A subset of these two variables, called the invariant set will map onto themselves.

Because of their unstable nature, it is difficult to access members of the invariant set or the stable manifold directly.

The invariant set is the intersection of the stable and unstable manifolds.

On this basis we can calculate the fractal dimension of the invariant set:

Similarly, the fractal dimension will give us information about the density of orbits in the invariant set.

(3) There is a very real sense in which the strange invariant set is a single object and not just the sum of its parts.

Some trajectories have been lost from our strange invariant set.

Then, near a hyperbolic invariant set, the following statement holds:

The proviso is that the absolute must be an invariant set of all hyperbolic motions.

An invariant set is a set that evolves to itself under the dynamics.