Degree growth of rational and meromorphic self-maps

Abstract

Let $f$ be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of $f$ differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic \[ \mathrm{deg}_{i,ω}(f^n)=C_{f,ω,i}\,λ_i(f)^n+O(η^n), \] for every $\sqrt{λ_{i-1}(f)λ_{i+1}(f)}<η<λ_i(f)$. This extends the degree-asymptotic theorem of Dang--Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom--Favre--Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge--Lorentz packages on varying finite-time orbit spaces.