A dynamical pairing between two rational maps

Given two rational maps $\phi$ and $\psi$ on ${\text{PP}}^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\phi ,\psi >$ which is closely related to the canonical height functions $h_{\phi }$ and $h_{\psi }$ associated to these maps. Our main results show a strong connection between the value of $<\phi ,\psi >$ and the canonical heights of points which are small with respect to at least one of the two maps $\phi$ and $\psi$. Several necessary and sufficient conditions are given for the vanishing of $<\phi ,\psi >$. We give an explicit upper bound on the difference between the canonical height $h_{\psi }$ and the standard height $h_{\text{st}}$ 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 $\psi$.