In what follows we give a compact introduction to the terminology used in the study of discrete dynamical systems. For more details, see [7].
To begin with, a discrete dynamical system is a recurrence
relation 
, with the index k playing the role
of a discrete ``time''.
Note that the points 
may be multi-dimensional.
The sequence 
is called the orbit
of the initial point 
under the map 
, and is
denoted as orb(
).
Any points x that satisfy 
are called fixed points
of themap, and more generally if 
where 
then x is called a periodic point of the map.
If N is the least such number n, then as usual we say x has
period N. The 
- limit set of orb(
) is the set
of all initial points whose orbits approach orb(
) as ``time''
increases; to be precise, an initial point 
is in the 
-limit
set of orb(
) if there exist m and n such that for all 
there exists K such that 
implies 
.
The 
-limit set of orb(
) is the set of accumulation
points of orb(
). An attractor of a map is a set of points which
``attracts'' orbits, from some set of initial points of nonzero probability
of being selected. To be precise, an attractor of a map is an indecomposable
closed invariant set 
with the property that, given 
,
there is a set U of positive Lebesgue measure in the 
-neighbourhood of
such that if x is in U then the 
-limit set of orb(x)
is contained in 
and the orbit of x is contained in U [9].
An invariant set is a set such that 
,
and an indecomposable set is one which cannot be broken into two or more
pieces which are distinct under G. A map G is said to be
sensitive to initial conditions (SIC) if initially close initial points
have orbits that separate at an exponential rate. A map that is SIC is
also said to be chaotic. The possible average exponents of these
rates of separation are called the Lyapunov exponents of the map.
Osledec's theorem [19] states that for a wide class of maps,
and for almost all initial points, there are only finitely many
possible Lyapunov exponents (in fact, only n for an n-dimensional map).