Continuity of core entropy of quadratic polynomials

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends the entropy theory for real unimodal maps to complex polynomials: the real segment is replaced by an invariant tree, known as the Hubbard tree, which lies inside the filled Julia set. We prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle in parameter space, answering a question of Thurston.