A dynamical pairing between two rational maps

Given two rational maps $\varphi$ and $\psi$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\varphi,\psi>$ which is closely related to the canonical height functions $h_\varphi$ and $h_\psi$ associated to these maps. Our main results show a strong connection between the value of $<\varphi,\psi>$ and the canonical heights of points which are small with respect to at least one of the two maps $\varphi$ and $\psi$. Several necessary and sufficient conditions are given for the vanishing of $<\varphi,\psi>$. We give an explicit upper bound on the difference between the canonical height $h_\psi$ and the standard height $h_\st$ in terms of $<\sigma,\psi>$, where $\sigma(x)=x^2$ denotes the squaring map. The pairing $<\sigma,\psi>$ is computed or approximated for several families of rational maps $\psi$.