On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties

2012/08/03 by Kawaguchi, Shu, Silverman, Joseph H. · 3 citations
#11G50 #37P30 #37P55 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary: 37P15 #Secondary: 37P05

paper · doi:10.48550/arxiv.1208.0815

Abstract

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(fⁿ(P)) << (d_f + e)ⁿ h_X(P), where the implied constant depends only on X, h_X, f, and e. As applications, we prove a fundamental inequality a_f(P) ≤ d_f for the upper arithmetic degree and we construct canonical heights for (nef) divisors. We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and we describe some cases for which we can prove our conjecture.

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