Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Abstract

Let {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over {\mathbb{K}}. Let {f:X\vdash X} and {g:Y\vdash Y} be dominant correspondences, and {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that {\pi\circ f=g\circ\pi}. We define relative dynamical degrees {\lambda_{p}(f|\pi)\geq 1} for any {p=0,\dots,\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 {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to {\pi_{1}:(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 {\mathbb{K}=\mathbb{C}}. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Étale dynamical systems and topological entropy

“…In previous work [26], we have defined (relative) dynamical degrees, for rational maps and more general correspondences, of algebraic varieties (not necessarily smooth or compact) over an arbitrary field, in terms of algebraic cycles modulo numerical equivalences.…”

“…In the algebraic setting, over fields K different from C, there is still no definition of cohomological entropy. There are indications [6,12,26] that one should be willing to work with objects such asétale topology -seemingly further away from dynamical systems -in order to come closer to satisfying Gromov -Yomdin 's theorem. In fact, for automorphisms of compact surfaces, the work in [12] relates the quantity in Gromov -Yomdin's theorem to l-adic cohomology, which suggests that we may even need to deal with stranger objects thanétale topology.…”

“…Even when K = C, the fact that the dynamical degrees are invariant under generically finite semi-conjugacies ( [6], see [26] for a generalisation to correspondences) is an evidence in support of the philosophy that to understand the dynamics of a rational map f : X → X, it is useful and necessary to study all rational maps f ′ : X ′ → X ′ (or even correspondences) which are semi-conjugate to f via a generically finite rational map…”

“…First, we consider the case where K = C. Let f : X → X be a dominant rational map or more generally dominant correspondence, where X is a complex projective. Let B(f ) be the set of all dominant correspondences f Z : Z → Z, where Z is a complex projective variety so that is in fact the same as that for each individual f Z , see [6,26]. We then define theétale analogue of entropy…”

“…Note that we can choose Z = X, and hence h eta (f ) ≥ 0. Moreover, whenever π Z : Z → X is a generically finite rational map, we can always find such a correspondence f Z (the pullback of f by π Z , see [26] for more details), where d Z = deg(π Z ). By the results in [10], we have that h eta (f ) is bounded from above in terms of the dynamical degrees of f .…”