ε − ( d − α ) M α ( A ) = lim exists with 0 < M α ( A ) < ∞ . In this case, M α ( A ) is the α -dimensional Minkowski content of A . One also has the average Minkowski content � 1 ε α − d V A ( ε ) d ε 1 M α ( A ) := lim | ln δ | ε δ → 0 + δ which always exists for self-similar sets [Gatzouras].
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski measurability Definition. The set A ⊆ R d is Minkowski measurable in dimension α iff ε → 0 + V A ( ε ) ε − ( d − α ) M α ( A ) = lim exists with 0 < M α ( A ) < ∞ . In this case, M α ( A ) is the α -dimensional Minkowski content of A . ∞ , α < D , Since M α ( A ) = M D ( A ) , α = D , let M ( A ) = M D ( A ) . 0 , D < α,
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Measuring with respect to different dimensions ¥ a ( A ) a -dimensional measure D ( A ) a = D 0 dimension a
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Hausdorff: � � ε → 0 + . A ⊆ | U j | α . H α,ε ( A ) = inf { U j , | U j | ≤ ε } − − − − − → H α ( A ) . j ∈J j ∈J
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Hausdorff: � � ε → 0 + . A ⊆ | U j | α . H α,ε ( A ) = inf { U j , | U j | ≤ ε } − − − − − → H α ( A ) . j ∈J j ∈J Minkowski: � � N ( ε ) ε α = inf { ε → 0 + . A ⊆ | U j | α . U j , | U j | = ε } − − − − − → ? M α ( A ) . j ∈J j ∈J Hausdorff allows countable covers by arbitrarily sized sets. Minkowski uses finite covers by uniformly sized sets. (Compare to partitions used in Lebesgue vs Riemann integration.)
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Why Minkowski dimension instead of Hausdorff?
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Why Minkowski dimension instead of Hausdorff? It is not as refined a notion as Hausdorff dimension!
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Hausdorff measure: σ -additive. Hausdorff dimension is sensitive to translations of connected components. dim ( ¶W ) = log 3 2 dim ( ¶W ) = 0
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Minkowski vs. Hausdorff Hausdorff measure: σ -additive. Hausdorff dimension is sensitive to translations of connected components. dim ( ¶W ) = log 3 2 dim ( ¶W ) = log 3 2 dim ( ¶W ) = 0 dim ( ¶W ) = log 3 2 Minkowski content: only finitely additive. Minkowski dimension is invariant under translations of connected components. = ⇒ dim M makes more sense for drums
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability What V A ( ε ) looks like, for open subsets of R Definition. A fractal string L is the complement of a fractal subset of R . L is a bounded open subset of R , so L = { L n } ∞ n = 1 , where each L n is an open interval with length denoted ℓ n .
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability What V A ( ε ) looks like, for open subsets of R Definition. A fractal string L is the complement of a fractal subset of R . L is a bounded open subset of R , so L = { L n } ∞ n = 1 , where each L n is an open interval with length denoted ℓ n . The Cantor string: CS = { 1 3 , 1 9 , 1 9 , 1 27 , 1 27 , 1 27 , 1 27 , 1 81 , . . . } L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability What V A ( ε ) looks like, for open subsets of R Definition. A fractal string L is the complement of a fractal subset of R . L is a bounded open subset of R , so L = { L n } ∞ n = 1 , where each L n is an open interval with length denoted ℓ n . The Cantor string: CS = { 1 3 , 1 9 , 1 9 , 1 27 , 1 27 , 1 27 , 1 27 , 1 81 , . . . } For fractal strings: � � V L ( ε ) = 2 ε + ℓ n . ℓ n > 2 ε ℓ n ≤ 2 ε
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability V ( e ) 1 1 1 18 54 18 6 6 � � V L ( ε ) = 2 ε + ℓ n ℓ n > 2 ε ℓ n ≤ 2 ε ε 2 ε + 2 ε + 2 ε + l 4 + l 5 + l 6 + l 7 + l 8 + l 9 + ...
