In this talk based on a joint work with Charles Favre, I will explain how, using techniques from functional analysis, one can understand some problems in algebraic dynamics: namely study the growth of the degrees of the iterates of a given rational self-map on the projective n-space. Our method relies on endowing certain space of cohomology classes of an "infinite blow-up space" with a Banach norm, on which we uncover some spectral gap phenomena.