Seminar

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A polynomial-like map f is a holomorphic branched covering map from a smaller topological disk U to a strictly larger topological disk V. By a theorem of Douady and Hubbard, f is topologically (in fact, “hybrid”) conjugate on the set K* of its non-escaping points to a polynomial. The modulus of the annulus A between U and V is what controls geometry of K*, and it often helps when the modulus mod(A) of A is large. We discuss conditions under which mod(A) must necessarily be small, and also how knowing this helps proving positive results. Roughly speaking, if f is a restriction of a polynomial P with connected filled Julia set K, and there are many periodic components of K-K*, then mod(A) is small. This is similar to the Pommerenke-Levin-Yoccoz inequality that implies, for a P-fixed point z with K-z having many components, that |P’(z)| is close to 1. We give several inequalities on mod(A) and state related conjectures. As an application, we consider parameter space results for the family of cubic polynomials.

The talk is based on joint projects with A. Blokh, G. Levin, and L. Oversteegen.

Upper Bounds for the Moduli of Polynomial-Like Maps 1.99 MB application/pdf

COMD Research Seminar Series: Upper Bounds For The Moduli Of Polynomial-Like Maps