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 $\frac{1}{4}$, $-\frac{3}{4}$, $-\frac{5}{4}$ and $-\frac{7}{4}$.