On Dense Orbit Transversality for Endomorphisms of Abelian Varieties

Let $X\mathrm{/}K$ be a smooth projective variety defined over a number field and $f:X\to X$ be a morphism defined over $K$. Assuming there exists a point in $X(K)$ whose $f$-orbit is Zariski dense in $X$ and up to replacing $K$ by a finite extension, Pasten and Silverman studied the distribution of grand $(f,K)$-orbits and proved that many sets of representatives of grand $(f,K)$-orbits on various classes of varieties are Zariski dense. In particular, they showed that if $X$ is a geometrically simple abelian variety, then all such sets of representatives are Zariski dense. We demonstrate the existence of a dense set of representatives for maps on all abelian varieties.