Relative dynamical degrees of correspondences over a field of arbitrary characteristic

TLDR: In this paper, the relative dynamical degrees of positive algebraic cycles are defined and a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semiconjugacy.

Abstract: Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant correspondences, and $\pi :X\rightarrow Y$ a dominant rational map such that $\pi \circ f=g\circ \pi$. We define relative dynamical degrees $\lambda _p(f|\pi )$ ($p=0,\ldots ,\dim (X)-\dim (Y)$). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy $(\varphi ,\psi )$ from $(X_2,f_2)\rightarrow (Y_2,g_2)$ to $(X_1,f_1)\rightarrow (Y_1,g_1)$ we have $\lambda _p(f_1|\pi _1)\geq \lambda _p(f_2|\pi _2)$ for all $p$. Many of our results are new even when $K=\mathbb{C}$. We make use of de Jong's alterations and Roberts' version of Chow's moving lemma. In the lack of resolution of singularities, the consideration of correspondences is necessary even when $f,g$ are rational maps. The case $K$ is not algebraically closed further requires working with correspondences over reducible varieties.