Cubic Critical Portraits and Polynomials with Wandering Gaps

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Thurston introduced -invariant laminations (where (z) coincides with zᵈ:→, d≥ 2) and defined wandering k-gons as sets ⊂ such that ⁿ() consists of k≥ 3 distinct points for all n≥ 0 and the convex hulls of all the sets ⁿ() in the plane are pairwise disjoint. He proved that ₂ has no wandering k-gons. Call a lamination with wandering k-gons a WT-lamination. In a recent paper it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding quotient spaces J, are realizable as cubic polynomials on their (locally connected) Julia sets. In the present paper we use a new approach to construct cubic WT-laminations with all of the above properties and the extra property that the corresponding wandering branch point of J has a dense orbit in each subarc of J (we call such orbits condense), and to show that critical portraits corresponding to such laminations are uncountably dense in the space ₃ of all cubic critical portraits.