The Gauss map preserves the Gauss measure
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.