Definition. A set or region 
is said to be invariant under the
map f if for any 
we have 
for all n.
The simplest example of an invariant set is the collection of points forming a periodic orbit. But, as we will see shortly, there are more complex examples, such as strange invariant sets, which are candidates for chaotic attractors or repellers .
Definition. For mappings of 
, a set
is a repelling (resp., attracting) hyperbolic
set for f if is closed, bounded, and invariant under f and there
exists an N ;SPMgt; 0 such that 
(resp., ;SPMlt; 1) for all 
and all 
[10].
This definition says that none of the derivatives of points in the invariant set are exactly equal to one. A simple example of a hyperbolic invariant set is a periodic orbit that is either repelling or attracting, but not neutral. In higher dimensions a similar definition of hyperbolicity holds, namely, all the points in the invariant set are saddles .
The existence of both a simple periodic regime and a complicated fully developed chaotic (yet well understood) hyperbolic regime turns out to be quite common in low-dimensional nonlinear systems. In Chapter 5 we will show how information about the hyperbolic regime, which we can often analyze in detail using symbolic dynamics, can be exploited to determine useful physical information about a nonlinear system.
In examining the dynamics of the quadratic map for 
we
proceed in two steps: first, we examine the invariant set, and second, we
describe how orbits meander on this invariant set. The set itself is a
fractal Cantor set [11], and to describe the dynamics on this fractal
set we employ the method of symbolic dynamics.
Since f(1/2) ;SPMgt; 1 for there exists an open interval centered at
1/2 with points that leave the unit interval after one iteration,
never to return. Call
this open set 
(see Figure 2.23).
Figure 2.23: Quadratic map for . (Generated by the Quadratic Map program.)
These are the points in whose image under f is greater than one.
On the second iteration,
more points leave the unit interval. In fact, these are the points that get
mapped to after the first iteration: 
. Inductively, define 
; that is, 
consists of all points
that escape from I at the n+1st iteration. Clearly, the invariant limit
set, call it 
, consists of all the remaining points
What does look like? First, note that consists of 
disjoint open intervals, so
is
disjoint closed intervals.
Second, 
monotonically maps each of these intervals onto I.
The graph of is a
polynomial with humps. The maximal sections of the humps are the
collection of intervals
that get mapped out of I, but more importantly this polynomial
intersects the y=x line times. Thus, 
has
periodic points in I.
The set is a Cantor set
if is a closed,
totally disconnected, perfect subset. A set
is disconnected if it contains no
intervals; a set is perfect if every point is a limit point. It is not too hard
to show that the invariant set defined by equation (2.33)
is a Cantor set [10].
Thus, we see that the invariant limit set
arising from the quadratic map for is a fractal Cantor set with
a countable infinity of periodic orbits.
