Totally real algebraic numbers in generalized Mandelbrot set

Abstract

In this article, we study some potential theoretical and topological aspects of the generalized Mandelbrot set introduced by Baker and DeMarco. For $\alpha$ real, we study the set of all totally real algebraic parameters $c$ such that $\alpha$ is preperiodic under the iteration of the one-parameter family $f_c(x) = x^2 + c$. We show that when $|\alpha| < 2$ and rational then the set of totally real algebraic parameters $c$ with this property is finite, whereas if $|\alpha| \geq 2$ and rational then this set is countably infinite. As an unexpected consequence of this study, we also show that when $|\alpha| \geq 2$ then parameters $c$ such that $\alpha$ is $f_c$-periodic are necessarily real. As a special case, we classify all totally real algebraic integers $c$ such that $\alpha = \pm1$ is preperiodic.