M-set interior point probability on the real axis For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$.  Some of these points are in the interior of the m-set, and some are on the boundary.  Those points in the interior are inside hyperbolic regions of one of the bulbs on the real axis.  What is the probability that a random point on the real axis, greater than -2, and less then 0.25, is in the interior of the m-set?  Is that value $\lt 1$, and perhaps arbitrarily small for other regions on the real axis?
Of course I realize, numerical calculations are not particularly relevant for a mathematical proof, but I did numerical experiments on the region from $[-2,-1.40115518909205060052]$, or from -2 to the Feigenbaum limit point of the tip of the main cardiod.  It seems like in this region, there's approximately a 10% chance that a random point is inside a hyperbolic region.  I experimented by iterating $z_n\mapsto z_{n-1}^2+z_0$, where z is a random real number in that region and $z_0=z+i10^{-30}$.  90% of the values quickly iterate to infinity. Using a much larger imaginary $\delta=i10^{-5}$ did not not change the results.  I did a similar analysis on the region from -1.625 to -1.615, which appears to have an even smaller probability of landing in the interior of a mini-Mandelbrot bulb.  There are an infinite number of mini-Mandelbrots in that region, but the largest bug in that region is around 1/1000th the size of the region, and the next two bugs are around 1/4000th the size of the region and the bugs rapidly get smaller after that.
Anyway, clearly someone much smarter than I am has proved something about this.  I conjecture that most of these random points are not inside the m-set, but rather are chaotic random points on the boundary of the m-set, and that these chaotic boundary points are a super-set of the Misiurewicz points, and that most of the points in the real valued region $[-2,-1.40115518909205060052]$ are chaotic, and not in the interior of the m-set.
 A: I found this link on the web, which agrees with my answer, and has foototes with a proof.
https://sites.google.com/site/fabstellar84/fractals/real_chaos
This page is about an investigation that I conducted on the Mandelbrot Needle... my initial hypothesis about the needle turned out to be false...
... The Mandelbrot Needle is the part of the Mandelbrot Set between the Myrberg-Feigenbaum point and c == -2. Also, for each window of non-chaotic behavior in the chaotic region of the Logistic map, there is a miniature copy of the Mandelbrot Set in the Mandelbrot Needle.
I hypothesized that the chaotic region of the Logistic map was made completely out of an infinite number of tiny little windows of non-chaotic behavior. If that were true, then the set of all chaotic parameters would have a Lebesgue measure of 0, meaning it would be somewhat similar to the Cantor set. And, the Mandelbrot Needle would be made completely out of tiny little Mandelbrot Set copies aligned back to front. After investigating the Mandelbrot Needle for a week, I learned that back in 1981, M. V. Jacobson proved that the chaotic parameters in the chaotic region have a finite Lebesgue measure. This means my hypothesis is completely false!
Although the chaotic parameters in the Logistic map have a finite Lebesgue measure, they are infinitely discontinuous, which was proved by J. Graczyk and G. Swiatek. This means that if you examine any interval of the chaotic region, no matter how small, there will always be windows of non-chaotic behavior. As a result, if you take a picture of the Mandelbrot Needle anywhere at any magnification, there will always be miniature Mandelbrot Set copies in the picture. This means that the chaotic points in the Mandelbrot Needle form a so called "fat" fractal. The chaotic point set is somewhat similar to a Smith-Volterra-Cantor set. But, instead of removing predetermined length pieces from line segments, you make the set by removing all points from the Mandelbrot Needle that are found in the interiors of the miniature Mandelbrot Set copies.
M. V. Jacobson, Commun. Math. Phys. 81, 39 (1981).
J.Graczyk, G.Swiatek, Hyperbolicity in the Real Quadratic Family, Report no. PM 192 PennState (1995).
