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

Assuming Vojta’s conjecture, we give a sufficient condition for the limit $$\underset{n\to \mathrm{\infty }}{\lim }\frac{h_Y(f^n(x))}{h_H(f^n(x))}$$ is equal to zero, where $f\text{\hspace{0.17em}⁣}:X⟶X$ 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 ${\mathbb{P}}_{\bar{\mathbb{Q}}}^2$. 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 $f$