ℚ-rational cycles for degree-2 rational maps having an automorphism

Let ϕ:ℙ1 → ℙ1 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 PGL2 (ℚ̄). 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 p...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2008-05, Vol.96 (3), p.669-696
Main Author: Manes, Michelle
Format: Article
Language:English
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Summary:Let ϕ:ℙ1 → ℙ1 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 PGL2 (ℚ̄). 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.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms/pdm044