Preperiodic points with local rationality conditions in the quadratic unicritical family

For rational numbers $c$, 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 $\frac {1}{4}.$