The $\mathbb{Z} /p \mathbb{Z}$-equivariant product-isomorphism in fixed point Floer cohomology

Let $p \geq 2$ be a prime, and $\mathbb{F}_p$ the field with $p$ elements. Extending result of Seidel for $p=2,$ we construct an isomorphism between Floer cohomology exact or Hamiltonian symplectomorphism $\phi,$ coefficients, $\mathbb{Z}/p \mathbb{Z}$-equivariant Tate its $p$-th power $\phi^p.$ The construction involves Kaledin-type quasi-Frobenius map, as well pants product: equivariant operation inputs $1$ output. Our method proof spectral sequence action filtration, local coproduct providing inverse on $E^2$-page. This strategy has advantage accurately describing effect filtration levels. We describe applications to symplectic mapping class group, develop Smith theory persistence module diffeomorphism $\phi$ symplectically aspherical manifolds. illustrate latter by giving new celebrated no-torsion theorem Polterovich, relating growth rate number periodic points $p^k$-th iteration distance identity. Along way, prove sharpening classical inequality actions \mathbb{Z}.$