A dynamical pairing between two rational maps

Given two rational maps $\varphi$ and $\psi$ on $\mathbb{P}^1$ of degree at least two, we study a symmetric, nonnegative real-valued pairing $\langle \varphi ,\psi \rangle$ 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 $\langle \varphi ,\psi \rangle$ 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 $\langle \varphi ,\psi \rangle$. We give an explicit upper bound on the difference between the canonical height $h_\psi$ and the standard height $h_{\mathrm {st}}$ in terms of $\langle \sigma ,\psi \rangle$, where $\sigma (x)=x^2$ denotes the squaring map. The pairing $\langle \sigma ,\psi \rangle$ is computed or approximated for several families of rational maps $\psi$.