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.

我们证明了射影变体上的有理自映射的动态度可以解释为合适的 Banach 空间上自然定义的算子的谱半径。推广Shokurov的$b$-除数概念，我们考虑$b$-高维循环的类空间，并赋予这个空间各种Banach范数。在这些结构的基础上，我们设计了由 Cantat 和 Boucksom-Favre-Jonsson 在曲面的情况下引入的 Picard-Manin 空间的更高维度的自然扩展。我们证明了 Hodge 指数定理的一个版本，并在这个 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在动态度上假设 $\lambda_1^2>\lambda_2$ 下对映射的迭代度序列的精确控制。作为结果，