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##
Continued Fractions and Chaos ~~~~~~ Robert M. Corless

This method is likely worse than nearly any
other in existence, since it does ** not** converge to the correct
value in any particular fixed-precision system, since all orbits are
ultimately periodic, and the Lyapunov exponent of a
periodic orbit is the logarithm of an algebraic number, which can't
be unless is a special algebraic
number, namely the **P**-th root (for some integer **P**) of a
quadratic irrational.
Yet this qualifies as a genuine method, since in principle you
could implement higher and higher precision floating-point systems
and achieve the desired accuracy by longer and longer runs with
this high-precision arithmetic. Of course this is impractical,
perhaps even ridiculous. There is also the problem of choosing
``good'' initial points---if we are lucky, the first initial point
we choose for whatever floating-point system we have will do the
trick---but there is no guarantee, and indeed the Lyapunov exponent
may converge to something totally different (or worse, something only
slightly different).

This method is clearly related to the Monte Carlo methods [23], with
the roundoff error associated with the floating-point arithmetic
playing the part of the random number generator required.

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