homoclinic class

Let $M$ be a compact smooth manifold and $f\colon M\to M$ a diffeomorphism. The homoclinic class of a hyperbolic periodic point $p$ of $f$ , denoted $H(p,f)$ , is the closure of the set of transverse intersections between the stable and unstable manifolds all points in the orbit of $p$ ; i.e.

H(p,f)=\operatorname{cl}\left(\bigcup_{n\in\mathbb{N}}W^{s}(p)\pitchfork% \bigcup_{n\in\mathbb{Z}}W^{u}(p)\right).

Homoclinic classes are topologically transitive, and the number of homoclinic classes is at most countable. Moreover, generically (in the $\mathcal{C}^{1}$ topology of $\operatorname{Diff}(M)$ ), they are pairwise disjoint and maximally transitive.