MLC at Feigenbaum points

TL;DR

This paper proves bounds for Feigenbaum quadratic polynomials, establishing local connectivity of Julia sets and the Mandelbrot set at specific parameters, and confirms universality in their combinatorics.

Contribution

It provides the first proof of a priori bounds for Feigenbaum polynomials of bounded type, advancing understanding of MLC and universality conjectures.

Findings

Proved a priori bounds for Feigenbaum quadratic polynomials.

Established local connectivity of Julia sets and Mandelbrot set at Feigenbaum parameters.

Confirmed universality in the combinatorics of these polynomials.

Abstract

We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.