In this lecture we define the limit set of a dynamical system $f \colon X \to X$ to be the set of all possible limit points of orbits:
$$
\mathsf{lim}(f) := \overline{\bigcup_{x \in X} \omega_f(x) \cup \alpha_f(x)}.
$$
We then introduce the notion of a cycle in the limit set, and explain how the only way to produce chain recurrent points that are not limit points is to exploit the existence of  cycles in the limit set.

Switching back to hyperbolic dynamics, we obtain the following pleasing result:

Theorem:

Let $f$ be a dynamical system on a compact manifold $M$. Then the following are equivalent:

  1. $f$ satisfies Axiom A and the set $\overline{\mathsf{per}(f)}$ has no cycles.
  2. $ \mathsf{lim}(f)$ is hyperbolic and has no cycles.
  3. $\mathsf{cha}(f)$ is hyperbolic.

Whilst this theorem is pretty in its own right, the most spectacular consequence is still to come. Next lecture we will prove that all three conditions are equivalent to $f$ being omega stable.



Comments and questions?