Geometry of quadratic polynomials: moduli, rigidity and local connectivity

TL;DR

This paper advances the understanding of the Mandelbrot set's local connectivity by establishing a priori bounds and rigidity results for certain infinitely renormalizable quadratic polynomials, extending previous work to new parameter regimes.

Contribution

It proves MLC for some infinitely renormalizable quadratic polynomials with bounded combinatorial rotation number and high combinatorial type, expanding the scope of known results.

Findings

Proved a priori bounds for a new class of quadratic polynomials.

Established quasi-conformal rigidity and topological rigidity results.

Extended the applicability of Sullivan's renormalization theory.

Abstract

A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at most finitely renormalizable parameter values. One of our goals is to prove MLC for some infinitely renormalizable parameter values. Loosely speaking, we need all renormalizations to have bounded combinatorial rotation number (assumption C1) and sufficiently high combinatorial type (assumption C2). For real quadratic polynomials of bounded combinatorial type the complex a priori bounds were obtained by Sullivan. Our result complements the Sullivan's result in the unbounded case. Moreover, it gives a background for Sullivan's renormalization theory for some bounded type polynomials outside the real line where the problem of a priori bounds was not handled before for any single polynomial. An important consequence of a priori bounds is absence of invariant measurable line fields on the Julia set (McMullen) which is equivalent to quasi-conformal (qc) rigidity. To prove stronger topological rigidity we construct a qc conjugacy between any two topologically conjugate polynomials (Theorem III). We do this by means of a pull-back argument, based on the linear growth of moduli and a priori bounds. Actually the argument gives the stronger combinatorial rigidity which implies MLC.