论文标题
动态程度和应用的光谱解释
Spectral interpretations of dynamical degrees and applications
论文作者
论文摘要
我们证明，可以将投影品种的理性自图的动态程度解释为在合适的Banach空间上自然定义的操作员的光谱半径。概括了Shokurov的B型分散概念，我们考虑了更高的编成循环的B类空间，并以各种Banach规范赋予了该空间。在这些构造的基础上，我们将自然扩展设计为Cantat和Boucksom-Favre-Jonsson在表面的情况下引入的Picard-Manin空间的更高尺寸。我们证明了Hodge索引定理的一个版本，并且在这个Banach空间中产生了令人惊讶的紧凑性。我们使用这两个定理来推断对图的迭代序列的精确控制，假设第一动力学的平方严格大于第二个动力学程度。结果，我们获得了仿射3空间自动形态的动力学程度都是代数数。
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.
