Wandering polygons and recurrent critical leaves

Abstract

Let $T$ be a finite subset of the complex unit circle ${\mathbb{S}^1}$ and define $f: {\mathbb{S}^1} \mapsto {\mathbb{S}^1}$ by $f(z) = z^d$. Let $\mathrm{CH}(T)$ denote the convex hull of $T$. If $\mathrm{card}(T) = N \geq 3$, then $\mathrm{CH}(T)$ defines a polygon with $N$ sides. The $N$-gon $\mathrm{CH}(T)$ is called a wandering N-gon if for every two non-negative integers $i \neq j,\ \mathrm{CH}(f^i(T))$ and $\mathrm{CH}(f^j(T))$ are disjoint $N$-gons. A non-degenerate chord of ${\mathbb{S}^1}$ is said to be 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, we call a critical chord 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, Levin and Thurston 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 $\mathrm{ord}(v) -1$ 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.