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)
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.
The object of Part II is to develop initial familiarity with Möbius transformations and
Exercise II.1 (i) Find the xed points of the Möbius transformation f : ^C ! ^C dened
2 (3 + i)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 .
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