Driven IFS and Financial Cartoons

Background

One of the most obvious departures from uniform fill occurs when a some combinations of transformations never occur.

For example, the left picture shows the result of driving the IFS with the DNA sequence for amylase, using this conversion:

apply T1 whenever a C is encountered, apply T2 whenever an A is encountered, apply T3 whenever a T is encountered, and apply T4 whenever a G is encountered.

The sequence consists of 3967 entries, certainly enough to believe that the departures from uniform fill are genuine.

Much of this structure can be understood from a single exclusion.

The middle picture contains the same number of points, with the Ti applied randomly with one exception:

T4 is never applied immediately after T1.

The largest empty square is T4(T1(S)), where S denotes the unit square.

The right picture is a map of the length 2 address subsquares. The meaning of this name should be apparent.

Looking at the middle figure, we see some consequences of the emptiness of square 41.

For example, the square T1(T4(T1(S))) is empty:

because T4(T1(S)) contains no points, T1(T4(T1(S))) cannot contain points.

We say 41 is a forbidden word, and in general it is true that any square whose address contains a forbidden word will be empty.

In the Sample, we shall discover another signature of departure from uniform fill.

Return to Background.