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

Abstract

AbstractLet φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit 𝒪φ(P) of P. We then prove our conjectures for monomial maps.