Dynamics of quadratic polynomials II: rigidity

TL;DR

This paper proves a rigidity theorem for quadratic polynomial dynamics, establishing uniqueness and local connectivity of the Mandelbrot set at certain parameters, advancing understanding of complex dynamical systems.

Contribution

It introduces a rigidity theorem showing at most one quadratic polynomial per combinatorial class satisfies secondary limbs conditions with bounds, confirming local connectivity of the Mandelbrot set.

Findings

Uniqueness of quadratic polynomials in combinatorial classes with secondary limbs conditions

Rigidity of maps leading to local connectivity of the Mandelbrot set

Establishment of a key theorem in complex dynamics

Abstract

This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a-priori bounds. As a corollary, such maps are combinatorially and topologically rigid, and as a consequence, the Mandelbrot set is locally connected at the correspoinding parameter values.