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

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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

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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

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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

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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

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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 Rd euclidean space, but I will only consider examples in R2 and R1.

Sascha Troscheit Assouad Dimension and Random Fractals

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Classical Dimensions

Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ, called Nδ to cover a set E. Nδ ∼ δ−s

Sascha Troscheit Assouad Dimension and Random Fractals

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Classical Dimensions

Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ, called Nδ to cover a set E. Nδ ∼ δ−s Hausdorff dimension Similar idea, but not restricted to boxes. You can take any open set Uδ of diameter less than δ and consider the ‘best’ cover:

  • |U|s ∼ 1

Sascha Troscheit Assouad Dimension and Random Fractals

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Classical Dimensions

Box counting dimension The box counting dimension s is the exponential factor between minimum number of boxes of side length δ, called Nδ to cover a set E. Nδ ∼ δ−s Hausdorff dimension Similar idea, but not restricted to boxes. You can take any open set Uδ 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

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Assouad dimension

Sascha Troscheit Assouad Dimension and Random Fractals

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Assouad dimension

Brace yourself!

Sascha Troscheit Assouad Dimension and Random Fractals

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Assouad dimension

Brace yourself! Definition Let (X, d) be a metric space and for any non-empty subset F ⊆ X and r > 0, let Nr(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

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Assouad dimension

Brace yourself! Definition Let (X, d) be a metric space and for any non-empty subset F ⊆ X and r > 0, let Nr(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, dimA F, is defined by

Sascha Troscheit Assouad Dimension and Random Fractals

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Assouad dimension

Brace yourself! Definition Let (X, d) be a metric space and for any non-empty subset F ⊆ X and r > 0, let Nr(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, dimA F, is defined by dimA F = inf

  • α
  • ∃ C, ρ > 0 such that, for all 0 < r < R ≤ ρ,

we have sup

x∈F

Nr

  • B(x, R) ∩ F
  • ≤ C

R r α

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimensions summarised

In general we have:

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimensions summarised

In general we have: dimH F ≤ dimBF ≤ dimBF ≤ dimA F

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimensions summarised

In general we have: dimH F ≤ dimBF ≤ dimBF ≤ dimA F dimH F ≤ dimP F ≤ dimBF

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimensions summarised

In general we have: dimH F ≤ dimBF ≤ dimBF ≤ dimA F dimH F ≤ dimP F ≤ dimBF For many settings with a lot of ‘regularity’, like self-similar fractals all notions coincide.

Sascha Troscheit Assouad Dimension and Random Fractals

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Random Fractals

We will introduce the following random models:

Sascha Troscheit Assouad Dimension and Random Fractals

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Random Fractals

We will introduce the following random models: Mandelbrot Percolation

Sascha Troscheit Assouad Dimension and Random Fractals

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Random Fractals

We will introduce the following random models: Mandelbrot Percolation 1-Variable Random Iterated Function System

Sascha Troscheit Assouad Dimension and Random Fractals

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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

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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

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.85

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Mandelbrot Percolation p = 0.65

Sascha Troscheit Assouad Dimension and Random Fractals

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Percolation Tree Structure, d = 2, n = 2, p = 1

Sascha Troscheit Assouad Dimension and Random Fractals

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Percolation Tree Structure, d = 2, n = 2, p = 0.7

Sascha Troscheit Assouad Dimension and Random Fractals

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Percolation Tree Structure, d = 2, n = 2, p = 0.3

Sascha Troscheit Assouad Dimension and Random Fractals

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Percolation Tree Structure, d = 2, n = 2, p = 0.3

Sascha Troscheit Assouad Dimension and Random Fractals

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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 dimH F = dimB F = dimP F = log ndp log n conditioned on F being non-empty.

Sascha Troscheit Assouad Dimension and Random Fractals

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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 dimH F = dimB F = dimP F = log ndp log n conditioned on F being non-empty. Theorem (Fraser-Miao-T. ’14) Almost surely, conditioned on F being non-empty, we have dimA F = d

Sascha Troscheit Assouad Dimension and Random Fractals

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Iterated Function Systems - (IFS)

Let I = {S1, S2, . . . , Sn} be a set of n contractions Si : R2 → R2.

