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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 (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.