  We prove the existence of the arithmetic degree for dominant rational
self-maps at any point whose orbit is generic. As a corollary, we prove the
same existence for \'etale morphisms on quasi-projective varieties and any
points on it. We apply the proof of this fact to dynamical Lang-Siegel problem.
Namely, we prove that local height function associated with zero-dimensional
subscheme grows slowly along orbits of a rational map under reasonable
assumption. Also if local height function associated with any proper closed
subscheme grows fast on a subset of an orbit of a self-morphism, we prove that
such subset has Banach density zero under some assumptions.
