Question 1. (i) Recall that an ane map in 2D may be represented by...

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Question 1. (i) Recall that an ane map in 2D may be represented by a matrix of the form 0 @ a b c c d g 0 0 1 1 A where a; b; c; d; e; g are real numbers. Let f : R2 ! R2 be the ane map represented by 0 @ 0:5 0:5 0 0:5 0 0:5 1 1 1 1 A 0 @ 1 0 1 0 1 1 1 1 1 1 A 􀀀1 : To where does f map the three points A = (1; 0);B = (0; 1);C = (1; 1)? (Hint: there is no need to calculate the inverse map?) Explain your answer. (ii) Calculate the Hausdor distance between B := f(x; y) 2 R2 : 0  x  1; 0  y  1g and C := f(x; y) : x = y; 0  x  1g: Explain your reasoning. Question 2. (i) Explain what is meant by a contractive IFS in R2, and what is meant by the attractor of a contractive IFS. (ii) Explain how an invertible ane transformation in 2D can be uniquely specied. (iii) The vertices of the big triangle H (red and green) in gure are labeled by symbols A;B;C and the midpoint of the top edge is labeled D. Using these symbols, dene ane maps fi : R2 ! R2; i = 1; 2 such that the attractor of the IFS fR2; f1; f2g is H. Explain why you believe that your maps are indeed contractive. (iv) Find all addresses of D (with respect to the IFS) (v) Identify the critical set and the dynamical boundary of H (with respect to the IFS) 1 Question 3. (i) Describe two methods for calculating a picture of the attractor of a con- tractive IFS of an ane maps in 2D. (ii) Next gure shows a screenshot from SFVideoShop. How is the image on the top right computed? In particular, explain why the top "sierpinski" on the right is represented in two colours. Question 4. (i) Dene a metric space (X; d). (ii) Dene a Cauchy sequences fxng1 n=1 in the metric space (X; d). (iii) What does it mean, to say that the metric space (X; d) is complete? (iv) Which of the following spaces are complete ? (R2; dEuclidean); ( P = f1; 2; : : : ;Ng1; dP) where dP is the code space metric and dH is the Hausdor metric. (v) Dene a contraction mapping f : X ! X. (vi) Dene a contractive iterated function system. (vii) State the Banach contraction mapping theorem, and explain why it plays a funda- mental role in fractal geometry. (viii) What is a group G of transformations acting on a space X ? Question 5. Prove Banach's Contraction Mapping Theorem. 2

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