# Cubic critical portraits and polynomials with wandering gaps

### Abstract

• Thurston introduced -invariant laminations (where -d(z) coincides with and defined wanderingâ ons as sets such that gma -d({T}}) consists of k\ge 3 distinct points for all n ge 0 and the convex hulls of all the sets sigma -d^n({\mathbf {T}}) in the plane are pairwise disjoint. He proved that \sigma -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. Here we use a new approach to construct cubic WT-laminations with the above properties so that any wandering branch point of J has a dense orbit in each subarc of J (we call such orbits condense), and show that critical portraits corresponding to such laminations are dense in the space {\mathcal A}-3of all cubic critical portraits. © Cambridge University Press, 2013.

### Author List

• Blokh A; Curry C; Oversteegen L

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