Open Conjectures
Robert P. Munafo, 2023 Jun 16.
There are several unproven conjectures related to the Mandelbrot set; most are fairly easy to state and have been investigated for 30 years or longer.
MLC
Perhaps the most famous open conjecture is "local connectivity". The conjecture was put forth by Douady and Hubbard, and many have worked on it. Some weaker conjectures (such as local connectivity at certain limit points) have been proven.
For example it is known to be locally connected at Misiurewicz points, see Dierk Schleicher.
Area of the Boundary
It is proven that the boundary of the Mandelbrot set has Hausdorff dimension exactly 2. This makes it possible that the boundary has a Lebesgue measure greater than zero, as is true for an Osgood curve. If that is true, then the area of the Mandelbrot set would be greater than the sum of the area of all mu-atoms. Certain points (such as generalised Feigenbaum points) definitely resemble the Osgood curve, but only in arbitrarily small neighborhoods of the Feigenbaum point itself; and these are not sufficiently dense in the boundary of the Mandelbrot set to prove anything about the Lebesgue measure.
Computability
According to some restrictive models of theoretical computation (such as Real computation in which each variable is a real number with unlimited precision), it is not possible in general to discover (in finite time) whether a point is a member point.
Islands in the Boundary
If MLC is true than there are an infinite number of small islands within any neighborhood of any boundary point (see the islands article for an explanation of why). If instead this could be proven false for neighborhoods of some points then it could probably lead to disproof of MLC. We already know it to be true for a dense subset of the boundary (see the boundary article, in the section "Density of Islands") but that's not quite the same thing.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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