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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