Papers › The Arakelov-Zhang pairing and Julia sets
The Arakelov-Zhang pairing and Julia sets
Andrew Bridy, Matt Larson
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The Arakelov-Zhang pairing ⟨ψ,ϕ⟩ is a measure of the "dynamical distance" between two rational maps ψ and ϕ defined over a number field K. It is defined in terms of local integrals on Berkovich space at each completion of K. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height h_ϕ and the standard Weil height h, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.
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