  We are interested in a non-local partial differential equation modeling equal
mitosis. We prove that the solutions present persistent asymptoticoscillations
and that the convergence to this periodic behavior, in suitable spaces of
weighted signed measures, occurs exponentially fast. It can beseen as a result
of spectral gap between the countable set of dominant eigenvalues and the rest
of the spectrum, which is to our knowledge completely new. The two main
difficulties in the proof are to define the projection onto the subspace of
periodic (rescaled) solutions and to estimate thespeed of convergence to this
projection. The first one is addressed by using the generalized relative
entropy structure of the dual equation, and thesecond is tackled by applying
Harris's ergodic theorem on sub-problems.