Sascha Troscheit Assouad Dimension and Random Fractals

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Iterated Function Systems - (IFS)

Let I = {S1, S2, . . . , Sn} be a set of n contractions Si : R2 → R2. Take a ‘nice’ compact set ∆, say the unit square ∆ = [0, 1]2, and define iteratively

Sascha Troscheit Assouad Dimension and Random Fractals

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Iterated Function Systems - (IFS)

Let I = {S1, S2, . . . , Sn} be a set of n contractions Si : R2 → R2. Take a ‘nice’ compact set ∆, say the unit square ∆ = [0, 1]2, and define iteratively F0 = ∆ Fn+1 =

n

  • i=1

Si(Fn)

Sascha Troscheit Assouad Dimension and Random Fractals

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Iterated Function Systems - (IFS)

Let I = {S1, S2, . . . , Sn} be a set of n contractions Si : R2 → R2. Take a ‘nice’ compact set ∆, say the unit square ∆ = [0, 1]2, and define iteratively F0 = ∆ Fn+1 =

n

  • i=1

Si(Fn) The ‘limit’ of these sets is the self-similar or self-affine fractal F, more precisely

Sascha Troscheit Assouad Dimension and Random Fractals

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Iterated Function Systems - (IFS)

Let I = {S1, S2, . . . , Sn} be a set of n contractions Si : R2 → R2. Take a ‘nice’ compact set ∆, say the unit square ∆ = [0, 1]2, and define iteratively F0 = ∆ Fn+1 =

n

  • i=1

Si(Fn) The ‘limit’ of these sets is the self-similar or self-affine fractal F, more precisely F = lim

N→∞ N

  • n=1

Fn

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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An example: Sierpi´ nski triangle

Sascha Troscheit Assouad Dimension and Random Fractals

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Equivalent notion

Remember I?

Sascha Troscheit Assouad Dimension and Random Fractals

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Equivalent notion

Remember I? Index elements by Λ = {1, 2, . . . , n} and consider the coding space Σ = ΛN, consisting of infinite sequences of entries in Λ.

Sascha Troscheit Assouad Dimension and Random Fractals

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Equivalent notion

Remember I? Index elements by Λ = {1, 2, . . . , n} and consider the coding space Σ = ΛN, consisting of infinite sequences of entries in Λ. For x = (x1, x2, . . .) ∈ Σ we define Sx = lim

k→∞ Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆)

Sascha Troscheit Assouad Dimension and Random Fractals

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Equivalent notion

Remember I? Index elements by Λ = {1, 2, . . . , n} and consider the coding space Σ = ΛN, consisting of infinite sequences of entries in Λ. For x = (x1, x2, . . .) ∈ Σ we define Sx = lim

k→∞ Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆)

We can code F with Σ as every y ∈ F has at least one coding x ∈ Σ such that Sx = y and Sx ⊆ F.

Sascha Troscheit Assouad Dimension and Random Fractals

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Equivalent notion

Remember I? Index elements by Λ = {1, 2, . . . , n} and consider the coding space Σ = ΛN, consisting of infinite sequences of entries in Λ. For x = (x1, x2, . . .) ∈ Σ we define Sx = lim

k→∞ Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆)

We can code F with Σ as every y ∈ F has at least one coding x ∈ Σ such that Sx = y and Sx ⊆ F. We have F = lim

k→∞ k

  • i=1
  • x1∈Λ, x2∈Λ,

..., xk−1∈Λ, xk∈Λ

Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆)

Sascha Troscheit Assouad Dimension and Random Fractals

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(1-variable) Random Iterated Function System (RIFS)

Start with a collection of IFSs I = {I1, I2, . . . , IN}, indexed by Λ and each IFS is indexed by Λi.

Sascha Troscheit Assouad Dimension and Random Fractals

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(1-variable) Random Iterated Function System (RIFS)

Start with a collection of IFSs I = {I1, I2, . . . , IN}, indexed by Λ and each IFS is indexed by Λi. Construct F by ‘randomly choosing’ an IFS at each step of the construction.

Sascha Troscheit Assouad Dimension and Random Fractals

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(1-variable) Random Iterated Function System (RIFS)

Start with a collection of IFSs I = {I1, I2, . . . , IN}, indexed by Λ and each IFS is indexed by Λi. Construct F by ‘randomly choosing’ an IFS at each step of the construction. More formally we define the realisation ω ∈ Ω as an element of the set of all possible outcomes Ω = ΛN.

