The Arakelov-Zhang pairing and Julia sets

The Arakelov-Zhang pairing $\langleψ,ϕ\rangle$ 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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