One can define the Frobenius operator on a commutative ring of characteristic p as the p th power operation, and this has generalizations to a larger class of commutative rings, and even to topological spaces and spectra. Spectra with circle actions and Frobenius operators are called cyclotomic spectra. The simplest example is the free loop space, and important examples arise in algebraic and arithmetic geometry as topological Hochschild homology of rings and categories. By topological reasons and mirror symmetry, it is natural to expect such a structure to arise in symplectic topology-- more precisely in ``closed string Floer theory''. In this talk, we will explain how to construct such spectra using Hamiltonian Floer theory, i.e. by using holomorphic cylinders in symplectic manifolds. Joint work in progress with Laurent Cote.