Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation

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