Spectral interpretations of dynamical degrees and applications

Abstract

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 $\lambda _1^2>\lambda _2$ on the dynamical degrees. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine $3$-space are all algebraic numbers.