A survey on MLC, Rigidity and related topics

TL;DR

This survey reviews recent progress on the Mandelbrot set's local connectivity and related conjectures in complex dynamics, covering various families of maps and their interconnections.

Contribution

It provides a comprehensive overview of recent advances and the relationships between key conjectures in complex dynamics, including MLC, hyperbolicity, and rigidity.

Findings

Progress on MLC and related conjectures

Connections between different conjectures in complex dynamics

Coverage of various families of maps including transcendental ones

Abstract

The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic family $\{z^2+c\}_{c\in\mathbb{C}}$. We describe recent advances on MLC and the relations between MLC, the Density of Hyperbolicity Conjecture, the Rigidity Conjecture, the No Invariant Line Fields Conjecture, and the Triviality of Fibers Conjecture. We treat families of unicritical polynomials and rational maps as well as the exponential family and families of transcendental maps with finitely many singular values.