ZENTRUM MATHEMATIK

DER TECHNISCHEN UNIVERSITÄT MÜNCHEN

Prof. Dr. Sandra Hayes

# SEMINAR SS2003

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=Π-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