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability V ( e ) 1 1 1 18 54 18 6 6 � � V L ( ε ) = 2 ε + ℓ n ℓ n > 2 ε ℓ n ≤ 2 ε ε 2 ε + 2 ε + 2 ε + 2 ε + l 5 + l 6 + l 7 + l 8 + l 9 + ...
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability e 1 - D V ( e ) 1 1 1 18 54 18 6 6 ε → 0 + V L ( ε ) ε − ( 1 − D ) M = lim
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability e 1 - D V ( e ) 1 1 1 18 54 18 6 6 ε → 0 + V L ( ε ) ε − ( 1 − D ) M = lim * V ( e ) e 1 - D e * 1 1 1 54 18 6
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability e 1 - D V ( e ) 1 1 1 18 54 18 6 6 ε → 0 + V L ( ε ) ε − ( 1 − D ) M = lim * V ( e ) e 1 - D log e * 1 1 1 1 54 18 6 2
Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability Using V L ( ε ) to see Minkowski measurability e 1 - D V ( e ) 1 1 1 18 54 18 6 6 There are “ geometric oscillations ” in V L ( ε ) of order D . This means lim ε → 0 + ε − ( 1 − D ) V L ( ε ) cannot exist. ε − ( 1 − D ) V L ( ε ) contains terms of the form c ω ε Im ( ω ) ✐ = c ω e Im ( ω ) ✐ log ε .
Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets How to compute the tube formula for a self-similar set? Use inner tube formula for A ∁ to find outer tube formula for A . = ∪
Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets How to compute the tube formula for a self-similar set? Use inner tube formula for A ∁ to find outer tube formula for A . = ∪ (1) Obtain the components of A ∁ from the IFS. (2) Determine compatibility conditions. (3) Compute V ( ε ) = vol d (( A ∁ ) − ε ) using complex dimensions.
Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets Self-similar sets Self-similarity sets and IFSs SG is invariant under Φ j ( x ) = x 2 + q j , j = 1 , 2 , 3 where { q 1 , q 2 , q 3 } are the vertices of a triangle. q 1 F 2 ( q 2 ) = q 2 q 2 q 3
Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets Self-similar sets Self-similarity sets and IFSs SG is invariant under Φ j ( x ) = x 2 + q j , j = 1 , 2 , 3 where { q 1 , q 2 , q 3 } are the vertices of a triangle. q 1 F 2 ( q 2 ) = q 2 q 2 q 3 Definition. A self-similar set F ⊆ C ( X ) is a fixed point � J F = Φ( F ) := j = 1 Φ j ( F ) , F � = ∅ . Each Φ j should be contractive: | Φ j ( x ) − Φ j ( y ) | < | x − y | .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Initiating the tiling construction For the construction to be possible, assume F satisfies the open set condition If O is a feasible open set for F , this means: 1 Φ j ( O ) ∩ Φ k ( O ) = ∅ for j � = k , 2 Φ j ( O ) ⊆ O for each j , 3 F ⊆ O ,
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Initiating the tiling construction For the construction to be possible, assume F satisfies the open set condition , and int F = ∅ . If O is a feasible open set for F , this means: 1 Φ j ( O ) ∩ Φ k ( O ) = ∅ for j � = k , 2 Φ j ( O ) ⊆ O for each j , 3 F ⊆ O , and 4 O � Φ( O ) . For a feasible open set O , construct a tiling of K := O .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket The Koch tiling and the Sierpinski gasket tiling for ξ = 1 ✐ 3 ∈ C . Φ 1 ( z ) = ξ z , Φ 2 ( z ) = ( 1 − ξ )( z − 1 ) + 1 , 2 + √ 2 Φ 1 ( x ) = x 2 + p 1 , Φ 2 ( x ) = x 2 + p 2 , Φ 3 ( x ) = x 2 + p 3 .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Definition of the tiling Each tile is the image of a generator Definition. The generators { G q } Q q = 1 are the connected components of O \ Φ( K ) . Some examples may have multiple generators.
