Totally real points in the Mandelbrot Set

Abstract

Recently, Noytaptim and Petsche proved that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c(z):=z^2+c$ is postcritically finite are $0$, $-1$ and $-2$. In this note, we show that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c$ has a parabolic cycle are $\frac14$, $-\frac34$, $-\frac54$ and $-\frac74$.