Quadratic maps with a periodic critical point of period 2

Abstract

We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are exactly 13 possible graphs, and that such maps have at most nine rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.

Quadratic rational functions with a rational periodic critical point of period 3

“…In this article we continue the classification of preperiodicity graphs of quadratic rational functions defined over Q with a Q-rational periodic critical point that was begun in [3]. Our aim in this article is to provide a complete classification of rational quadratic functions defined over Q with a Q-rational periodic critical point of period 3.…”

“…Theorem (Canci and Vishkautsan [3]). Assuming the conjecture above, there are exactly 13 possible preperiodicity graphs for quadratic rational functions defined over Q with a Q-rational periodic critical point of period 2.…”

“…In particular, they showed that a quadratic rational function defined over Q with a Q-rational critical 3-cycle is PCF if and only if both of its critical points lie in the same 3-cycle, and there is a unique conjugacy class of quadratic rational functions (over Q) with this property, realizing the graph R3P0 in Table 1.1. Table 1 [3] implies that starting from the critical cycle graph R3P1, one can obtain any realizable graph of a non-PCF rational function defined with a rational critical 3-cycle, by recursively adding vertices and arrows to the graph using either of the following two recursion steps:…”

“…In the article [3], J.K. Canci and this author proposed a generalization of Poonen's classification of preperiodicity graphs of quadratic polynomials to the classification of preperiodicity graphs of quadratic rational functions with a rational periodic critical point. Recall that a point P ∈ P 1 is called critical if φ (P ) = 0 (if P or φ(P ) are ∞ then we conjugate φ by an automorphism of P 1 , in order to move P and φ(P ) away from ∞).…”

“…In [3], Canci and the author provided a full classification of preperiodicity graphs of quadratic rational functions defined over Q with a Q-rational periodic critical point of period 2, up to the following conjecture. [3]). Let φ be a quadratic rational function defined over Q having a Q-rational periodic critical point of period 2.…”