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Definition of the tiling Each tile is the image of a generator Definition. The generators { G q } Q q = 1 are the connected components of O \ Φ( K ) . Definition. The self-similar tiling associated with Φ and O is . w ∈ W , q = 1 , . . . , Q } , T = T ( O ) = { Φ w ( G q ) . . where W := � ∞ k = 0 { 1 , . . . , N } k is all finite strings on { 1 , . . . , N } , and Φ w := Φ w 1 ◦ Φ w 2 ◦ . . . ◦ Φ w n . Let T = � R ∈T R denote the union of the tiles.
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators Choosing a good O (or K ) Theorem [Compatibility Theorem]: Let int F = ∅ satisfy OSC with feasible set O and associated tiling T ( O ) . Then TFAE: 1 bd T = F . 2 bd G q ⊆ F , for all q ∈ Q . 3 F ε ∩ K = T − ε for all ε ≥ 0 . 4 F ε ∩ K ∁ = K ε ∩ K ∁ for all ε ≥ 0 . So for a given Φ and F , check that one of 1–4 is satisfied. Then 5–6 ensure the inner/outer decomposition:
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators Choosing a good O (or K ) Specific possibilities: (1) Choose K = [ F ] and O = int K . Feasible iff int Φ j ( K ) ∩ Φ k ( K ) = ∅ for j � = k . (Tileset condition) In this case, int F � = ∅ iff F is convex. (Nontriviality condition)
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators Choosing a good O (or K ) Specific possibilities: (1) Choose K = [ F ] and O = int K . Feasible iff int Φ j ( K ) ∩ Φ k ( K ) = ∅ for j � = k . (Tileset condition) In this case, int F � = ∅ iff F is convex. (Nontriviality condition) (2) Let U be the unbounded component of F ∁ . Choose K = U ∁ (the envelope of F ) and O = int K . For the envelope, one always has bd K ⊆ F ⊆ K ⊆ [ F ] , and K is convex iff K = [ F ] .
Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators Choosing a good O (or K ) Specific possibilities: (1) Choose K = [ F ] and O = int K . Feasible iff int Φ j ( K ) ∩ Φ k ( K ) = ∅ for j � = k . (Tileset condition) In this case, int F � = ∅ iff F is convex. (Nontriviality condition) (2) Let U be the unbounded component of F ∁ . Choose K = U ∁ (the envelope of F ) and O = int K . For the envelope, one always has bd K ⊆ F ⊆ K ⊆ [ F ] , and K is convex iff K = [ F ] .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Encoding the data into a formula We have seen A decomposition of the set to be measured (via tiling). Conds on { Φ 1 , . . . , Φ J } that ensure tiling is possible. Conds on O (or K = O ) that allow vol d ( F ε ) = V T ( ε ) + vol d ( K ε ) .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Encoding the data into a formula We have seen A decomposition of the set to be measured (via tiling). Conds on { Φ 1 , . . . , Φ J } that ensure tiling is possible. Conds on O (or K = O ) that allow vol d ( F ε ) = V T ( ε ) + vol d ( K ε ) . Next: how to encode the resulting data from The scaling factors r w , where r w = r w 1 . . . r w n is scaling ratio of Φ w := Φ w 1 ◦ . . . ◦ Φ w n . The generator(s) G = O \ Φ( K ) .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling The scaling data of the tiling Definition. Let r w = r w 1 r w 2 . . . r w k be the scaling ratio of Φ w . The scaling zeta function is given by the scaling ratios of Φ via � 1 r s for s ∈ C . ζ s ( s ) = w = , 1 − � J j = 1 r s w ∈W j ζ s records the sizes (and multiplicities) of the tiles T = { Φ w ( G ) } . ∞ � { 1 , . . . , J } k is all finite seqs of indices, and Φ w := Φ w 1 ◦ Φ w 2 ◦ . . . ◦ Φ w k . W := k = 0
