Addendum to “Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties"

By Shu Kawaguchi, Joseph H. Silverman. Transactions of the American Mathematical Society, 2013

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Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. Further, if X is an abelian variety defined over a number field and D is a nonzero nef divisor, we characterize points satisfying h_D(x)=0, and we use this characterization to prove that if the f-orbit of x is Zariski dense in X, then a_f(x) is equal to the dynamical degree of f.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c130e37832d389918676a73f7d5cf98ccb32c3d4

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Current trends and open problems in arithmetic dynamics (2018, background): “Parts (a), (b), and (c) are true when f is a morphism [136].”