Existence of polynomial dynamical degrees for rational and meromorphic maps

Determine whether, for every dominant rational self-map of a normal projective variety or dominant meromorphic self-map of a compact Kähler manifold and every intermediate codimension 1 <= i <= d-1, the limit defining mu_i(f) exists.

Background

The paper defines the upper and lower polynomial dynamical degrees by taking the limsup and liminf, respectively, of the normalized logarithmic degree growth divided by log n. It establishes general upper and lower bounds but does not prove that these two quantities coincide.

The limit is known in functorial settings such as regular or holomorphic self-maps and in several specialized classes, but the general rational and meromorphic case is explicitly left unresolved, particularly at resonant dynamical-degree indices.

References

This leaves the following finer questions; cf.Conjecture~2 and the subsequent paragraph and Question~13.16.

\begin{question}\label{question:polynomial-dyn-deg} Assume that either \cref{setting:projective} or \cref{setting:kahler} holds, and let $1\leq i\leq d-1$.

\begin{enumerate}[label=\textnormal{(\arabic*)}] \item Existence. Does the limit

\mu_i(f)\coloneqq\lim_{n\to\infty}\frac{\log\deg_{i,\omega}(fn)-n\log\lambda_i(f)}{\log n}

exist?

— Degree growth of rational and meromorphic self-maps  (2610.18094 - Hu, 13 Oct 2026) in Question polynomial-dyn-deg, Section Further consequences and sharpness