whenever this limit exists. Note that 
exists even at the
jump discontinuities 
, but there is a real singularity at the
origin. Nearby orbits will separate from
the orbit of 
at an average rate of 
,
after k iterations of G.
Khintchin [15]
derived a remarkable theorem with which we could
show the common Lyapunov exponent of
almost all (in the sense of Lebesgue
measure) orbits is 
. Easier ways
have since been found to establish this result, using ergodic
theory. We summarize the ergodic results in the next section.
Note that for any rational initial point,
the above limit does not exist (because 
is
eventually 0 and the derivative blows up there). Further, for any
periodic orbit the calculation can be made explicitly, to give
Lyapunov exponents that differ from the almost-everywhere
value.
For example, the fixed points 
have Lyapunov exponents
so there are orbits with arbitrarily large Lyapunov exponents,
i.e., orbits that are arbitrarily sensitive to perturbations in the
initial point. The asymptotic formula above was derived from
the explicit form for 
obtained by solving 
for its positive root, and then using Maple's ![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/sshow-81.gif)
asympt command.
It is not too hard to show, because the limit can be written down
explicitly, that for the orbit of e, the
limit defining the Lyapunov exponent is infinite.
On the other end of the scale, the
special case N=1 of 
gives 
, the golden ratio. Thus 
, which is smaller than the almost-everywhere
Lyapunov exponent. In fact, we have the following:
