On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties.

Let f : X → X be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K, let δf be the dynamical degree of f, and let hX: X(K̅) → [1, ∞) be a Weil height relative to an ample divisor. We prove that for every ϵ > 0 there is a height bound hX ∘ fn ⪡ (δf + ϵ)nhX, valid for all points whose f-orbit is well-defined, where the implied constant depends only on X, hX, f, and ϵ. An immediate corollary is a fundamental inequality α̅f (P) ≤ δf for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence f* D ≡ βD for some β> √δf, we prove that the canonical height hf,D = lim β-nhD ∘ fn converges and satisfies hf,D ∘ f = βhf,D and hf,D = hD + O(√hX). We also prove that the arithmetic degree αf(P), if it exists, gives the main term in the height counting function for the f-orbit of P. We conjecture that αIf(P) = δf whenever the f-orbit of P is Zariski dense and describe some cases for which we can prove our conjecture. [ABSTRACT FROM AUTHOR]