Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space

Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup h(F^n(P))^(1/n) and the canonical height of P to be h_F(P) = limsup h(F^n(P))/n^k d_F^n for an appropriately chosen integer k = k_F. In this article we prove some elementary relations and make some deep conjectures relating d_F, a_F(P), and h_F(P). We prove our conjectures for monomial maps.