The following theorem justifies the remarks of the previous section.
The basic idea of its proof is that given some initial point 
the floating-point Gauss map also generates an initial point y
whose continued fraction representation is exactly equal to 
, where the 
are all (machine representable) integers.
This initial point y has a G-orbit that is everywhere within a
small multiple of u, the machine epsilon, of the 
-orbit
of 
. The technique of the proof is of interest for more
than just the Gauss map,
because similar techniques can be used to prove that numerical simulations
of orbits of some continuous systems are machine close to exact orbits
of some nearby initial point (for a descriptive review of work by
Yorke, Grebogi, and Hammel establishing similar results for continuous
maps see [5]).
Theorem. If 
, 
, 
, 
, 
is the sequence
of iterates of 
, and 
, 
, 
, 
is the sequence of (machine representable) integers that arise in the
process, then 
has an orbit under G whose
elements are close to 
, 
, 
in a sense to be made
precise, and, in particular, y is close to 
.
We will show first that we may approximate an element of the orbit
of y by a certain rational number. We then show, using a common
model of floating-point arithmetic, that the corresponding 
is ``machine close'' to this same rational number.
This last will be seen to depend on the fact that if you run the
Gauss map backwards, errors are damped instead of amplified.
Proof. Consider 
.
The rational numbers 
satisfy
and 
where 
is the nth Fibonnacci number, from
elementary properties of simple continued fractions (see [18]
or [10] for details).
This means that given an 
we can find an n so that
.
To prove the second part, we use the common model of floating-point
division that states that if the floating-point numbers a, b,
and c satisfy 
, where the division takes place
over the floating point numbers, then there is a number 
with 
u so that 
exactly.
Note that we do not model the addition, since this will be seen
to be unnecessary.
If the orbit 
has been produced by a
floating-point system satisfying this model, then for each n
there is a number 
with 
u
such that
where we may consider the addition as exact, since 
is a machine representable number, defined by this process, and
is a machine representable floating-point number.
If we put 
then we have
Now put 
for
, and put 
for 
. Note that 
is
the error we wish to estimate, since by the first part we can estimate
the error 
.
So now
from whence, on cross-multiplying and expanding, we get the recurrence relation
from which we may derive an upper bound on 
, and we
note at this point that 
is one of the rationals which approximates
. Note that the first term in this recurrence relation is
essentially the roundoff error introduced at this particular step, while
the second term is the error from one level below in the continued
fraction, multiplied by a ``shrinkage factor'' 
.
As in the proof that 
has the minimum Lyapunov exponent, we are
unable to say anything useful about 
directly, but we are
able to bound 
, which is easily shown to be
less than 
. With some simple estimates on the above recurrence
this gives
and since as 
, 
, we have at last
Thus there is a nearby initial point 
whose orbit under G
follows as near as can be expected the computed orbit 
of the floating-point Gauss map.
Our earlier example of 
gave a periodic orbit on the HP28S,
which has u
. The nearby initial point with this
orbit under 
is
As a further curiousity, we note that the machine representation of
on the HP28S is an actual fixed point of 
, allowing
us to calculate the exact continued fraction of 
on a finite
machine.