  We compute the product and coproduct structures on the fixed point Floer
homology of iterations on the single Dehn twist, subject to some mild
topological restrictions. We show that the resulting product and coproduct
structures are determined by the product and coproduct on Morse homology of the
complement of the twist region, together with certain sectors of product and
coproduct structures on the symplectic homology of $T^*S^1$. The computation is
done via a direct enumeration of $J-$holomorphic sections: we use a local
energy inequality to show that some of the putative holomorphic sections do not
exist, and we use a gluing construction plus some Morse-Bott theory to
construct the sections we could not rule out.
