As discussed in section 2.5, the exact location of a period n orbit is
determined by the roots of the fixed point equation, 
. This naive
method of
locating the periodic points
is impractical in general because it requires finding
the roots of an arbitrarily
high-order polynomial. We now
show that the problem is easy to solve using symbolic dynamics if we ask not
for the exact location, but only for the location relative to all the other
periodic orbits.
If 
, then there exists an interval centered about x=1/2 for
which f(x) ;SPMgt; 1. Call this interval 
(see Figure 2.23). Clearly, no
periodic orbit exists in since all points in leave the unit
interval I at the first iteration, and thereafter escape to 
. As we
argued in section 2.11.1, the periodic points must be part of the invariant
set, those points that are never mapped into .
As shown in Figure 2.25,
Figure 2.25: Symbolic coordinates and the alternating binary tree.
the points in the invariant set can be constructed by
considering the preimages of the unit interval found from the inverse
map 
. The first iteration of produces two
disjoint intervals,
which are labeled 
(the left interval) and 
(the right interval). As
indicated by the arrows in Figure 2.25, preserves
orientation, while reverses orientation. The
orientation of the interval is simply determined by the slope
(derivative) of 
,
We view 
as a first-level approximation to the invariant set
. In particular, gives us a very rough idea as to the
location of both period one orbits, one of which is located somewhere in ,
while the other is located somewhere in .
To further refine the location of these periodic orbits, consider the
application of to both and ,
The second iteration gives rise to four disjoint intervals. Two of these contain the distinct period one orbits, and the remaining two intervals contain the period two orbit,
In general we can define 
disjoint intervals at the nth level of
refinement by
With each new refinement, we hone in closer and closer to the periodic orbits.
The one-to-one correspondence between 
and 
is easy to see
geometrically by observing that, as 
,
forms an infinite intersection of nested nonempty closed intervals that
converges to a unique point in the unit interval.
The invariant limit set is the
collection of all such limit points, and the periodic points are all those
limit points indexed by periodic symbolic strings.
