Density of orbits in laminations and the space of critical portraits

Academic Article

Abstract

  • Thurston introduced σ d-invariant laminations (where σ d(z) coincides with zd : S → S, d ≥ 2). He defined wandering k-gons as sets T c S such that σ nd (T) consists of k ≥ 3 distinct points for all n ≥ 0 and the convex hulls of all the sets σ nd (T) in the plane are pairwise disjoint. Thurston proved that 2 has no wandering k-gons and posed the problem of their existence for σ d, d ≥ 3. Call a lamination with wandering k-gons a WT-lamination. Denote the set of cubic critical portraits by A 3. A critical portrait, compatible with a WT-lamination, is called a WT-critical portrait; letWT 3 be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with condense orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in A 3 (D c X is condense in X if D intersects every subcontinuum of X). Here we show that WT 3 is a dense first category subset of A 3, that critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of A 3, and that the existence of a condense orbit in the Julia set J implies that J is locally connected.
  • Digital Object Identifier (doi)

    Author List

  • Blokh A; Curry C; Oversteegen L
  • Start Page

  • 2027
  • End Page

  • 2039
  • Volume

  • 32
  • Issue

  • 6