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ℚ‐rational cycles for degree‐2 rational maps having an automorphism
Author(s) -
Manes Michelle
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm044
Subject(s) - mathematics , degree (music) , automorphism , rational point , rational number , period (music) , point (geometry) , periodic point , combinatorics , pure mathematics , discrete mathematics , mathematical analysis , geometry , physics , acoustics , algebraic number
Let ϕ:ℙ 1 → ℙ 1 be a rational map of degree d = 2 defined over ℚ and assume that f −1 °ϕ° f = ϕ for exactly one nontrivial f ε PGL 2 (ℚ − ). We describe families of such maps that have ℚ‐rational periodic points of period 1, 2, and 4, and we prove that no such map has a ℚ‐rational periodic point of exact period 3. We give a complete description of the ℚ‐rational preperiodic points with period at most 4 and show in particular that there are at most 12 such points.