ZENTRUM MATHEMATIK
DER TECHNISCHEN
UNIVERSITÄT MÜNCHEN
Prof. Dr. Sandra Hayes
Random
Fractals And Multifractals
I. The Unique Invariant measure μ and the Unique Invariant Set A, the
attractor, for n Random Contractions.
1. Motivation (K. Reifinger)
The Chaos Game,
Random Cantor Sets, Measures and Integration on Fractals, Sierpinski Triangle,
Code Space Σn = {1, …, n}À.
[P, 3.1] [H,
p.728, 734, P p.47-8] [B, p.334-54, Theorem 1, Exercises 4.2, 4.3, 4.6, 4.7,
4.9]
2. Existence of the Measure μ (T. Haensse)
Hutchinson metric
dH on the set P(X) of probability measures on a compact metric space
X makes P(X) compact. Weak Topology. The Marhov Operator M: P(X)O for random
contractionsis a contraction. Examples. The code map j : Σn
® A.
[B, p.355-7, Theorems, Exercises 6.3-6] [H, p.719-20, 731-4] [B, p.120,
Exercises 1.2, 1.5]
3. The measure μ and the Attractor A (A. Nowak)
Product Measure Π on Σn,
A is the support of μ=Πoφ-1. Coding examples,
examples of measures. σÎΣn defines an ergodic Markov chain converging to a
random variable with distribution μ.
[B, p.118-29, 365,
Theorem 2, Exercises 6.11, Fig.9.66(a),370-1] [H, p.715, 733-4] [P, p.39-48,
58-67]
4. Dimension of Statistically Self-Similar Fractals
(M. Hirmer)
Mass Distribution,
Hausdorff and Box Dimension of Random Cantor Sets, von Koch Curves, Sierpinski
Triangles.
[Fa, p.224-31,
14-15, Exercises 15.1-5]
5. Random Spread: Percolation (H. Hagedorn)
The model Ap=ÇAk,
Ak+1ÌAk for an occupation probability
0<p<1 with Ao=[0,1]2 divided into n2
subsquares. Condition on p so that (i)
Ap=Æ with probability q
(ii) Ap is totally disconnected a. s. (iii) Ap is connected. Hausdorff
and Box Dimension of Ap.
[C, p113-22] [Fa, p.231-6, Exercises 15.4, 15.5]
II. Attractors and Graphs of Continuous Curves
6. Examples (M. Goy)
The von Koch curve
and Heighway’s Dragon as graphs of continuous functions f:[0,1]®Â2.
The box dimension of the graph of Weierstraß’s function and its generalization.
Kieswetter’s and Hironaka’s curve. Self affine dust.
[E, p18-22, 62-4, 200-5] [F, p.146-55, Exercises 11.1, 11.4, 11.6]
7. Data Interpolation with affine contractions (M.
Wünsch)
Construction of n
affine maps for n data points in Â2 (contractions for a certain metric on Â2) and of a function whose graph is the attractor interpolating
the data.
[B, p207-33,
Theorems 6.2.1, 6.2.2, 6.3.1, Exercises 2.1, 2.2, 2.4, 2.16] [Fa, Fig.11.5]
8. Brownian Paths and Graphs of Brownian Functions (J.
Hoeglund)
Potential Theory.
Hausdorff and Box Dimension of Brownian paths [0,1]®Ân and Graphs of Brownian functions [0,1]®Â. Temperature
on a fractal.
[Fa, p64-6,
237-43, Exercises 16.1-3] [Fe, p.163-70, 184-8] [Mc, 212-224]
9. Hurst’s R/S Analysis of Time Series (A. Müller)
Rescaled Range Analysis applied to Wave Height Time
Series.
[Fe, p.149-63,
193-9]
10. Generalization: Fractional Brownian Motion (N.
Kessinger)
Hausdorff and Box
Dimension of Index α-Brownian functions on [0, 1] .Randomized Weierstrass
functions and mountain modeling. Autocorrelation function of f:ÂO. Box Dimension of graph f.
[Fa, p.245-53, 155-8, exercises 16.5-6] [Fe, p170-8]
11. Multifractals (M. Kraus)
Triadic Cantor Set
C, the Devil’s Staircase. Two-scale Cantor Sets and the asymmetric Tent Map.
Random Mass Distribution on C. Calculation of f(α)-spectrum of μ for the attractor of n random contracting
similarities using concave functions. Generalized Dimensions.
[Mc, p.136-42,
194-208] [Fe, p.66-70] [Fa, p.254-64, Example 17.1] [L, p.58-63, 74-8: Two
proofs of Theorem 5.6]
Literature
|
[B] |
Barnsley, M.: Fractals
Everywhere, Academic Press, 1988 |
|
[E] |
Edgar, G.: Measure,
Topology and Fractal Geometry, Springer |
|
[B/G/Z] |
Fractal
Geometry and Stochastics,
ed. C. Bandt,
S.Graf, M. Zähle, Birkhäuser, 1995 |
|
[Fa] |
Falconer, K.: Fractal
Geometry (Math. Foundations and Applications), Wiley 1990 |
|
[Fe] |
Feder, J.: Fractals,
Plenum 1988 |
|
[H] |
Hutchinson, J.: Fractals
and Self Similarity, Indiana Univ. Math. J. 30, 5, 713-47 |
|
[Mc] |
McCauley J.: Chaos,
Dynamics and Fractals, Cambridge Univ. Press 1993 |
|
[C] |
Chayes, in
[B/G/Z] |
|
[L] |
Lau,
in [B/G/Z] |
|
[P] |
Peruggia, M.: Discrete
Iterated Function Systems, AK Peters, 1993 |
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