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling The scaling data of the tiling Definition. Let r w = r w 1 r w 2 . . . r w k be the scaling ratio of Φ w . The scaling zeta function is given by the scaling ratios of Φ via � 1 r s for s ∈ C . ζ s ( s ) = w = , 1 − � J j = 1 r s w ∈W j ζ s records the sizes (and multiplicities) of the tiles T = { Φ w ( G ) } . Poles of ζ s encode scaling information: . ω is a pole of ζ s } . D s = { ω . .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling The scaling data of the tiling Definition. Let r w = r w 1 r w 2 . . . r w k be the scaling ratio of Φ w . The scaling zeta function is given by the scaling ratios of Φ via � 1 r s for s ∈ C . ζ s ( s ) = w = , 1 − � J j = 1 r s w ∈W j ζ s records the sizes (and multiplicities) of the tiles T = { Φ w ( G ) } . Poles of ζ s encode scaling information: . ω is a pole of ζ s } . D s = { ω . . The complex dimensions D s generalize the Minkowski dimension. . ζ s ( σ ) < ∞} . Theorem [Lapidus et al.] D ∈ D s and D = inf { σ ∈ R . .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Consider a self-similar set with mappings Φ 1 , Φ 2 having scaling ratios r 1 = 2 − 1 and r 2 = 2 − φ . D is the set of solutions of the transcendental equation 2 − s + 2 − φ s = 1 ( s ∈ D ) ,
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Consider a self-similar set with mappings Φ 1 , Φ 2 having scaling ratios r 1 = 2 − 1 and r 2 = 2 − φ . D is the set of solutions of the transcendental equation 2 − s + 2 − φ s = 1 ( s ∈ D ) , The fractal can be approximated by a sequence with scaling ratios 2 1 , 3 2 , 5 3 , 8 5 , 13 8 , 21 13 , 34 21 , 55 34 , . . . − → φ. For any such approximation, r 1 and r 2 are integer powers of some common base r ( F is lattice type ) r 1 = r k 1 , r 2 = r k 2 , k 1 , k 2 ∈ N
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling 50 50 50 p p p p 700 D D D D − 1 1 − 1 1 − 1 1 − 1 1 f » 2/1 f » 3/2 f » 5/3 f » 8/5 p p p 100 100 100 100 p 50 0 D 1 D D D D − 1 1 − 1 1 − 1 1 − 1 1 f » 13/8 f » 21/13 f » 34/21 f » 55/34
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Lattice vs. Nonlattice For self-similar strings, a dichotomy exists. The lattice case : { log r 1 , . . . , log r J } are rationally dependent. # { Re ω } is finite. ∃ a row of dimns on Re s = D . Re ω = D for infinitely many ω ∈ D s . ∂ L is not Minkowski measurable. Example: Cantor String CS .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Lattice vs. Nonlattice For self-similar strings, a dichotomy exists. The lattice case : The nonlattice case : { log r 1 , . . . , log r J } are rationally Some log r j are rationally dependent. independent. # { Re ω } is finite. # { Re ω } is infinite. ∃ a row of dimns on Re s = D . Re ω dense in parts of [ σ l , D ] . Re ω = D for infinitely many ω ∈ D s . Re ω = D = ⇒ ω = D . ∂ L is not Minkowski measurable. ∂ L is Minkowski measurable. Example: Cantor String CS . Example: The golden string GS .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Conjecture: Lattice/Nonlattice Dichotomy Theorem. A self-similar subset F ⊆ R is Minkowski measurable ⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.)
