Abstract: A celebrated result of Franks shows that a Hamiltonian diffeomorphism of the sphere with more than two fixed points must have infinitely many periodic points. We present a symplectic variant of this phenomenon for symplectomorphisms of surfaces of higher genus that are isotopic to the identity; it implies an upper bound for the Floer-homological count of the number of fixed points of a symplectomorphism with finitely many periodic points. From a higher dimensional viewpoint, this can be understood as evidence for a non-Hamiltonian variant of Shelukhin's result on the Hofer-Zehnder conjecture. Furthermore, we discuss the construction of a symplectic flow on a surface of any positive genus having a single fixed point and no other periodic orbits. This is joint work with Marta Batoréo and Brayan Ferreira.