In the previous section we attempted to capture all the
recurrent behavior of the mapping 
.
Let
Clearly,
There is no universally accepted notion of a set that
contains all the recurrence, but this set ideally ought to be
closed and invariant. Per(f) is too small--periodicity is
too limited a kind of recurrence. 
is not necessarily
closed. 
, while closed and invariant, has the drawback
that:
The mapping 
makes sense, of course,
because of 
's invariance.
In certain contexts the chain recurrent set, which is bigger than
, has all the desirable properties. According to the ``chain recurrent
point of view,'' all the interesting dynamics take place in the chain
recurrent set [4].
Let 
. An 
-pseudo orbit is a finite sequence such that
where 
is a metric on the manifold M.
An -pseudo orbit can be thought of as a ``computer-generated orbit''
because of the slight roundoff error the computer makes at each stage
of an iteration (see Fig. 4.4(a)).
Figure 4.4: (a) An -pseudo orbit, or computer orbit, of a map;
(b) a chain recurrent point.
A point 
is chain recurrent if, for all
, there exists an -pseudo orbit
such that 
(see Fig. 4.4(b)). The chain recurrent
set is defined as
The chain recurrent set R(f) is closed and f-invariant. Moreover,
. For proofs, see reference [1] or [4].
As an example consider the quadratic map, 
, for 
and 
.
It is easy to see from graphical analysis that if 
then 
is the attracting limit set. When 
, the limit set 
is a Cantor set (see section 2.11.1) and
the dynamics on are topologically conjugate to a full shift on two
symbols, so the periodic orbits are dense in
. The chain recurrent set 
.
As another example, consider the circle map 
shown in Figure 4.5, in which
Figure 4.5: Circle map with two fixed points.
the only fixed points are x and y. The arrows indicate the direction a point goes in when it is iterated. It is easy to see that
However, because an -pseudo orbit can jump
across the fixed points. Chain recurrence is a very weak form
of recurrence.