Two-sided asymptotic growth of degree sequences

Establish whether, whenever the polynomial dynamical degree mu_i(f) exists for a dominant rational self-map of a normal projective variety or a dominant meromorphic self-map of a compact Kähler manifold, the degree sequence satisfies deg_{i,omega}(f^n) asymp n^{mu_i(f)} lambda_i(f)^n.

Background

The paper asks whether the existence of a limiting polynomial exponent is sufficient to yield a two-sided asymptotic estimate for the normalized degree sequence. Its main theorem provides an upper polynomial bound, while the general matching lower bound required for asymp notation is not established.

This question is unresolved in the general non-functorial setting, although affirmative results are recorded for regular or holomorphic maps and for several special classes, including surface maps, certain polynomial maps, monomial maps, and strict dynamical-degree breaks.