Zero Lebesgue measure of postcritical sets for all quadratic polynomials

Prove that the postcritical set of every quadratic polynomial has zero Lebesgue measure, including quadratic polynomials that are neither hyperbolic nor covered by the currently known finite- or infinite-renormalization results.

Background

The paper proves that the postcritical set of every neutral quadratic polynomial has zero Lebesgue measure, thereby removing all arithmetic restrictions on the neutral fixed point’s rotation number. The proposed conjecture asks for the analogous conclusion for arbitrary quadratic polynomials.

The paper lists several cases in which the conjecture is already known, including hyperbolic polynomials, polynomials with a neutral periodic point, at most finitely renormalizable polynomials with all periodic points repelling, and robust infinitely renormalizable polynomials. The unresolved difficulty is associated primarily with quadratic polynomials requiring more general, potentially unbounded types of quadratic-like renormalization; the text suggests that uniform bounds for all such renormalization types may be needed.