Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

Let phi: P-N -> P-N be a dominant rational map. The dynamical degree of phi is the quantity delta(phi) = lim(deg phi(n))(1/n). When phi is defined over (Q) over bar, we define the arithmetic degree of a point P is an element of P-N ((Q) over bar) to be alpha(phi) (P) = lim sup h (phi(n) (P))(1/n) and the canonical height of P to be (h) over cap (phi) = lim sup delta(-n)(phi)n(-l phi) h (phi(n) (P)) for an appropriately chosen l(phi). We begin by proving some elementary relations and making some deep conjectures relating delta(phi), alpha(phi) (P), (h) over cap (phi) (P), and the Zariski density of the orbit O-phi (P) of P. We then prove our conjectures for monomial maps.