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A dynamical pairing between two rational maps
Clayton Petsche, Lucien Szpiro, Thomas J. Tucker
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Given two rational maps φ and ψ on 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 hᵩ 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)=x² denotes the squaring map. The pairing <σ,ψ> is computed or approximated for several families of rational maps ψ.
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