Unformatted text preview: scale of the attractor by the same
factor w. If we are interested only in the shape, not the size, of the attractor
of an IFS {T1 , . . . Tn }, then we can restrict each ei and fi to lie in the square
{(ei , fi ) : −1 ≤ ei ≤ 1, −1 ≤ fi ≤ 1}. Simply replace each ei and fi by ei /m
and fi /m, where m = max{e1 , . . . , en , f1 , . . . , fn }.
Certainly, θ and ϕ can take on any value in [0, 2π ], but we need to think a
bit diﬀerently to understand the eﬀect of a small change of parameters. These
variables represent angles, so θ = 1.9π is close to θ = 0.1π . Consequently, we
shall think of both θ and ϕ as representing points on a circle, S 1 . Together, θ
and ϕ range over the product of two circles, that is, a torus T 2 .
For the r and s parameters, a ﬁrst thought is that a transformation is a
contraction if max{r, s} < 1, and so the parameter space of values giving a
contraction map is
((−1, 1) × (−1, 1)) × T 2 × R2
or if we are concerned only with the shape of the attractor P0 = ((−1, 1) × (−1, 1)) × T 2...
View
Full
Document
This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

Click to edit the document details