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.


2 Answers 2


My goal was to have someone else tell my why my intuition was correct, along with a suggested book/link with a proof in it. But I think the key to answering the question is to analyze the behavior of the m-set near a Misiurewicz on the real axis. I think this can be used to show that a random point on the real axis $[-2,0.25]$ has a probability of less than one of being in the interior of the m-set, since I can show that in the neighborhood of Misiurewicz points on the real axis, that probability can get arbitrarily small as you zoom in on the Misiurewicz point.

This isn't a proof because I assume a lemma that the length of the mini-Mandelbrot is proportional to the square of the reciprocal of the slope of $z_n(x)$ at the real axis, where the location of the bugs are determined by $z_n(x)=0$. This explains why, as you zoom in to the Misiurewicz point, the mini-Mandelbrot bugs get smaller in a very predictable way. The Misiurewicz point I analyzed is C=-1.54368901269207636157, which is the real root of $z^3 + 2z^2 + 2z + 2$, generated by iterating starting with $z_1(x)=x$ and $z_{n+1}\mapsto z_n^2+x$, and solving $z_3=z_2$. Lets look at the behavior on the m-set on the real axis in the neighborhood of this Misiurewicz point; the first graph goes from -1.645 to -1.47, which contains two main Mandelbrot bugs, with period5 on the left, and period6 on the right. We can solve numerically for the hyperbolic centers of these two bugs by solving the algebraic equations, $z_5=0$ and solving $z_6=0$. In the window of interest, there is one solution for each. I marked the solutions the six largest bugs, for $z_5=0$, $z_6=0$, $z_7=0$ and $z_8=0$, $z_9=0$, and $z_{10}=0$. There are two additional solutions for $z_9=0$ and $z_{10}=0$, so I marked the closer in slightly smaller bug as well, as z9b and z10b. I also marked the Misiurewicz point, in yellow. Below this image, I show a similar image, zoomed in two steps towards the bugs z9b and z10b, showing the largest mini-Mandelbrots in that zoomed in region. These mini-Mandelbrots are observably much smaller. But how much smaller? And why?

mset from -1.645 to -1.47 The next image is zoomed in by two steps, between the main bugs with period9=z9b above and period10=z10b above. mset z9 to z10

To start, lets look at two more images. The first shows the region from z9b to z10b; same as the second Mandelbrot image. I show the graph of $z_9(x)$, $z_{10}(x)$, $z_{11}(x)$, $z_{12}(x)$, $z_{13}(x)$, and $z_{14}(x)$, and the real axis. The zeros of these six graphs are the locations of the biggest mini-Mandelbrots in the zoomed in region. Next, I show another graph zoomed in two more steps, zooming in on the z13 to z14 bug region. The second graph is close to converging, and similar graphs can be made, zooming in arbitrarily. I marked the mini-Mandelbrot locations as zero crossings. Also, notice that all of the curves intersect at the Misiurewicz point, which is also marked. One can imagine an infinite number of these curves, of which only the first six with zero-crossings with the smallest slopes are shown in each graph. The scale factor converges to approximately 0.35491x, as you zoom in. The scale factor can be computed by generating the Taylor series for $z_n(x)$ for x centered at the Misiurewicz point, and generating the ratio of the derivative to the derivative of $z_{n+2}(x)$. With a little algebra this can be shown to be equal to the reciprical of $4z_3^2$, by developing the Taylor series at c, and iterating $z_{n+1}=z_n^2+x+c$, with $z_0=0$. plot from z9 to z10 zooming in another two steps, self similarity takes over. plot from z13 to z14

