  We study arithmetic degree of a dominant rational self-map on a smooth
projective variety over a function field of characteristic zero. We see that
the notion of arithmetic degree and some related problems over function fields
are interpreted into geometric ones. We give another proof of the theorem that
the arithmetic degree at any point is smaller than or equal to the dynamical
degree. We give a sufficient condition for an arithmetic degree to coincide
with the dynamical degree, and prove that any self-map has so many points whose
arithmetic degrees are equal to the dynamical degree. We study dominant
rational self-maps on projective spaces in detail.
