ℚ‐rational cycles for degree‐2 rational maps having an automorphism
ℚ‐rational cycles for degree‐2 rational maps having an automorphism
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ℚ-具有自同构的 2 度有理图的有理循环
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发表时间：
2008
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通讯作者：
M. Manes
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作者：
M. Manes
设ϕ:ℙ1→ℙ1是定义在ℚ上的d=2次有理映射，并假设f−1°ϕ°f=ϕ恰好有一个非平凡的fεPGL2(ℚ−).我们刻画了周期为1，2，4的ℚ-有理周期的映射族，证明了不存在周期为3的ℚ-有理周期的映射族，给出了周期至多为4的ℚ-有理预周期点的完整刻画，特别是证明了至多有12个这样的点.
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 (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.