Kawaguchi–Silverman conjecture on arithmetic degrees

Determine, for every projective variety X over the algebraic closure of Q and every dominant rational self-map f: X → X, that for every point x whose forward orbit is well-defined the arithmetic degree α_f(x) exists, and that if the forward orbit O_f(x) is Zariski dense in X then α_f(x) equals the first dynamical degree λ_1(f).

Background

The arithmetic degree measures the exponential growth rate of heights along orbits, while the first dynamical degree measures the asymptotic growth of pullbacks in codimension one. The conjecture predicts a precise equality linking arithmetic and geometric complexity for points with Zariski dense orbits.

The conjecture is known in several special cases (e.g., polarized endomorphisms, many surface endomorphisms, certain regular affine automorphisms). This paper proves the conjecture for a broad class of rational self-maps on projective surfaces, including all birational ones, but the full generality remains open.

References

\begin{con}[Kawaguchi-Silverman conjecture]\label{kscin} Let $X$ be a projective variety over $\overline{}$. Let $f: X\dashrightarrow X$ be a dominant rational self-map. Then for every $x\in X_f(\overline{})$, $\alpha_f(x)$ is well defined. Moreover, if $O_f(x)$ is Zariski dense, then we have $\alpha_f(x)= _1(f).$ \end{con}

— Algebraic dynamics and recursive inequalities  (2402.12678 - Xie, 2024) in Subsection “Kawaguchi-Silverman conjecture”, Conjecture