homoclinic class

Let $M$ be a compact smooth manifold and $f:M\to M$ 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.

$$H(p,f)=\mathrm{cl}\left(\bigcup _{n\in \mathbb{N}}{W}^s(p)⋔\bigcup _{n\in \mathbb{Z}}{W}^u(p)\right).$$

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