Let $f \in \text{Diff}_{\omega}^1$ be a $C^1$ symplectomorphism of an $n-$dimensional symplectic manifold $(M, \omega)$.
$\textit{Question}$: If $x$ is any point point in $M$ and $k$ is any positive integer, does there exist a $g \in \text{Diff}_{\omega}^1$ $C^1-$close to $f$ with a periodic point $p$ of minimal period greater than $k$?
If this is true, what else can we assume about $p$? Can it be hyperbolic for $g$, have transversal homoclinic points, etc?
Any related results would also be welcome!
