The observation that we can get an approximate value for the
Lyapunov exponent of the exact Gauss map by calculating the
average exponent from the first N elements of a numerically generated
orbit gives us a new and interesting, though completely impractical,
method for calculating .
We simply choose some initial point more or less at random, say ,
and produce the first N iterates under the floating-point Gauss map, and
accumulate the average Lyapunov exponent.
At the end, this is supposed to be close to the exact almost-everywhere
Lyapunov exponent of the exact Gauss map, .
Well, if we know and can take square roots, this gives us the
value of . Using the HP28S and 100,000 iterates of the floating-point
Gauss map with the above initial point, we get .
Note that this method relies on roundoff error, since without
it this orbit terminates!
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