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Conjecture: Lattice/Nonlattice Dichotomy Theorem. A self-similar subset F ⊆ R is Minkowski measurable ⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.) Lapidus et. al conjectured that (under suitable hypotheses): 1 Lattice-type fractals are not Minkowski measurable. 2 Nonlattice-type fractals are Minkowski measurable.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Conjecture: Lattice/Nonlattice Dichotomy Theorem. A self-similar subset F ⊆ R is Minkowski measurable ⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.) Lapidus et. al conjectured that (under suitable hypotheses): 1 Lattice-type fractals are not Minkowski measurable. 2 Nonlattice-type fractals are Minkowski measurable. Theorem. A self-similar subset F ⊆ R is Minkowski measurable if F is nonlattice. (Proved by Gatzouras.) The converse has kept us busy.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Converting “scales” to “sizes” Definition. The inradius of A ⊆ R d is the radius of the largest metric ball contained in A . . A − ε = A } . Equivalently, ρ ( A ) := inf { ε > 0 . . For A = G , write g := ρ ( G ) . The tile Φ w ( G ) has inradius r w g .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling Converting “scales” to “sizes” Definition. The inradius of A ⊆ R d is the radius of the largest metric ball contained in A . . A − ε = A } . Equivalently, ρ ( A ) := inf { ε > 0 . . For A = G , write g := ρ ( G ) . The tile Φ w ( G ) has inradius r w g . We now have all the scaling data we need. − → Next, we look at the geometry of the generator.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators Monophase and pluriphase generators Definition. G is monophase iff V G ( ε ) is polynomial on [ 0 , g ] : d − 1 � κ k ε d − k , V G ( ε ) = 0 ≤ ε ≤ g . k = 0 Motivation: Steiner’s Theorem. d − 1 � κ k ε d − k for ε ≥ 0 . If A is convex, then vol d ( A ε ) = k = 0 Here, κ k = µ k ( A ) vol d - k ( B d − k ) , and µ k are the intrinsic volumes .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators Monophase and pluriphase generators Definition. G is monophase iff V G ( ε ) is polynomial on [ 0 , g ] : d − 1 � κ k ε d − k , V G ( ε ) = 0 ≤ ε ≤ g . k = 0 Definition. G is pluriphase iff V G ( ε ) is polynomial on each subinterval of [ 0 = a 0 , a 1 , a 1 , . . . , a M = g ] .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators How to compute the tube formula Suppose you have Φ satisfying OSC, with int F = ∅ , and a feasible open set O satisfying the compatibility theorem.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators How to compute the tube formula Suppose you have Φ satisfying OSC, with int F = ∅ , and a feasible open set O satisfying the compatibility theorem. � r s Scaling data: ζ s ( s ) = w . w ∈W d − 1 � κ k ε d − k . Generator geometry: V G ( ε ) = k = 0 We compute V T ( ε ) = vol d ( T − ε ) , the inner tube formula for the tiling, in terms of these ingredients.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The fractal tube formula Fractal tube formula Theorem [Pearse 2006], [Lapidus, Pearse, Winter 2011] Suppose T has a single monophase generator and ζ s has only simple poles. Then for 0 ≤ ε ≤ g , � c ω ε d − ω , V T ( ε ) = ω ∈D s ∪{ 0 , 1 ,..., d − 1 } where � res ( ζ s ; ω ) � d g ω − k ω − k κ k , ω ∈ D s , k = 0 c ω = κ k ζ s ( k ) , ω = k ∈ { 0 , 1 , . . . , d } . Recall D s := { poles of ζ s } .
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results Lattice type fractals are not Minkowski measurable Theorem [Lapidus, Pearse, Winter 2013]. Suppose F is a fractal with one monophase generator, d − 1 < D < d , and satisfying some technical conditions. If F is lattice, then F is not Minkowski measurable.
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results Lattice type fractals are not Minkowski measurable Theorem [Lapidus, Pearse, Winter 2013]. Suppose F is a fractal with one monophase generator, d − 1 < D < d , and satisfying some technical conditions. If F is lattice, then F is not Minkowski measurable. Proof uses the pointwise tube formula, with an estimate based on careful choice of a window (contour). W n +1 W n 0 D
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results The screen and the window Choose a vertical contour . t ∈ R } . S = { S ( t ) + ✐ t . where S : R → R is Lipschitz. S is the screen . . Re z ≥ S ( Im z ) } . . Define the window W = { z ∈ C . W D
Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results Fractal tube formula with error term Theorem [Lapidus, Pearse, Winter 2013] Suppose T has a single monophase generator and ζ s has only simple poles. Then for 0 ≤ ε ≤ g , � � c ω ε d − ω + c k ε d − k + R ( ε ) , V T ( ε ) = ω ∈D s ∩ W k ∈{ 0 , 1 ,..., d − 1 }∩ W where D s := { poles of ζ s } and d � g ω − k c ω = res ( ζ s ; ω ) ω − k κ k , c k = κ k ζ s ( k ) , k = 0 as before,
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