  Let f : X --> X be a dominant rational map of a projective variety defined
over a global field, let d_f be the dynamical degree of f, and let h_X be a
Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) <<
(d_f + e)^n h_X(P), where the implied constant depends only on X, h_X, f, and
e. As applications, we prove a fundamental inequality a_f(P) \le d_f for the
upper arithmetic degree and we construct canonical heights for (nef) divisors.
We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and
we describe some cases for which we can prove our conjecture.
