Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
Abstract
Let F : PN --> PN be a dominant rational map. The dynamical degree of F is the quantity dF = lim (deg Fn)(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be aF(P) = limsup h(Fn(P))(1/n) and the canonical height of P to be hF(P) = limsup h(Fn(P))/nk dFn for an appropriately chosen integer k = kF. In this article we prove some elementary relations and make some deep conjectures relating dF, aF(P), and hF(P). We prove our conjectures for monomial maps.
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