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Post-critically finite maps on P^n for n≥2 are sparse
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Ingram, Patrick, Ramadas, Rohini and Silverman, Joseph (2023) Post-critically finite maps on P^n for n≥2 are sparse. Transactions of the American Mathematical Society, 376 . pp. 3087-3109. doi:10.1090/tran/8871 ISSN 0002-9947.
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Official URL: https://doi.org/10.1090/tran/8871
Abstract
Let f : P^n → P^n be a morphism of degree d ≥ 2. The map f is said to be post-critically finite (PCF) if there exist integers k ≥ 1 and l ≥ 0 such that the critical locus Critf satisfies f^(k+l)(Critf ) ⊆ f^l(Critf ). The smallest such l is called the tail-length. We prove that for d ≥ 3 and n ≥ 2, the set of PCF maps f with tail-length at most 2 is not Zariski dense in the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with l = 0, are not Zariski dense.
Item Type: | Journal Article | |||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||
Library of Congress Subject Headings (LCSH): | Differentiable dynamical systems, Ergodic theory, Mappings (Mathematics) | |||||||||||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | |||||||||||||||
Publisher: | American Mathematical Society | |||||||||||||||
ISSN: | 0002-9947 | |||||||||||||||
Official Date: | 2023 | |||||||||||||||
Dates: |
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Volume: | 376 | |||||||||||||||
Page Range: | pp. 3087-3109 | |||||||||||||||
DOI: | 10.1090/tran/8871 | |||||||||||||||
Status: | Peer Reviewed | |||||||||||||||
Publication Status: | Published | |||||||||||||||
Re-use Statement: | First published in Transactions of the American Mathematical Society in [volume/issue number and year], published by the American Mathematical Society,” and the copyright notice in proper form must be placed on all copies. | |||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
Copyright Holders: | © Copyright 2023 American Mathematical Society | |||||||||||||||
Date of first compliant deposit: | 21 February 2023 | |||||||||||||||
Date of first compliant Open Access: | 21 February 2023 | |||||||||||||||
RIOXX Funder/Project Grant: |
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