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A Numberphile video piqued my interest regarding the hyperbolicity property of points in the Mandelbrot set. But I can't seem to find a concise statement about the conjecture about hyperbolic points and their distribution in the interior of the set and its various bulbs.

So is it the case that:

  1. It is conjectured that all points in the Mandelbrot set are hyperbolic, but there are no (connected) sub-regions of known hyperbolicity within the set.

  2. It is conjectured that all points in the Mandelbrot set are hyperbolic, but there are regions where it is known that all the points are hyperbolic (e.g. main cardioid, period-2 bulb etc.)

  3. It is conjectured that all points in the Mandelbrot set are hyperbolic, and it is not known if regions like the main cardioid, and other bulbs are dense in hyperbolic points.

  4. Something more precise than the above.

I also can't quite find any other source for the statement that the limit cycle of $c=-3/2$ is not known (or as described in the video, it is unknown if $c=-3/2$ is hyperbolic).

Hope someone could shed some light on the above, thanks in advance.

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    $\begingroup$ The hyperbolicity conjecture is that all interior points are hyperbolic. Certainly not all points are hyperbolic, as hyperbolic points are always interior, and the Mandelbrot set has a boundary (of dimension 2, even). $\endgroup$
    – Geoffrey Irving
    Commented Aug 30 at 22:15
  • $\begingroup$ It is hard to know when the "conjectures" you mention end and the next phrase (if any) begins. $\endgroup$
    – Daniel Asimov
    Commented Aug 31 at 19:10
  • $\begingroup$ @DanielAsimov Consider each "conjecture" to comprise the entire line, unless you find the sub-clauses to be inconsistent. $\endgroup$
    – nonreligious
    Commented Sep 1 at 14:06
  • $\begingroup$ Since each of "conjectures' 1., 2., 3. are multi-line, there is no way to know what "the entire line" refers to. A much better idea would be for you to edit the question so that it is unambiguous. $\endgroup$
    – Daniel Asimov
    Commented Sep 1 at 18:17
  • $\begingroup$ @DanielAsimov I'm not sure where you're from, but syntactically, each of those statements is one line. If you select each point, hit the Ctrl key (that's usually at the bottom left or right of your keyboard) and hold it while then pressing the "C" key, you can copy the text. If you open up a program like Microsoft Word, you can "paste" the text into it and see that it will fit in one line. $\endgroup$
    – nonreligious
    Commented Sep 1 at 19:51

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It is conjectured that hyperbolic maps are dense in the family of quadratic polynomials $Q_c(z) = z^2+c$. And it is known

  • if $c$ is not in the Mandelbrot set, $Q_c$ is hyperbolic.
  • if $c$ is in the boundary of the Mandelbrot set, $Q_c$ is NOT hyperbolic.

Moreover, it is known that if a parameter in the interior of the Mandelbrot set is hyperbolic, then every parameter in the same interior component is hyperbolic.

Therefore, the conjecture is equivalent that every interior component of the Mandelbrot set is hyperbolic.

This is written, for example, in McMullen's book.

So far, no one found any non-hyperbolic interior parameter of the Mandelbrot set. In other words, all known parameters in the interior of the Mandelbrot set are hyperbolic.

For the real case, it is already known that hyperbolic maps are dense. Hence for the case $c=-\frac32$, it is either hyperbolic or in the boundary, but it is often quite hard to judge for a specific parameter.

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