Assouad Dimension and Random Fractals (Contains joint work with - PowerPoint PPT Presentation

Assouad Dimension and Random Fractals (Contains joint work with Jonathan M. Fraser and Jun J. Miao) Sascha Troscheit University of St Andrews October 3, 2014 Pure Postgraduate Seminar Sascha Troscheit Assouad Dimension and Random Fractals

Dimension Theory Dimension Theory Dimension theory is, broadly speaking, the study of the relationship between content of a set and its size. Sascha Troscheit Assouad Dimension and Random Fractals

Dimension Theory Dimension Theory Dimension theory is, broadly speaking, the study of the relationship between content of a set and its size. Exponential ratio In particular it looks at the exponent α , called the dimension, such that content ∼ size − α Sascha Troscheit Assouad Dimension and Random Fractals

Dimension Theory Dimension Theory Dimension theory is, broadly speaking, the study of the relationship between content of a set and its size. Exponential ratio In particular it looks at the exponent α , called the dimension, such that content ∼ size − α A glimmer of hope Everything in the following slides extends to R d euclidean space, but I will only consider examples in R 2 and R 1 . Sascha Troscheit Assouad Dimension and Random Fractals

Classical Dimensions Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ , called N δ to cover a set E . N δ ∼ δ − s Sascha Troscheit Assouad Dimension and Random Fractals

Classical Dimensions Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ , called N δ to cover a set E . N δ ∼ δ − s Hausdorff dimension Similar idea, but not restricted to boxes. You can take any open set U δ of diameter less than δ and consider the ‘best’ cover: | U | s ∼ 1 � Sascha Troscheit Assouad Dimension and Random Fractals

Classical Dimensions Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ , called N δ to cover a set E . N δ ∼ δ − s Hausdorff dimension Similar idea, but not restricted to boxes. You can take any open set U δ of diameter less than δ and consider the ‘best’ cover: | U | s ∼ 1 � Packing dimension I don’t care. Sascha Troscheit Assouad Dimension and Random Fractals

Assouad dimension Sascha Troscheit Assouad Dimension and Random Fractals

Assouad dimension Brace yourself! Sascha Troscheit Assouad Dimension and Random Fractals

Assouad dimension Brace yourself! Definition Let ( X , d ) be a metric space and for any non-empty subset F ⊆ X and r > 0, let N r ( F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . Sascha Troscheit Assouad Dimension and Random Fractals

Assouad dimension Brace yourself! Definition Let ( X , d ) be a metric space and for any non-empty subset F ⊆ X and r > 0, let N r ( F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . The Assouad dimension of a non-empty subset F of X , dim A F , is defined by Sascha Troscheit Assouad Dimension and Random Fractals

Assouad dimension Brace yourself! Definition Let ( X , d ) be a metric space and for any non-empty subset F ⊆ X and r > 0, let N r ( F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . The Assouad dimension of a non-empty subset F of X , dim A F , is defined by � � dim A F = inf α � ∃ C , ρ > 0 such that, for all 0 < r < R ≤ ρ , � � α � � R � � we have sup N r B ( x , R ) ∩ F ≤ C r x ∈ F Sascha Troscheit Assouad Dimension and Random Fractals

Dimensions summarised In general we have: Sascha Troscheit Assouad Dimension and Random Fractals

Dimensions summarised In general we have: dim H F ≤ dim B F ≤ dim B F ≤ dim A F Sascha Troscheit Assouad Dimension and Random Fractals

Dimensions summarised In general we have: dim H F ≤ dim B F ≤ dim B F ≤ dim A F dim H F ≤ dim P F ≤ dim B F Sascha Troscheit Assouad Dimension and Random Fractals

Dimensions summarised In general we have: dim H F ≤ dim B F ≤ dim B F ≤ dim A F dim H F ≤ dim P F ≤ dim B F For many settings with a lot of ‘regularity’, like self-similar fractals all notions coincide. Sascha Troscheit Assouad Dimension and Random Fractals

Random Fractals We will introduce the following random models: Sascha Troscheit Assouad Dimension and Random Fractals

Random Fractals We will introduce the following random models: Mandelbrot Percolation Sascha Troscheit Assouad Dimension and Random Fractals

Random Fractals We will introduce the following random models: Mandelbrot Percolation 1-Variable Random Iterated Function System Sascha Troscheit Assouad Dimension and Random Fractals

Random Fractals We will introduce the following random models: Mandelbrot Percolation 1-Variable Random Iterated Function System Self-similar graph directed random Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation Notation Let F be the limit set of a Mandelbrot percolation of a d dimensional cube, dividing each side into n pieces with retaining probability p for each subcube in the construction. Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 85 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Mandelbrot Percolation p = 0 . 65 Sascha Troscheit Assouad Dimension and Random Fractals

Percolation Tree Structure, d = 2, n = 2, p = 1 Sascha Troscheit Assouad Dimension and Random Fractals

Percolation Tree Structure, d = 2, n = 2, p = 0 . 7 Sascha Troscheit Assouad Dimension and Random Fractals

Percolation Tree Structure, d = 2, n = 2, p = 0 . 3 Sascha Troscheit Assouad Dimension and Random Fractals

Percolation Tree Structure, d = 2, n = 2, p = 0 . 3 Sascha Troscheit Assouad Dimension and Random Fractals

Dimension of Mandelbrot percolation Theorem (Kahane-Peyriere ’76, Hawkes ’81, Falconer ’86, Mauldin-Williams ’86) Almost surely the Hausdorff, box and packing dimension is given by dim H F = dim B F = dim P F = log n d p log n conditioned on F being non-empty. Sascha Troscheit Assouad Dimension and Random Fractals

Dimension of Mandelbrot percolation Theorem (Kahane-Peyriere ’76, Hawkes ’81, Falconer ’86, Mauldin-Williams ’86) Almost surely the Hausdorff, box and packing dimension is given by dim H F = dim B F = dim P F = log n d p log n conditioned on F being non-empty. Theorem (Fraser-Miao-T. ’14) Almost surely, conditioned on F being non-empty, we have dim A F = d Sascha Troscheit Assouad Dimension and Random Fractals

Iterated Function Systems - (IFS) Let I = { S 1 , S 2 , . . . , S n } be a set of n contractions S i : R 2 → R 2 . Sascha Troscheit Assouad Dimension and Random Fractals

Iterated Function Systems - (IFS) Let I = { S 1 , S 2 , . . . , S n } be a set of n contractions S i : R 2 → R 2 . Take a ‘nice’ compact set ∆, say the unit square ∆ = [0 , 1] 2 , and define iteratively Sascha Troscheit Assouad Dimension and Random Fractals

Iterated Function Systems - (IFS) Let I = { S 1 , S 2 , . . . , S n } be a set of n contractions S i : R 2 → R 2 . Take a ‘nice’ compact set ∆, say the unit square ∆ = [0 , 1] 2 , and define iteratively F 0 = ∆ n � F n +1 = S i ( F n ) i =1 Sascha Troscheit Assouad Dimension and Random Fractals

Iterated Function Systems - (IFS) Let I = { S 1 , S 2 , . . . , S n } be a set of n contractions S i : R 2 → R 2 . Take a ‘nice’ compact set ∆, say the unit square ∆ = [0 , 1] 2 , and define iteratively F 0 = ∆ n � F n +1 = S i ( F n ) i =1 The ‘limit’ of these sets is the self-similar or self-affine fractal F , more precisely Sascha Troscheit Assouad Dimension and Random Fractals