Spectral interpretations of dynamical degrees and applications

TL;DR

This paper links dynamical degrees of rational maps on projective varieties to spectral radii of operators on Banach spaces, extending classical concepts to higher dimensions and providing new bounds and algebraicity results.

Contribution

It introduces a spectral interpretation of dynamical degrees using Banach space operators, generalizes the Picard-Manin space, and proves new theorems on degree growth and algebraicity.

Findings

Dynamical degrees are spectral radii of specific operators.

Established a higher-dimensional analogue of the Picard-Manin space.

Proved algebraicity of dynamical degrees for automorphisms of affine 3-space.

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 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.