Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

Let $f\text{\hspace{0.17em}⁣}:X\to X$ be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of $f$ may be studied through the dynamics of the linear action $f^*\colon Pic(X)$\mathbb{R}$\rightarrow Pic(X)$\mathbb{R}$$, which are governed by the spectral theory of $f^{\ast }$. Let ${\lambda }_1(f)$ be the spectral radius of $f^{\ast }$. We study $\mathbb{Q}$-divisors $D$ with $f^{\ast }D={\lambda }_1(f)D$ and $\kappa (D)=0$ where $\kappa (D)$ is the Iitaka dimension of the divisor $D$. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. Finally we identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture.