Preperiodic points with local rationality conditions in the quadratic unicritical family

TL;DR

This paper classifies preperiodic points with local rationality conditions for quadratic unicritical polynomials, providing a trichotomy and explicit examples based on adelic and Julia set dynamics.

Contribution

It introduces a trichotomy for preperiodic points under local conditions and classifies quadratic polynomials with finitely many such points, using adelic and Julia set analysis.

Findings

Classified preperiodic points with local rationality conditions in quadratic unicritical family.

Established a trichotomy for totally real and totally p-adic preperiodic points.

Explicitly computed preperiodic points for specific parameters c = -1, 0, 1/5, 1/4.

Abstract

For rational numbers $c$, we present a trichotomy of 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 $c\in\mathbb{Q}$ so 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 dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totally real $f_c$-preperiodic points when $c=-1,0,\frac{1}{5}$ and $\frac{1}{4}.$