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We consider the limit   $$ \begin{align*} &amp; \lim_{n\to \infty} \sum_{v\in S} \lambda_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) \end{align*}$$  where   $f \colon X \longrightarrow X$   is a surjective self-morphism on a smooth projective variety   $X$   over a number field,   $S$   is a finite set of places,   $ \lambda _{Y,v}$   is a local height function associated with a proper closed subscheme   $Y \subset X$  , and   $h_{H}$   is an ample height function on   $X$  . We give a geometric condition that ensures that the limit is zero, unconditionally when   $\dim Y=0$   and assuming Vojta’s conjecture when   $\dim Y\geq 1$  . In particular, we prove (one is unconditional, one is assuming Vojta’s conjecture) dynamical Lang–Siegel type theorems, that is, the relative sizes of coordinates of orbits on   ${{\mathbb {P}}}^{N}$   are asymptotically the same with trivial exceptions. These results are higher dimensional generalization of Silverman’s classical result.

Growth of Local Height Functions Along Orbits of Self-Morphisms on Projective Varieties