The dual nest for degenerate Yoccoz puzzles

TL;DR

This paper extends the Yoccoz puzzle technique in Holomorphic Dynamics to cases lacking non-degenerate annuli, enabling new applications like proving local connectedness of Julia sets for certain rational maps.

Contribution

It introduces a dual nest approach that generalizes the combinatorial argument for degenerate Yoccoz puzzles, broadening their applicability.

Findings

Established a new combinatorial framework for degenerate puzzles

Proved local connectedness of Julia sets in new rational map families

Extended the utility of Yoccoz puzzles to matings of quadratic polynomials

Abstract

The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the Branner-Hubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.