  For a dominant rational self-map on a smooth projective variety defined over
a number field, Kawaguchi and Silverman conjectured that the (first) dynamical
degree is equal to the arithmetic degree at a rational point whose forward
orbit is well-defined and Zariski dense. We prove this conjecture for
surjective endomorphisms on smooth projective surfaces. For surjective
endomorphisms on any smooth projective varieties, we show the existence of
rational points whose arithmetic degrees are equal to the dynamical degree.
Moreover, we prove that there exists a Zariski dense set of rational points
having disjoint orbits if the endomorphism is an automorphism.
