Uniform Tur\'an estimates and sharp bounds for degree and mixed orbit growth

Abstract

The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $\lambda_i$, so that $\textrm{deg}_i(f^n)\asymp\lambda_i^n n^{\mu_i}$ for a unique integer $\mu_i\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(\lambda_i)_i$ at $i$ forces $\mu_i=0$, while equality gives \[ 0\ \le\ 2\mu_i-\mu_{i-1}-\mu_{i+1}\ \le\ 4 . \] Consequently, $\mu_i\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $\gamma$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\sum_j\gamma_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact K\"ahler $d$-fold $Y$, the same method gives \[\left\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.