A dynamical pairing between two rational maps

2009/11/10 by Petsche, Clayton, Szpiro, Lucien, Tucker, Thomas J. · 2 citations
#11G50 #14G40 #37P15 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0911.1875

Abstract

Given two rational maps φ and ψ on \PP¹ of degree at least two, we study a symmetric, nonnegative-real-valued pairing 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 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 . We give an explicit upper bound on the difference between the canonical height h_ψ and the standard height h_\st in terms of, where σ(x)=x² denotes the squaring map. The pairing is computed or approximated for several families of rational maps ψ.

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