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

Abstract

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.

45 pages (substantially revised from first version)

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Cited by in corpus (6)

• On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
• Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties
• Examples of dynamical degree equals arithmetic degree
• Canonical Height Functions For Monomial Maps
• Silverman's conjecture for additive polynomial mappings
• On variation of dynamical canonical heights, and Intersection numbers