  The Kawaguchi-Silverman conjecture relates two different invariants of a
surjective endomorphism, the dynamical and arithmetic degrees. As the
Kawaguchi-Silverman conjecture is only meaningful when a morphism has a Zariski
dense orbit, it has no content for varieties with positive Kodaira dimension. A
generalization of the Kawaguchi-Silverman conjecture which is meaningful in
positive Kodaira dimension is the so called sAND conjecture, which involves the
set of "small" arithmetic degrees. Kawaguchi and Silverman showed that a small
arithmetic degree is the modulus of an eigenvalue of $f^*\colon
N^1(X)\rightarrow N^1(X)$. In this article we investigate which possible
eigenvalues arise as an arithmetic degree. We show that surjective
endomorphisms of abelian varieties may have eigenvalues which are not
arithmetic degrees. Conversely, we show that every eigenvalue of a surjective
endomorphism of a toric variety is an arithmetic degree using the minimal model
program. Finally, we investigate how the minimal model program may be applied
to study this realizability question for varieties that admit an int-amplified
endomorphism.
