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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 theThis 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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