we have 
for all 
. S is an invariant set of a map if for any , 
for all n. We also speak of a positively invariant set when
we restrict the definition to positive times, 
or 
.
Invariant sets are important because they give us a means of decomposing phase space. If we can find a collection of invariant sets, then we can restrict our attention to the dynamics on each invariant set and then try to sew together a global solution from the invariant pieces. Invariant sets also act as boundaries in phase space, restricting trajectories to a subset of phase space.