On the correspondence of external rays under renormalization

Let $P$ be a monic polynomial of degree $D\ge 3$ whose filled Julia set $K_P$ has a non-degenerate periodic component $K$ of period $k\ge 1$ and renormalization degree $2\le d<D$. Let $I=I_K$ denote the set of angles $\theta$ on the circle $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ for which the (smooth or broken) external ray $R_{\theta }^P$ for $P$ accumulates on $\partial K$\mathrm{.}Weprovethefollowing:$\bullet$ $I$ is a compact set of Hausdorff dimension $<1$ and there is an essentially unique degree $1$ monotone map $\mathrm{\Pi }:I\to \mathbb{T}$ which semiconjugates $\theta \mapsto D^k\theta$ (mod 1) on $I$ to $\theta \mapsto d \theta$(mod1)on$\mathbb T$\mathrm{.}$\bullet$Anyhybridconjugacy$\varphi$betweenarenormalizationof$P^{\circ k}$onaneighborhoodof$K$andamonicdegree$d$polynomial$Q$inducesasemiconjugacy$\Pi: I \to {\mathbb T}$ with the property that for every $\theta \in I$ the external ray $R_{\theta }^P$ has the same accumulation set as the curve ${\phi }^{-1}(R_{\mathrm{\Pi }(\theta )}^Q)$. In particular, $R_{\theta }^P$ lands at $z\in \mathrm{\partial }K$ if and only if $R_{\mathrm{\Pi }(\theta )}^Q$ lands at $\phi (z)\in \mathrm{\partial }{K}_Q$. $\bullet$ The ray correspondence established by the above result is finite-to-one. In fact, the cardinality of each fiber of $\mathrm{\Pi }$ is $\le D-d+2$, and the inequality is strict when the component $K$ has period $k=1$. Using a new type of quasiconformal surgery we construct a class of examples with $k=1$ for which the upper bound $D-d+1$ is realized and the set $I$ has isolated points.