On the correspondence of external angles under renormalization

Abstract

Let P $P$ be a monic polynomial of degree D ⩾ 3 $D \geqslant 3$ whose filled Julia set K P $K_P$ has a non-degenerate periodic component K $K$ of period k ⩾ 1 $k \geqslant 1$ and renormalization degree 2 ⩽ d < D $2 \leqslant d<D$ . Let I = I K $I=I_K$ denote the set of angles θ $\theta$ on the circle T = R / Z ${\mathbb {T}}={\mathbb {R}}/{\mathbb {Z}}$ for which the (smooth or broken) external ray R θ P $R^P_\theta$ for P $P$ accumulates on ∂ K $\partial K$ . We prove the following. I $I$ is a compact set of Hausdorff dimension ⩽ log d / ( k log D ) < 1 $\leqslant \log d/(k \log D)<1$ and there is a continuous surjection Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ which semiconjugates θ ↦ D k θ ( mod Z ) $\theta \mapsto D^k \theta \ (\operatorname{mod} {\mathbb {Z}})$ on I $I$ to θ ↦ d θ ( mod Z ) $\theta \mapsto d \theta \ (\operatorname{mod} {\mathbb {Z}})$ on T ${\mathbb {T}}$ . Moreover, Π $\Pi$ is unique up to postcomposition with a power of the rotation θ ↦ θ + 1 / ( d − 1 ) ( mod Z ) $\theta \mapsto \theta +1/(d-1) \ (\operatorname{mod} {\mathbb {Z}})$ . Any hybrid conjugacy φ $\varphi$ between a renormalization of P ∘ k $P^{\circ k}$ on a neighborhood of K $K$ and a degree d $d$ monic polynomial Q $Q$ induces a semiconjugacy Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ as above with the property that for every θ ∈ I $\theta \in I$ the external ray R θ P $R^P_\theta$ has the same accumulation set as the curve φ − 1 ( R Π ( θ ) Q ) $\varphi ^{-1}(R^Q_{\Pi (\theta )})$ . In particular, R θ P $R^P_\theta$ lands at z ∈ ∂ K $z \in \partial K$ if and only if R Π ( θ ) Q $R^Q_{\Pi (\theta )}$ lands at φ ( z ) ∈ ∂ K Q $\varphi (z) \in \partial K_Q$ . The projection Π $\Pi$ is uniformly finite-to-one. In fact, the cardinality of each fiber of Π $\Pi$ is ⩽ D − d + 2 $\leqslant D-d+2$ , and the inequality is strict when the component K $K$ has period k = 1 $k=1$ . The upper bound in the above result is sharp. Using a new type of quasiconformal surgery, we construct polynomials P $P$ with a prescribed hybrid class near K = P ( K ) $K=P(K)$ and a prescribed set of D − d + 1 $D-d+1$ consecutive fixed points of θ ↦ D θ ( mod Z ) $\theta \mapsto D \theta \ (\operatorname{mod} {\mathbb {Z}})$ in the same fiber of Π $\Pi$ .