To begin this discussion, we will examine two similar-seeming iterated function systems on \(\mathbb{R}^2\) whose functions are compositions of the following transformations:
Given \(\rho > 0\) and \(N = 6\),
Let \(T_1(x,y) = \frac{1}{2}(x,y) + \frac{1}{2}(\rho,0)\).
Let \(T_2(x,y) = \frac{1}{4}(x,y) + \frac{3}{4}(\rho,0)\).
Let \(R_k(x,y) = (x\cos{\frac{2\pi k}{N}} - y \sin{\frac{2\pi k}{N}}, y\cos{\frac{2\pi k}{N}} + x \sin{\frac{2\pi k}{N}})\).
Let \(F(x,y) = (x,-y)\).
\(T_1\) and \(T_2\) are contractions followed by translations, the \(R_k\) are rotations, and \(F\) is a reflection about the x-axis.
The iterated function systems are as follows:
Let \(\mathcal{F}_1 = \{\mathbb{R}^2, T_1, R_1T_2, R_2T_1, R_3T_2, R_4T_1,R_5T_2\}\).
Let \(\mathcal{F}_2 = \{\mathbb{R}^2, T_1R_1, R_1T_2R_1, R_2T_1R_1, R_3T_2R_1, R_4T_1R_1,R_5T_2R_1\}\).
Consider the regular hexagon \(S \subseteq \mathbb{R}^2\) centered at the origin, and with vertex at \((\rho,0)\).
A regular hexagon set \(S\), (shown in white).
Let \(H_1(E) = \bigcup_{f \in \mathcal{F}_1}{f(E)}\) be the Hutchinson operator for the IFS \(\mathcal{F}_1\).
Let \(H_2(E) = \bigcup_{f \in \mathcal{F}_2}{f(E)}\) be the Hutchinson operator for the IFS \(\mathcal{F}_2\).
Observe that applying either Hutchinson operator to \(S\) yields the same result. That is, \(H_1(S) = H_2(S)\).
The image of \(S\) under \(H_1\), (shown in red), is the same as the image under \(H_2\), (shown in blue).
Naively, we might assume this implies that \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have the same attractor, but applying the hutchinson operators one more time reveals that this is probably not the case, since \(H_1^2(S) \neq H_2^2(S)\).
The image of \(S\) under \(H_1^2\), (shown in red), is not the same as the image under \(H_2^2\), (shown in blue).
Sure enough, \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have distinct attractors, as shown below.
The attractor of \(\mathcal{F}_1\), (shown in red), and the attractor of \(\mathcal{F}_2\), (shown in blue).
In this topic, we will be examining families of IFS's, (like \(\mathcal{F}_1\) and \(\mathcal{F}_2\)), whose Hutchinson operators all give the same image for some set \(S\). We will call such families, Hutchinson equivalent. More precisely, we will say that two IFS with Hutchinson operators \(H_1\) and \(H_2\) are nth-order Hutchinson equivalent over some set S if and only if \(H_1^n(S) = H_2^n(S)\).
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Next: Using Symmetry To Generate Hutchinson Equivalent Families