  Let $f\colon X\rightarrow X$ 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 $f^*$. Let $\lambda_1(f)$ be the spectral radius of $f^*$.
We study $\mathbb{Q}$-divisors $D$ with $f^*D=\lambda_1(f) D$ 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.
