PART I The object of Part I is to practice your knowledge IFS attr...

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PART I The object of Part I is to practice your knowledge IFS attractors , the Collage Theorem and addresses of points on fractals. Exercise 1: Consider the two IFSs F = fR2 : f1(x; y) = (x=2; y=2); f2(x; y) = (x=2 + 1=2; y=2)g: G = fR2 : f1(x; y) = (􀀀x=2 + 1=2; y=2); f2(x; y) = (x=2 + 1=2; y=2)g: In each case, identify the attractor, the critical set, and the dynamical boundary. In each case, nd a point whose address is each of 1, 2, 2212, i.e. calculate F(1); F(2); F(2212); G(1); G(2) and G(2212). In each case, nd all of the addresses of (0:5; 0), if this point lies on the attractor. Exercise 2: Pick out a couple of interesting Collages in Figure 1. Your pair should dierent from the ones of your colleagues. Identify an IFS for each of the objects. Then, using that IFS, identify the points whose addresses are 1, 2, 2212, the critical set, the dynamical boundary, and nd the addresses of an interesting point on the attractor. Specically calculate one of the IFSs. Figure 1: Use this in connection with Exercise 2. 1 PART II The object of Part II is to develop initial familiarity with Möbius transformations and dynamical systems. Exercise II.1 (i) Find the xed points of the Möbius transformation f : ^C ! ^C dened by: f(z) = 2 􀀀 (3 + i)z 1 􀀀 z Identify which xed point is attractive and which is repulsive. (ii) Describe the behavior of a typical orbit that starts close to the attractive xed point. (iii) Change coordinates so that the attractive xed point is at origin. Exercise II.2 The Möbius transformation f : ^C ! ^C dened by f(z) = 2 + (3 + i)z has a xed point at zF = 1. By means of the change of coordinates T(z) = 1=z; describe the behaviour of f in the vicinity of zF . 2

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