The prototypical example of a unimodal map is the
*quadratic map*:

Figure 1 shows , and
includes a few iterations of the point .
In general, a *unimodal map* maps an interval *I*
into intself, has a single critical point *C* in *I*, and
is monotone increasing on the left of *C* and decreasing on the right.
For the quadratic map, *I* = [0,1] and *C*=1/2.
The key topological property of a unimodal map
is that it stretches and folds the interval *I* into itself.

**Figure 1:** The quadratic map, with .
The vertical and horizontal lines show five
iterates of the point .

One of the motivating reasons for studying the quadratic map is to understand the diagram shown in Figure 2. A closer view of part of this plot is shown in Figure 3. In each of these plots, a large sequence of increasing values of are generated. For each value of , we set , and iterate the quadratic map 1000 times. We then plot the next 32 values in the iteration versus . These plots provide a rough picture of the asymptotic behavior of the quadratic map. For example, Figure 2 shows that for , there is a stable equilibrium. (Indeed, it is easy to prove that for , is an equilibrium, and it is stable for .) At , the equilibrium becomes unstable, and a stable period 2 orbit is born. As is increased, the period 2 orbit eventually becomes unstable, and a period 4 orbit is born. This sequence of period doubling bifurcations appears to continue as is increased, but there is not sufficient resolution in the figures to see what happens. We can see a stable period 6 orbit at , and a stable period 3 orbit at .

In principle, we could determine all the periodic orbits (and
determine their stability) by solving the appropriate formula.
For example, period 3 orbits satisfy
*x* = *f*(*f*(*f*(*x*))).
However, these equations quickly become intractable as the
period increases.
It has also been observed that
very similar bifurcation diagrams can be
found with other unimodal maps, suggesting that the
qualitative structure of the bifurcation diagram
is not unique to the quadratic map.
It turns out that the phenonema illustrated in the
bifurcation diagrams can be elucidated with symbolic
dynamics and kneading theory.

**Figure 2:** Bifurcation diagram for the quadratic map,
.

**Figure 3:** A closer look at the bifurcation diagram for the quadratic map,
.

To set up the *symbolic dynamics* of this system, we must first
define a *partition*.
In general, a partition is the separation of
phase space into disjoint regions.
For the quadratic map with , *I*=[0,1].
We define , *C*=1/2, and .
Given , we define by (1).
We now define

The *symbol sequence* or *itinerary* of the point
is the sequence .
For example, if and (see Figure 1),
then .
The quadratic map acts on the sequence
as a *shift*:

where operates on sequences by discarding the left-most symbol,
and shifting the rest of the sequence to the left.

We present a brief discussion of
*kneading theory* [6, 7].
(A nice introduction to the theory is given by Devaney [8].)
First, we define an ordering on the itineraries which preserves
the ordering on the interval.
More precisely,
if , then *x* < *y*, and if *x* < *y*,
then .
Note that for a unimodal map, *f* is increasing on and
decreasing on .
After one iteration, the order of points in is maintained,
while the order of points in is reversed.
That is, if and , and *x* < *y*, then
*f*(*x*) > *f*(*y*).

Define the order of the symbols to be 0 < *C* < 1.
Let's compare the symbol sequences
and .
Clearly, if , then *x* < *y*, and
if then *x* > *y*.
If , we compare and .
However, we have to take into account that if , then
the order of the points has been reversed by the first iteration
of the mapping.
Therefore, if and , then it must be
that *x* > *y* (and *x* < *y* if ).
Now suppose that the first two symbols agree,
but, say, .
If the first two symbols are 00, then there have been no
reversals, so *x* < *y*.
If the first two symbols are 01 or 10, then
there has been one reversal, so *x* > *y*.
Finally, if the first two symbols are 11, then there
have been *two* reversals, so *x* < *y*.

In general, the order is determined by the order of the first
pair of symbols that differ in the two sequences.
In order to know the correct direction of the inequalities,
we must keep track of how many times the order has been reversed.
But this is just the number of 1s that appear in the segment
of the sequences that agree, because a 1 means that the points
are to the right of *C*, and their order will be reversed
by the next iteration. If there are an *even* number of
1s, then we use the natural order of the first symbols that
differ. If there are an *odd* number of ones, then the
order of *x* and *y* is the reverse of the order of the first
pair of different symbols.

These simple observations on the ordering of symbol sequences
lead to some remarkable consequences.
An illustration of the power of the kneading theory are the following
theorems, which
hold for a large class of unimodal maps.
We define the *kneading sequence* *K*(*f*) of the unimodal map *f*
to be the
itinerary of *x*=*f*(*C*), i.e. *K*(*f*) = *S*(*f*(*C*)).

**Theorem.**
Let be a symbol sequence.
If *K*(*f*) is not periodic, and
for all ,
then there is a point such that .

This means we can construct an arbitrary symbol sequence , and if this sequence precedes the kneading sequence of the map, then there is a point in the interval with symbol sequence . A generalization is the following.

**Theorem.** Suppose we are given two symbol sequences
and , and there is a such that
.
If there is an such that
for all ,
then there is such that .

These results prove especially useful when we consider
periodic orbits. (See, for example, Chapter 1.19 of
Devaney [8] for a more detailed introduction.)
As we vary , the kneading sequence *K*(*f*) changes,
and the above results shows that the
kneading sequence ``determines'' which periodic orbits
exist.
It turns out that for the quadratic map, the kneading sequence
increases (in the sense of the order defined above) as
increases.
By combining the kneading theory with an additional property
of the quadratic map (namely that is has a negative
Schwarzian derivative), we obtain a detailed
description of how periodic orbits arise as
(and hence the kneading sequence) increases.
This theory explains the qualitative features of the
bifurcation diagram in Figure 2.