Arithmetic degrees for dynamical systems over function fields of characteristic zero

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