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

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