{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-dynamical-pairing-between-two-rational-maps","title":"A dynamical pairing between two rational maps","arxiv_id":"0911.1875","date":"2009-11-10","proceeding":null,"authors":["Clayton Petsche","Lucien Szpiro","Thomas J. Tucker"],"abstract":"Given two rational maps $\\varphi$ and $\\psi$ on $\\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\\varphi,\\psi>$ 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 $<\\varphi,\\psi>$ 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 $<\\varphi,\\psi>$. We give an explicit upper bound on the difference between the canonical height $h_\\psi$ and the standard height $h_\\st$ in terms of $<\\sigma,\\psi>$, where $\\sigma(x)=x^2$ denotes the squaring map. The pairing $<\\sigma,\\psi>$ is computed or approximated for several families of rational maps $\\psi$.","url_abs":"https://arxiv.org/abs/0911.1875v1","url_pdf":"https://arxiv.org/pdf/0911.1875v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-dynamical-pairing-between-two-rational-maps","repo_url":"https://github.com/MattLarson2399/Arakelov-Zhang-Pairing","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}