Driven IFS and Financial Cartoons

Sample

The left picture shows the plot of the (log of the) daily closing prices of the Frankfurt DAX from Jan 6, 1986, through Dec 31, 1997, giving 3128 data points.
The equal size bins are indicated by the horizontal lines.

The backwards Z is a common feature for these driven IFS.
The first several hundred data points hop across the boundary between bin 1 and bin 2 giving IFS points along the line between (0, 0) and (1, 0).
Then several hundred data points are in bin 1, producing a sequence of IFS points along the line from (1, 0) to (0, 0), converging to (0, 0).
Next some data points hop between bin 1 and bin 2, giving more IFS points along the line between (0, 0) and (1, 0).
Then data points in bin 2 produce a sequence of IFS points converging to (1, 0).
Next the data points hop between bin 2 and bin 3, producing a sequence of IFS points along the line between (1, 0) and (0, 1).
These are followed by data points in bin 3, giving a sequence of IFS points converging to (0, 1).
The next data points make a fairly quick transition to bin 4, producing a few IFS points on the line between (0, 1) and (1, 1).
Finally, the data points in bin 4 produce a sequence of IFS points converging to (1, 1).
Only points jumping more than a quarter of the range of the data values would give rise to IFS points not on this backwards Z.
Certainly, this is not an interesting representation of the data.

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