Preperiodic points for quadratic polynomials with small cycles over quadratic fields

摘要

Given a number field K and a polynomial , one can naturally construct a finite directed graph G(f, K) whose vertices are the K-rational preperiodic points of f, with an edge if and only if . The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers and , there are only finitely many isomorphism classes of directed graphs G(f, K) as one ranges over all number fields K of degree n and polynomials of degree d. In the case , Poonen has given a complete classification of all directed graphs which may be realized as for some quadratic polynomial , under the assumption that f does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case . By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen's-with a similar assumption on points of large period-but over all quadratic extensions of Q.