The Gauss map is well-known in ergodic theory (see [1] or [16]). 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.