We study the parameter space Sp 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 Sp. For every escape region E⊂Sp, every parameter ray in E with rational parameter angle lands at some uniquely defined point in the boundary ∂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 Sp, where maps in the same face of this tessellation always have the same period q orbit portrait.

Submitted 11 March, 2025; originally announced March 2025.

arXiv:2503.08868