  Let f : X --> X be a dominant rational map of a projective variety defined
over a number field. An important geometric-dynamical invariant of f is its
(first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic
points P of X whose forward orbits are well-defined, there is an analogous
(upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an
ample Weil height on X. In an earlier paper, we proved the fundamental
inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit
of P is Zariski dense. In this paper we show that the conjecture is true for
several types of maps. In other cases, we provide support for the conjecture by
proving that there is a Zariski dense set of points with disjoint orbits and
satisfying a_f(P) = d_f.
