Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle

TL;DR

This paper investigates the combinatorial and geometric structure of quadratic polynomials using Yoccoz puzzles, proving linear growth of moduli and establishing local connectivity of Julia sets for many infinitely renormalizable cases.

Contribution

It introduces new results on the growth of moduli in Yoccoz puzzles and derives complex bounds and local connectivity for a broad class of quadratic Julia sets.

Findings

Moduli of the principal nest grow linearly

Established complex a priori bounds

Proved local connectivity of Julia sets for many infinitely renormalizable quadratics

Abstract

This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the ``principal nest'' of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.