Totally real algebraic numbers in generalized Mandelbrot set

In this article, we study some potential theoretical and topological aspects of the generalized Mandelbrot set introduced by Baker and DeMarco. For α real, we study the set of all totally real algebraic parameters c such that α is preperiodic under the iteration of the one-parameter family fc(x) = x² + c. We show that when |α| < 2 and rational then the set of totally real algebraic parameters c with this property is finite, whereas if |α| ≥ 2 and rational then this set is countably infinite. As an unexpected consequence of this study, we also show that when |α| ≥ 2 then parameters c such that α is fc-periodic are necessarily real. As a special case, we classify all totally real algebraic integers c such that α = ±1 is preperiodic.