A spectral sequence of the Floer cohomology of symplectomorphisms of trivial polarization class

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $\varphi$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\widehat{\varphi }$ such that neither $\widehat{\varphi }$ nor ${\widehat{\varphi }}^2$ has fixed points. We give conditions under which Seidel and Smith’s localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $$HF$(\phi^2)\otimes \mathbb Z_2((\theta))$to$$HF$(\phi)\otimes \mathbb Z_2((\theta))$.