Manifolds of a periodic orbit

Question

Are there stable and or unstable manifolds transverse to a periodic orbit. I know there are stable and unstable manifold tangent to the eigenvectors of a fixed point.

Answer 1

There are (transverse) stable and unstable invariant manifolds for all points in any hyperbolic set.

Now let $x$ be a periodic point with period $p$ of a diffeomorphism $f$. Then the periodic orbit $$\{f^k(x):k=0,\ldots,p-1\}$$ is a hyperbolic set for $f$ if and only if $d_xf^p$ has no eigenvalues with modulus $1$.

So that's when there are (transverse) stable and unstable invariant manifolds for the points in a periodic orbit.