Spectral interpretations of dynamical degrees and applications

• 成果类型: Article
• 署名作者: Dang, Nguyen-Bac; Favre, Charles
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.194.1.5
• 发表日期: 2021
• 页码: 299-359
• 关键词: degree growth; birational maps; topological-entropy; arithmetic degrees; rational mappings; degree complexity; periodic points; AUTOMORPHISMS; compactifications; VALUATIONS

摘要

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(2)(1) > lambda(2) on the dynamical degrees. As a consequence, we obtain that the dynamical all algebraic of the affine 3-space are all algebraic numbers.