In the previous section we showed that when 
, the dynamics
of the quadratic map restricted to the invariant set 
are ``the same''
as those given by the shift map 
on the sequence space on two symbols,
. We established this correspondence by partitioning the unit interval
into two halves about the maximum point of the quadratic map, x = 1/2. The
left half of the unit interval is labeled 0 while the right half is
labeled 1, as illustrated in Figure 2.25. To any orbit of the quadratic
map 
we assign a sequence of symbols
--for example, 101001...--called the
itinerary , or
symbolic future , of the orbit.
Each 
represents the half of the unit interval in which the ith
iteration of the map falls.
In part, the theorem of
section 2.11.2
says that knowing an orbit's initial condition is exactly equivalent to
knowing an orbit's itinerary. Indeed, if we imagine that the itinerary is
simply an expression for some binary number, then perhaps the correspondence is
not so surprising. That is, the mapping 
takes some initial
coordinate number 
and translates it to a binary number
constructed from the symbolic future, which can be thought of as a
``symbolic coordinate.''
From a practical point of view, the renaming scheme described by symbolic dynamics is very useful in at least two ways:
and .
In practical applications we shall be most concerned with keeping track of the periodic orbits. Symbolic itineraries of periodic orbits are repeating finite strings, which can be written in various forms, such as
To see the usefulness of the symbolic description, let us consider the
following problem: for , find the approximate location of
all the periodic orbits in the quadratic map.