Data Analysis by Driven IFS

Exercise Answers

2. (a) Here is the Driven IFS.

The data points alternate between bins 1 and 2, so the driven IFS points converge to points (x1, y1) and (x2, y2) with addresses (12)infinity and (21)infinity.

Then we see T2(x1, y1) has address 2(12)infinity = (21)infinity, so T2(x1, y1) = (x2, y2). Similarly, T1(x2, y2) = (x1, y1). Combining these, we see

T₁T₂(x₁, y₁) = (x₁, y₁) and T₂T₁(x₂, y₂) = (x₁, y₁)

The first equation becomes (x1/4 + 1/4, y1/4) = (x1, y1),

so x1 = 1/3 and y1 = 0.

The second equation gives x2 = 2/3 and y2 = 0.

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