From pseudo-rotations to holomorphic curves via quantum Steenrod squares

TL;DR

This paper links pseudo-rotations in symplectic dynamics to the existence of non-trivial holomorphic spheres by using quantum Steenrod squares, revealing new connections between dynamics and symplectic topology.

Contribution

It demonstrates that symplectic manifolds with pseudo-rotations necessarily have non-trivial holomorphic spheres via quantum Steenrod squares, extending recent results in the field.

Findings

Pseudo-rotations imply non-trivial quantum Steenrod squares.

Existence of holomorphic spheres in manifolds with pseudo-rotations.

Generalization of Shelukhin's work on symplectic invariants.

Abstract

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 non-degenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitting pseudo-rotations.