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A dynamical pairing between two rational maps

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  • Given two rational maps φ and ψ on Ρ¹ 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 hψ 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 hψ and the standard height hₛₜ in terms of〈σ, ψ〉, where σ(χ) = χ² denotes the squaring map. The pairing〈σ, ψ〉is computed or approximated for several families of rational maps ψ.
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  • Petsche, C., Szpiro, L., & Tucker, T. J. (2012, April). A dynamical pairing between two rational maps. Transactions of the American Mathematical Society, 364(4), 1687-1710. Retrieved July 23, 2012
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  • 364
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  • 4
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  • The published article can be found at the American Mathematical Society.
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  • The first author was partially supported by NSF Grant DMS-0901147.The second author was supported by NSF Grants DMS-0854746 and DMS-0739346.The third author was supported by NSF Grants DMS-0801072 and DMS-0854839.
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