Totally real points in the multibrot sets

TL;DR

The paper determines all totally real parabolic parameters for unicritical multibrot families $f_c(z)=z^d+c$. It combines real-slice analysis, Milnor’s arithmetic constraints, and capacity-theoretic bounds to control Galois conjugates and discriminants of possible parameters. The main results are: $Par_3\cap\mathbb{Q}^{\mathrm{tr}}=\{\pm\tfrac{2\sqrt{3}}{9}\}$ and $Par_d\cap\mathbb{Q}^{\mathrm{tr}}=\emptyset$ for all even $d\ge4$, with the $d=2$ case previously known. The approach extends previous work for $d=2$ and leverages Kronecker-type arguments alongside capacity theory to avoid purely analytic capacity estimates in higher degrees.

Abstract

We classify all totally real parabolic parameters in the multibrot sets, extending a theorem of Buff and Koch.

Figures (1)

• Figure 1: The multibrot sets $\mathrm{M}_2$, $\mathrm{M}_3$ and $\mathrm{M}_4$. The images have been produced with Mathematica.

Theorems & Definitions (18)

• Theorem 1.1: MR4710201
• Theorem 1.2: buff2022totallyrealpointsmandelbrot
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3: MR3289903
• Corollary 2.4
• proof
• Corollary 2.5