Cyclotomic structure in the topological Hochschild homology of DX

Let X be a finite CW complex, and let DX be its dual in the category of spectra. We demonstrate that the Poincare/Koszul duality between THH(DX) and the free loop space Sigma(infinity)(+) LX is in fact a genuinely S-1 -equivariant duality that preserves the C-n - fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor Phi(G) of orthogonal G-spectra.