Sascha Troscheit Assouad Dimension and Random Fractals

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(1-variable) Random Iterated Function System (RIFS)

Start with a collection of IFSs I = {I1, I2, . . . , IN}, indexed by Λ and each IFS is indexed by Λi. Construct F by ‘randomly choosing’ an IFS at each step of the construction. More formally we define the realisation ω ∈ Ω as an element of the set of all possible outcomes Ω = ΛN. The set F(ω) is then defined as F(ω) = lim

k→∞ k

  • i=1
  • x1∈Λω1, x2∈Λω2,

..., xk−1∈Λωk−1, xk∈Λωk

Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆)

Sascha Troscheit Assouad Dimension and Random Fractals

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(1-variable) Random Iterated Function System (RIFS)

Start with a collection of IFSs I = {I1, I2, . . . , IN}, indexed by Λ and each IFS is indexed by Λi. Construct F by ‘randomly choosing’ an IFS at each step of the construction. More formally we define the realisation ω ∈ Ω as an element of the set of all possible outcomes Ω = ΛN. The set F(ω) is then defined as F(ω) = lim

k→∞ k

  • i=1
  • x1∈Λω1, x2∈Λω2,

..., xk−1∈Λωk−1, xk∈Λωk

Sx1 ◦ Sx2 ◦ . . . ◦ Sxk(∆) ‘Randomly choosing’ ω gives us a random fractal F(ω).

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Examples

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimension of Random Fractals

Theorem Assuming some ‘nice’ conditions (Uniform Open Set Condition), the almost sure Hausdorff, box and packing dimension for random self-similar sets is given by dim F(ω) = E{dim F(i, i, . . .)} =

  • i∈Λ

pi dim F(i, i, . . .) where pi is the probability of choosing the digit i.

Sascha Troscheit Assouad Dimension and Random Fractals

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Dimension of Random Fractals

Theorem (Fraser-Miao-T. ’14) Assuming some ‘nice’ conditions (Uniform Open Set Condition), the almost sure Assouad dimension for random self-similar sets is given by dimA F(ω) = max

i∈Λ dimA F(i, i, . . .)

assuming pi > 0 for all digits i.

Sascha Troscheit Assouad Dimension and Random Fractals

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An interesting self-affine example

Sascha Troscheit Assouad Dimension and Random Fractals

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An interesting self-affine example

Sascha Troscheit Assouad Dimension and Random Fractals

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Deterministic Graph Directed Construction

Take the infinite words idea and restrict the letter combinations.

Sascha Troscheit Assouad Dimension and Random Fractals

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Deterministic Graph Directed Construction

Take the infinite words idea and restrict the letter combinations. In particular take a finite directed strongly connected graph, where each of the edges represent maps.

Sascha Troscheit Assouad Dimension and Random Fractals

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Deterministic Graph Directed Construction

Take the infinite words idea and restrict the letter combinations. In particular take a finite directed strongly connected graph, where each of the edges represent maps. The fractal Ki is the limit set taking all possible infinite edge combinations of maps starting at vertex i.

Sascha Troscheit Assouad Dimension and Random Fractals

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Deterministic Graph Directed Construction

Take the infinite words idea and restrict the letter combinations. In particular take a finite directed strongly connected graph, where each of the edges represent maps. The fractal Ki is the limit set taking all possible infinite edge combinations of maps starting at vertex i. The limit set of an IFS is a graph directed fractal for the (almost) trivial graph consisting of one vertex and an edge for every map in the IFS.

Sascha Troscheit Assouad Dimension and Random Fractals

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Random Graph Directed Construction

There are two different construction I will introduce: The ‘Random Graph Directed Iterated Function Scheme’ (Lars’ version)

Sascha Troscheit Assouad Dimension and Random Fractals

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Random Graph Directed Construction

There are two different construction I will introduce: The ‘Random Graph Directed Iterated Function Scheme’ (Lars’ version) My very own model.

Sascha Troscheit Assouad Dimension and Random Fractals

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HD and AD of Random Graph Directed Construction

For Hausdorff, box and packing dimension of Lars’ version, see his book.

Sascha Troscheit Assouad Dimension and Random Fractals

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HD and AD of Random Graph Directed Construction

For Hausdorff, box and packing dimension of Lars’ version, see his book. For my construction, ask me in a couple of months.

Sascha Troscheit Assouad Dimension and Random Fractals