  We prove that dynamical degrees of rational self-maps on projective varieties
can be interpreted as spectral radii of naturally defined operators on suitable
Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the
space of b-classes of higher codimension cycles, and endow this space with
various Banach norms. Building on these constructions, we design a natural
extension to higher dimensions of the Picard-Manin space introduced by Cantat
and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the
Hodge index theorem, and a surprising compactness result in this Banach space.
We use these two theorems to infer a precise control of the sequence of degrees
of iterates of a map under the assumption that the square of the first
dynamical degree is strictly larger than the second dynamical degree. As a
consequence, we obtain that the dynamical degrees of an automorphism of the
affine 3-space are all algebraic numbers.
