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.
Consider 
.
The rational numbers
satisfy
and 
where 
is the nth Fibonnacci number, from
elementary properties of simple continued fractions (see [20]
or [11] 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.
