Wandering polygons and recurrent critical leaves

Abstract

Let T be a finite subset of the complex unit circle S^1, and define f: S^1 -> S^1 by f(z) = z^d. Let CH(T) denote the convex hull of T. If card(T) = N > 2, then CH(T) defines a polygon with N sides. The N-gon CH(T) is called a \emph{wandering N-gon} if for every two non-negative integers i \neq j, CH(f^i(T)) and CH(f^j(T)) are disjoint N-gons. A non-degenerate chord of S^1 is said to be \emph{critical} if its two endpoints have the same image under f. Then for a critical chord, it is natural to define its (forward) orbit by the forward iterates of the endpoints. Similarly, call a critical chord \emph{recurrent} if one of its endpoints is recurrent under f. The main result of our study is that a wandering N-gon has at least N-1 recurrent critical chords in its limit set (defined in a natural way) having pairwise disjoint, infinite orbits. Using this result, we are able to strengthen results of Blokh, Kiwi and Levin about wandering polygons of laminations. We also discuss some applications to the dynamics of polynomials. In particular, our study implies that if v is a wandering non-precritical vertex of a locally connected polynomial Julia set, then there exists at least ord(v)-1 recurrent critical points with pairwise disjoint orbits, all having the same omega-limit set as v. Thus, we likewise strengthen results about wandering vertices of polynomial Julia sets.