The following classical theorem, interpreted in a modern dynamical sense, identifies the fixed and periodic points of the Gauss map.
Theorem. (Galois) The number 
has a purely periodic continued
fraction, including the first integer 
, if and only if 
is
a
reduced quadratic irrational.
Corollary. The periodic points of the Gauss map are the reciprocals of the reduced quadratic irrationals.
For a proof of the theorem, see [18], or [10].
To prove the corollary, we note that 
is periodic under the Gauss map if and only if its continued fraction is
periodic, starting at 
, by the shift property mentioned in the
previous section.
An example of particular interest is 
, the golden ratio, which
satisfies 
. The other root of this quadratic
is 
which is in the desired interval. The continued fraction
of 
is 
, so 
has the
continued fraction 
, which shows that 
is a point of period 1 of the Gauss map. We will return to this example
later.
There are general results in the theory of chaotic dynamical systems, with which we could hope to establish the character of the set of periodic points of the Gauss map [21,24,15]. However, these results deal with the characterisation of the behaviour of continuous maps of the interval, extended by Block et. al. to maps of the circle [3], and the Gauss map has a singularity at the origin. Thus the hypotheses of these theorems are not weak enough to apply. However, the results of these theorems hold, as will be seen by direct methods.
We note here that there are infinitely many points of each period. For
example, 
has
period k, for any choice of integers 
, 
, 
, 
.
Having points of arbitrary period is one characteristic of a chaotic
map [15]. However, we would like to see if the map
is sensitive to initial conditions (SIC) in that nearby initial points
have orbits that separate at an exponential rate. This again can be
established in an elementary fashion by using a classical result.
Theorem. (Lagrange) 
has an ultimately periodic continued
fraction, which means that 
with transients 
,
, 
, 
at the start of a periodic continued
fraction, if and only if 
is a quadratic irrational (
is a root of a quadratic with integer coefficients).
Corollary. The Gauss map is SIC.
For a proof of Lagrange's theorem, see [10]. To prove
the corollary, we note that every rational initial point is ``attracted''
to the artificial fixed point at 0, while every quadratic irrational
is ultimately ``attracted'' to a periodic orbit. Both sets are dense
in the interval 
. The rate of separation may be checked by
considering all points in a small interval I, of width 
. By the
pigeonhole principle, this interval must contain a rational number of
the form 
, where n is the smallest integer larger than 
.
The number of iterations of the Gauss map required to reach zero from
this initial point is, by the speed of the Euclidean algorithm, 
and thus 
. To construct a specific initial point in this
interval that does something different under G, first expand 
into its finite continued fraction: 
.
Then for large enough N, the following infinite continued fraction
is the continued fraction expansion of a point in I: 
. Clearly, the orbit of G
starting at this initial point winds up on the fixed point at
.
Q.E.D.