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We introduce a metric of mutual energy for adelic measures associated to theArakelov-Zhang pairing. Using this metric and potential theoretic techniquesinvolving discrete approximations to energy integrals, we prove an effectivebound on a problem of Baker and DeMarco on unlikely intersections of dynamicalsystems, specifically, for the set of complex parameters $c$ for which $z=0$and $1$ are both preperiodic under iteration of $f_c(z)=z^2 + c$.

A metric of mutual energy and unlikely intersections for dynamical systems