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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 tozero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on asmooth 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 conjectureon a sufficient condition for the limit to be zero. We point out that ourconjecture implies Dynamical Mordell-Lang conjecture for endomorphisms on$\mathbb{P}^{2}_{\overline{\mathbb{Q}}}$. We also discuss applications ofVojta's conjecture with truncated counting function to the problem of theexistence of primitive prime divisors of coordinates of orbits of $f$

Vojta's conjecture, heights associated with subschemes, and primitive prime divisors in arithmetic dynamics