, 
, and Nonwandering
We begin by introducing
the -limit set , which starts us down the road
toward defining an attractor.
Let p be a point of a map 
of the manifold M. Then the
-limit set of p is
Conversely, by going backwards in time we get the
-limit set of p ,
These limit sets are the closure of the ends of the orbits. For example,
if p is a periodic point, then 
. It is not difficult to
show that 
is a closed subset of M and that it is
invariant under f, i.e., 
. Finally, a point 
is called recurrent if it is part of the
-limit set, i.e., 
.
The forward limit set , 
,
is defined as the union of all -limit sets;
likewise the
backward limit set , 
, is defined as the
union of all -limit sets:
The forward and backward limit sets are not necessarily closed. This
observation motivates yet another useful notion of recurrence, the nonwandering
set. A point wanders if there exist a neighborhood U of p
and an m ;SPMgt; 0 such
that 
for all n ;SPMgt; m.
A nonwandering
point is one that does not wander. This brings us to
the nonwandering set , 
, which is a
closed, invariant (under f) subset of M:
The dynamical decomposition of a set into its wandering and nonwandering parts separates, in mathematical terms, a dynamical system into its transient behavior --the wandering set--and long-term or asymptotic behavior --the nonwandering set.
The -limit set and the nonwandering set do not address the
question of the stability of an asymptotic motion. To get to the
idea of an attractor, we begin with the idea of an attracting set.
A closed invariant set 
is an attracting set if there is some neighborhood U of A such that for all 
and
all 
,
Moreover, the domain or basin of attraction of A is given by
The attracting set can consist of a collection of different sets that are dynamically disconnected. For instance, a single attracting set could consist of two separate periodic orbits. To overcome this last difficulty, we will say that an attractor is an attracting set that contains a dense orbit. Conversely, a repeller is defined as a repelling set with a dense orbit.
Although this is a reasonable definition mathematically, we will see that this is not the most useful definition of an attractor for physical applications or numerical simulations. In these circumstances it will turn out that a more useful (albeit mathematically naive) definition of an attractor or repeller is the closure of a certain collection of periodic orbits. The proper definition of an attractor is yet another hot spot in the creative tension between the rigor demanded by a mathematician and the utility required by a physicist.