Random IFS

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The diameter of a set is the maximum distance between any pair of points in the set.

For example, the diameter of a circle is just the common notion of diameter; the diameter of a square is the diagonal length of the square.

Some diameters

Because all the IFS rules are contractions, the diameter of a region of address length N goes to 0 as N goes to infinity.

We illustrate this with the four transformations

T3(x, y) = (x/2, y/2) + (0, 1/2)	T4(x, y) = (x/2, y/2) + (1/2, 1/2)
T1(x, y) = (x/2, y/2)	T2(x, y) = (x/2, y/2) + (1/2, 0)

As an IFS, these generate the unit square, S. We see

diam(S) = sqrt(2)

diam(T_i(S)) = sqrt(2)/2

diam(T_jT_i(S)) = sqrt(2)/4

and in general

diam(T_iN...T_i1(S)) = sqrt(2)/(2^N)

Consequently, diam(T_iN...T_i1(S)) -> 0 as N -> infinity.

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