Let $f \in \text{Diff}^1_{\omega}$ be a symplectomorphism of a symplectic $n-$dimensional manifold $M$. Let $T(f)$ be the closure of the set of all transversal homoclinic points of $f$.
$\textit{Question 1}:$ What are some known properties of $T(f)$? I would be interested in any special cases also.
Also, consider $T : \text{Diff}^1_{\omega} \to C(M)$ as a set valued function, where $C(M)$ is the set of all closed subsets of $M$ with the Hausdorff metric.
$\textit{Question 2}:$ What are some continuity properties of this function? Is it at least lower semi-continuous?