This last example suggests the following notion of equivalence of dynamical systems, which was originally put forth by Smale [12] and is fundamental to dynamical systems theory.
Definition. Let 
and 
be two maps. The functions f and g are said to be topologically
conjugate if there exists a homeomorphism 
such that 
.
The homeomorphism is called a topological conjugacy, and is more commonly defined by simply stating that the following diagram commutes:
Using the theorem of the previous section,
we know that if 
then 
(the quadratic map) is
topologically conjugate to 
(the shift map).
Topologically
conjugate systems are the same system insofar as there is a one-to-one
correspondence between the orbits of each system. Sometimes this
is too restrictive and we only require that the mapping between orbits
be many-to-one. In this latter case we say the two dynamical
systems are semiconjugate .
In nonlinear
dynamics, it is often advantageous to establish a conjugacy or a semiconjugacy
between
the dynamical system in question and the dynamics on some symbol space. The
properties of the dynamical system are usually easy to see in the symbol space
and, by the conjugacy or semiconjugacy,
these properties must also exist in the original dynamical
system. For instance, the following properties are easy to show in 
and
must also hold in 
, namely:
) is 
.
is dense in
. has a dense orbit in [10].
Although there is no universally accepted definition of chaos, most definitions incorporate some notion of sensitive dependence on initial conditions. Our notions of topological conjugacy and symbolic dynamics give us a promising way to analyze chaotic behavior in a specific dynamical system.
In the context of one-dimensional maps, we say
that a map 
possesses sensitive
dependence on initial conditions if there
exists a 
such that,
for any 
and any neighborhood N of x, there exist
and 
such that
. This
says that small errors due either to measurement or round-off
errors become magnified upon iteration--they cannot be ignored.
Let 
denote the unit circle.
Here we will think of the members of as being
normalized to the range [0, 1).
A simple example of a map that is chaotic
in the above sense is given by 
defined by 
. As we saw in section 2.10,
when 
is written in base two, 
is simply a shift map on the unit
circle. In ergodic theory the above shift map is known as a Bernoulli process.
If we think of each symbol 0 as a Tail (T), and each symbol 1 as a Head (H), then
the above shift map is topologically conjugate to a coin toss, our intuitive model
of a random process. Each shift represents a toss of the coin.
We now show that the shift map is essentially the same as the
quadratic map for 
; that is, the quadratic map
(a fully deterministic
process) can be as random as a coin toss.
If 
, then the limit set is the
whole unit interval I = [0,1] since the maximum 
;
that is, the map
is strictly onto (the map is measure-preserving and is roughly
analogous to a Hamiltonian system that conserves energy).
To continue with the analysis,
define 
by 
. Also define
. Then
so 
conjugates g and q. Note, however, that is two-to-one at most
points so that we only have a semiconjugacy. To go further,
if we define 
by 
, then 
.
Then 
is a topological semiconjugacy
between g and 
;
we have established the semiconjugacy between the
chaotic linear circle map and the quadratic map when .
The reader is invited to work through a few examples
to see how the orbits of the quadratic map, the linear circle map, and a coin
toss can all be mapped onto one another.