In this talk I will survey some results on the existence of periodic points of symplectomorphisms defined on closed orientable surfaces of positive genus g. Namely, I will describe some symplectic flows on such surfaces possessing finitely many periodic points and describe a non-Hamiltonian variant of the Hofer-Zehnder conjecture for symplectomorphisms defined on surfaces; this conjecture provides a quantitative threshold on the number of fixed points (possibly counted homologically) which forces the existence of infinitely many periodic points. This is joint work in progress with Marcelo Atallah and Brayan Ferreira.

A natural question bridging the celebrated Gromov–Eliashberg theorem and the C⁰-flux conjecture is whether the identity component of the group of symplectic diffeomorphisms is C⁰-closed in Symp(M,ω). Beyond surfaces and the cases in which the Torelli subgroup of Symp(M,ω) coincides with the identity component, little is known. In joint work with Cheuk Yu Mak and Wewei Wu, we show that, for all but a few positive rational surfaces, the group of Hamiltonian diffeomorphisms is the C⁰-connected component of the identity in Symp(M,ω), thereby giving a positive answer in this setting. Here, positive rational surface essentially means a k-point blow-up of CP² whose symplectic form evaluates positively on the first Chern class.