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arXiv · 0911.1875

A dynamical pairing between two rational maps

Clayton Petsche·Lucien Szpiro·Thomas J. Tucker
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Abstract

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

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Clayton Petsche, Lucien Szpiro, Thomas J. Tucker. 2009-11-10. A dynamical pairing between two rational maps. https://arxiv.org/abs/0911.1875

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