On Approximations of Functions Preserving Symplectic Forms

The problem of approximating a volume-preserving $C^k$ diffeomorphism (resp. flow) ($k\geq1$) on a compact manifold with or without boundary by a diffeomorphism (resp. flow) was originally motivated by considerations in dynamical systems theory and first posed by Palis and Pugh. This problem, despite its apparent simplicity for those less familiar with the subject matter, in fact hides an extremely nuanced technical complexity and difficulty. Zehnder's work on symplectic approximation techniques provides a compelling avenue to re-examine the foundational results in this area as established by Palis and Pugh. Revisiting their seminal contributions through the lens of Zehnder's symplectic framework could yield novel insights and advance the state-of-the-art. With this in mind, we will revisit the classical results on approximation and a symplectic approximation following Zehnder's ideas.