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Independence of multipliers via degenerations of rational maps
Abstract
Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every d≥ 2, the multipliers of any 2d−2 distinct periodic orbits of degree d rational maps are algebraically independent over ℂ, provided that at most d of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the d+1 fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.
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BibTeX
@misc{gorbovickis2026independence,
title = {{Independence of multipliers via degenerations of rational maps}},
author = {Igors Gorbovickis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06200},
eprint = {2609.06200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06200},
}
RIS
TY - UNPB
TI - Independence of multipliers via degenerations of rational maps
AU - Igors Gorbovickis
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06200
UR - https://arxiv.org/abs/2609.06200
AB - Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every d≥ 2, the multipliers of any 2d−2 distinct periodic orbits of degree d rational maps are algebraically independent over ℂ, provided that at most d of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the d+1 fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.
ER -