From Pseudo-Rotations to Holomorphic Curves via Quantum Steenrod Squares

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element and hence nontrivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on nonvanishing Gromov–Witten invariants of manifolds admitting pseudo-rotations.