FractionalGeometry-Chap2

# The continuity in theorem 291 is revealed 29 some

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Unformatted text preview: × ([−1, 1] × [−1, 1]) In fact, the situation is a bit more complicated. An aﬃne transformation T : R2 → R2 is a contraction if and only if the eigenvalues λ1 and λ2 satisfy |λ1 | &lt; 1 and |λ2 | &lt; 1. Note we do not want |λ1 λ2 | &lt; 1, which would give a contraction of area, but that both directions are contractions. The matrix r · cos(θ) r · sin(θ) −s · sin(ϕ) s · cos(ϕ) has eigenvalues 1 s · cos ϕ + r · cos θ ± (s · cos ϕ − r · cos θ)2 − 4 · r · s · sin ϕ sin θ 2 Say λ1 is the eigenvalue with the positive square root, and λ2 with the negative square root. Ideally, we would plot those r, s, θ, and ϕ values for which max{|λ1 |, |λ2 |} &lt; 1. But this would be a 4-dimensional plot, diﬃcult to visualize. Some sections of P1 = {(r, s, θ, ϕ) : max{|λ1 |, |λ2 |} &lt; 1} 2.9. SOME TOPOLOGY OF IFS ATTRACTORS 87 are shown in Fig. 2.54. All pictures are the torus unwrapped. That is, the horizontal axis is 0 ≤ θ ≤ 2π , the vertical axis is 0 ≤ ϕ ≤ 2π . Each picture is taken from the...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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