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Proceedings of the London Mathematical Society Advance Access originally published online on November 24, 2007
Proceedings of the London Mathematical Society 2008 96(3):669-696; doi:10.1112/plms/pdm044
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© 2007 London Mathematical Society

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

Michelle Manes

University of Southern California
Department of Mathematics
3620 South Vermont Avenue
KAP 108
Los Angeles
CA 90089-2532
USA

Received 11 February 2007. Revision received 10 October 2007.

Let {phi}:P1 -> P1 be a rational map of degree d = 2 defined over Q and assume that f–1°{phi}° f = {phi} for exactly one nontrivial f {epsilon} PGL2 (Formula). 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.

2000 Mathematics Subject Classification 11G99, 14G25.


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