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

The Gauss map is well-known in ergodic theory (see [1]
or [18]). The results are summarized here, for contrast with the
results of the sections previous and following. This section is meant
more as incentive for the reader to investigate ergodic theory than as
exposition.

The Gauss map
preserves the Gauss measure

where is the Lebesgue measure. Thus the Gauss map is
ergodic, and almost all (in the sense of either the Lebesgue
or Gauss measure) initial points have orbits which have the
interval as -limit set. Thus the * only*
attractor whose basin of attraction
has nonzero measure is the
interval . By the ergodicity of the map, we may explicitly
calculate the Lyapunov exponent as follows:
which holds for almost all initial points . This is of interest,
since there are few nontrivial maps for which the Lyapunov exponent
can be calculated explicitly.
This section, more than any other in the paper, provoked puzzlement
on the part of readers of the original. The original purpose was to
show the reader how powerful ergodic methods were: in one line
we establish the `almost-everywhere' value of the Lyapunov exponent
for the Gauss map, using the ergodic result that the `time-average'
of along the
orbit is equal to the (properly-weighted) `space-average' given
by the integral of with respect to the Gauss measure.

I have since also learned that this Lyapunov exponent can
be explicitly connected with Khintchin's constant (mentioned
previously as the geometric mean of the partial
quotients, which
turns out to be the same for almost all **x** in ).
Khintchin's constant is **K**, where .
(By the same arguments). The only difference is the
presence of the `fractional part'.
This integral gives a simple convergent series
for which can be transformed into a quickly convergent one.
(The integral becomes a series because frac(1/x) is piecewise
constant). See [7] for details.

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** Next:** The Floating-Point Gauss
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** Previous:** Remark.