Totally real points in the Mandelbrot Set

TL;DR

This paper classifies totally real parameters in the Mandelbrot set where the quadratic polynomial has either postcritically finite or parabolic cycles, extending understanding of real dynamics in complex quadratic maps.

Contribution

It identifies all totally real algebraic parameters with parabolic cycles in quadratic polynomials, complementing previous results on postcritically finite parameters.

Findings

Only four totally real parameters have parabolic cycles: 1/4, -3/4, -5/4, -7/4.

Confirmed the exclusive nature of these parameters for parabolic behavior.

Extended the classification of real parameters 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 $-\frac74$.