On the correspondence of external angles under renormalization

Let (Formula presented.) be a monic polynomial of degree (Formula presented.) whose filled Julia set (Formula presented.) has a non-degenerate periodic component (Formula presented.) of period (Formula presented.) and renormalization degree (Formula presented.). Let (Formula presented.) denote the set of angles (Formula presented.) on the circle (Formula presented.) for which the (smooth or broken) external ray (Formula presented.) for (Formula presented.) accumulates on (Formula presented.). We prove the following. (Formula presented.) is a compact set of Hausdorff dimension (Formula presented.) and there is a continuous surjection (Formula presented.) which semiconjugates (Formula presented.) on (Formula presented.) to (Formula presented.) on (Formula presented.). Moreover, (Formula presented.) is unique up to postcomposition with a power of the rotation (Formula presented.). Any hybrid conjugacy (Formula presented.) between a renormalization of (Formula presented.) on a neighborhood of (Formula presented.) and a degree (Formula presented.) monic polynomial (Formula presented.) induces a semiconjugacy (Formula presented.) as above with the property that for every (Formula presented.) the external ray (Formula presented.) has the same accumulation set as the curve (Formula presented.). In particular, (Formula presented.) lands at (Formula presented.) if and only if (Formula presented.) lands at (Formula presented.). The projection (Formula presented.) is uniformly finite-to-one. In fact, the cardinality of each fiber of (Formula presented.) is (Formula presented.), and the inequality is strict when the component (Formula presented.) has period (Formula presented.). The upper bound in the above result is sharp. Using a new type of quasiconformal surgery, we construct polynomials (Formula presented.) with a prescribed hybrid class near (Formula presented.) and a prescribed set of (Formula presented.) consecutive fixed points of (Formula presented.) in the same fiber of (Formula presented.).