The main cubioid (Q2877446)

The authors study the principal hyperbolic domain \(P_{d}\) of affine conjugacy classes \([f]\) of a degree-\(d\) polynomial \(f\), that is, all critical points of \(f\) are in the immediate attracting basis of the same attracting (or super-attracting) fixed point. Their first main result is to show that if \([f]\) belongs to the closure \(\overline{P_{d}}\) of \(P_{d}\), then \(f\) has a fixed non-repelling point, \(f\) has no repelling periodic cutpoint in its Julia set, and all non-repelling periodic cutpoints (except at most one fixed point) of \(f\) have multiplier equal to \(1\). The paper continues by defining the main cubioid \(C_{3}\) of classes of cubic polynomials \(f\) with connected Julia sets \(J(f)\) such that \(f\) has at least one non-repelling fixed point, \(f\) has no repelling periodic cutpoints in \(J(f)\), and all non-repelling periodic cutpoints (except at most one fixed point) of \(f\) have multiplier equal to \(1\). With \(L_{3}\) referring to the set of classes of all cubic polynomials with connected Julia sets, the second main result of the paper is to establish that \(\overline{P_{d}} \subset C_{3}\) and \(L_{3} \cap C_{3} = L_{3} \cap \overline{P_{d}}\). The paper closes with some interesting observations concerning leafs of laminations arising from elements of \(C_{3}\).