Does Smale-Birkhoff homoclinic theorem ("given a transverse homoclinic intersection for a diffeomorphism $f$, there exists an integer $n \geq 1$ such that $f^{n}$ has an invariant Cantor set on which it is topologically conjugate to a shift on $N$ symbols") imply the existence of infinitely many $\textbf{distinct}$ homoclinic orbits?
My reasoning being is that then $f^{n}$ will have an countable infinity of saddle periodic orbits, each with their homoclinic intersections... is this right?
There are infinitely many distinct homoclinic orbits, caused by the secondary intersections of the stable and unstable manifolds of the fixed points. It follows from the lambda-lemma.