A survey on MLC, Rigidity and related topics

Abstract

The 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∈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. 1

A brief introduction to one-dimensional complex dynamics

“…关于 MLC 猜想及其相关理论, 可参见 Benini 的综述 [66] (也可参见 Branner 的介绍 [122]). 特别 地, 除了有理函数, Benini 还将超越整函数动力系统的相应理论进行了平行归纳.…”

An optimal Yoccoz inequality for near-parabolic quadratic polynomials

“…It remains unknown whether the Mandelbrot set is locally connected. This question, called the MLC conjecture, has driven a great deal of research over the past forty years, see for example [Hub93], [KL08], [KL09], [CS15], [Ben17], and [DL18]. The following theorem due to Yoccoz [Hub93], with similar versions proved independently by Pommerenke [Pom86] and Levin [Lev91], is a key tool used in many results related to MLC: Theorem (Yoccoz inequality).…”