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

Published online by Cambridge University Press:  29 November 2012

JOSEPH H. SILVERMAN
JOSEPH H. SILVERMAN*
Affiliation:
Mathematics Department, Box 1917, Brown University, Providence, RI 02912, USA (email: [email protected])
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Abstract

Let 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 647 - 678
Copyright
©2012 Cambridge University Press 

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