Equivariant identification of iterated local symplectic homology with Z_k-invariant local Floer homology

Prove that for any isolated closed Reeb orbit x and any integer k≥1, the equivariant local symplectic homology CH(x^k;Q) is isomorphic to the Z_k-invariant part of the local Floer homology HF(ϕ^k;Q) of the Poincaré return map ϕ of x, and that the non-equivariant local symplectic homology SH(x^k;Q) decomposes as HF(ϕ^k;Q)^{Z_k} ⊕ HF(ϕ^k;Q)^{Z_k}[−1].

Background

Lemma 2.9 provides Z_k-actions on local symplectic homology via k-fold covers, and known identifications relate CH(x;Q) and HF(ϕ;Q) for prime orbits. The conjectured extension to iterates asserts precise Z_k-invariant identifications for both equivariant and non-equivariant local symplectic homology.

Establishing these isomorphisms would clarify the relation between Reeb dynamics and discrete-time return map dynamics under iteration, and unify local invariants across frameworks.