Cubic Polynomial Maps with Periodic Critical Orbit, Part III: Tessellations and Orbit Portraits

We study the parameter space ${\mathcal{S}}_p$ for cubic polynomial maps with a marked critical point of period $p$. We will outline a fairly complete theory as to how the dynamics of the map $F$ changes as we move around the parameter space ${\mathcal{S}}_p$. For every escape region $\mathcal{E}\subset {\mathcal{S}}_p$, every parameter ray in $\mathcal{E}$ with rational parameter angle lands at some uniquely defined point in the boundary $\mathrm{\partial }\mathcal{E}$. This landing point is necessarily either a parabolic map or a Misiurewicz map. The relationship between parameter rays and dynamic rays is formalized by the period $q$ tessellation of ${\mathcal{S}}_p$, where maps in the same face of this tessellation always have the same period $q$ orbit portrait.