  Assuming Vojta's conjecture, we give a sufficient condition for the limit \[
\lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to
zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a
smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and
$h_{Y}$ is a global height function associated with a closed subscheme $Y
\subset X$ of codimension at least two. Based on this, we propose a conjecture
on a sufficient condition for the limit to be zero. We point out that our
conjecture implies Dynamical Mordell-Lang conjecture for endomorphisms on
$\mathbb{P}^{2}_{\overline{\mathbb{Q}}}$. We also discuss applications of
Vojta's conjecture with truncated counting function to the problem of the
existence of primitive prime divisors of coordinates of orbits of $f$
