Preperiodic points with local rationality conditions in the quadratic unicritical family
Volume 212 / 2024
Abstract
For rational numbers , we present a trichotomy for the set of totally real (totally p-adic, respectively) preperiodic points for maps in the quadratic unicritical family f_c(x)=x^2+c. As a consequence, we classify quadratic polynomials f_c with rational parameters c\in \mathbb {Q} such that f_c has only finitely many totally real (totally p-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and the dynamics of the filled Julia set of f_c. Moreover, using a numerical criterion introduced by the author and Petsche (2024), we make explicit calculations of the set of totally real f_c-preperiodic points when c=-1,0,\frac {1}{5} and \frac {1}{4}.