UC Santa Cruz

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic points. These maps are of interest in both dynamics and symplectic topology. In this thesis, principally in relation with the Conley conjecture, we study pseudo-rotations from two different perspectives. In the first part, we prove a variant of the Chance--McDuff conjecture. 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. In the second part, we give a simple proof of a slightly weaker version of a recent theorem by Shelukhin which extends Franks' "two-or-infinitely-many'' theorem to Hamiltonian diffeomorphisms in higher dimensions. More precisely, we show that for a certain class of closed monotone symplectic manifolds (e.g. $\CP^n$) pseudo-rotations are the only strongly non-degenerate counterexamples to the Conley conjecture. In addition, we show that every non-degenerate pseudo-rotation of $\CP^2$ is balanced by using equivariant pair-of-pants product and quantum Steenrod squares.