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 dimension 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 λ 2 1 > λ 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. Contents5 The spectral gap on N 1 Σ (X ) when λ 1 (f ) 2 > λ 2 (f ) 43 5.

Regularizations of positive entropy pseudo-automorphisms

“…In [DF20] was proved a generalization of this theorem in the case of any birational automorphism. In case when the birational automorphism ϕ is regular or if dim(X) = 2, then the class θ 1 (ϕ) is nef.…”

Pluripotential-theoretic stability thresholds

“…In this section, we apply the intersection theory of Shokurov's b-divisors to the study of singularities of psh functions. Due to the technical assumptions in [DF20a] and [DF20b], we can not apply Dang-Favre's intersection theory directly. Although it seems possible to remove the technical assumptions in Dang-Favre's theory, we do not pursue this most general theory here.…”