I think these last two graphs give more of a sense of the self-similar behavior of the Mandelbrot set as you zoom in; but the graphs don't show the size of the mini-Mandelbrots, only the location of the mini-Mandelbrots, which are the zero crossings. Now, I stated an unproven lemma for the length of a mini-Mandelbrot $l(z_n=0) \approx \frac{1}{(z_n')^2}$. My rational is that at the hyperbolic center, we know $z_n(x)$ has a zero. Next, a Taylor series for $z_n$ is centered at the hyperbolic center itself, $z_n(x) = a_1 x+ a_2 x^2 + ....$. A similar Taylor series, also centered at the same hyperbolic center is generated for 2n, which by definition also has a zero at the hyperbolic center; with a little algebra, we can show it has the same first derivative, $z_{2n}(x) = a_1 x+ b_2 x^2 + b_3 x^3....$. Numerically, I observe that $b_2 \approx k a_1^3$, where k is a number greater than 1.5 and less than 2. $z_{2n}$ has two zeros; the second zero is very close to the first zero, and gives the length of the mini-Mandelbrot since it is the distance between the main cardioid, and the 2x cardioid. Since the Taylor series for $z_{2n}$ is centered at the first zero, we need an estimate for the second zero which is $z_{2n 2} \approx -\frac{a_1}{b_2} \approx \frac{a_1}{1.8a_1^3} \approx \frac{0.6}{a_1^2}$.

More pictures might help, showing the mini-Mandelbrot cardioid, and graphs of the two functions, $z_n$ and $z_{2n}$; and there's a lot of algebra I didn't show, that explains the lemma, but doesn't prove it. The lemma then predict that each of the bugs would scale by $\approx \frac{1}{(a_1)^2}$, where $a_1$ is the derivative of $z_n$ at the zero crossing. This matches the numerical data closely. The window size is scaling by approximately 0.35491x each iteration, but the bug length scaling by the square, 0.12596x, so relatively speaking, the bugs are getting smaller by 0.35491x each iteation. If the relative length of all the bugs in the window gets smaller by approximately 0.354921 each iteration, than the sum of the lengths of all of the bugs in the window also get smaller by the same ratio, and the total length of the bugs, relative to the window size, goes to zero as you zoom in. Assuming my logic is correct, than most of the points on the real axis, between $[-2,-1.4011552]$ are neither inside hyperbolic components, nor are they Misiurewicz points. They are simply random points, with zero width, on the real axis filigree that is the boundary of the m-set. There will be a hyperbolic component arbitrarily close to such random points, but I don't think these random points are adjacent to a hyperbolic component.

Numerical data showing the size of the six biggest bugs, in the window,
relative to the window size, as you zoom in by 2.81761x each iteration.

window -1.64500000000000000000 to -1.47000000000000000000
z5     -1.62541372512330373744     0.0378913101884657946067
z6     -1.47601464272842989752     0.0463018986609053128810
z7     -1.57488913975230096982     0.00655041322134289247183
z8     -1.52181723167125099520     0.00414679373793390113281
z9     -1.59568096343974577478     0.00119381469602047654212
z10    -1.50171683940963313793     0.00210630478418442001747

window -1.57964594647609667764 to -1.51753557148707102871
z7     -1.57488913975230096982     0.0184562130551868531476
z8     -1.52181723167125099520     0.0116838596493203506804
z9     -1.55528270076858318477     0.00276214675652270816184
z10    -1.53624327128231050351     0.00126848512492673185395
z11    -1.56362963108720118812     0.000659268351688881292083
z12    -1.53005645596148015452     0.000452102609375223546519

window -1.55645071921082531485 to -1.53440672674638312697
z9     -1.55528270076858318477     0.00778252719415849789938
z10    -1.53624327128231050351     0.00357403890898100087413
z11    -1.54790376180395502683     0.00106684073199442759725
z12    -1.54109410649138282038     0.000422764278591095275479
z13    -1.55107190019127551232     0.000289916617048784854801
z14    -1.53901837190200957968     0.000134635051612048004546

window -1.54821835199138716484 to -1.54039457691583425088
z11    -1.54790376180395502683     0.00300589278573849461110
z12    -1.54109410649138282038     0.00119116570728342799192
z13    -1.54520178169265663017     0.000393399151035709301528
z14    -1.54277519227761074816     0.000146158186626937596148
z15    -1.54636280570666495530     0.000113098359842964666488
z16    -1.54205531050306852211     0.0000447031218672100244229

window -1.54529654961969193113 to -1.54251976331242116562
z13    -1.54520178169265663017     0.00110842756050681389305
z14    -1.54277519227761074816     0.000411810147084660629635
z15    -1.54422856011952779613     0.000141929100569532803183
z16    -1.54336574221033890777     0.0000513163804751993036231
z17    -1.54464648152450039760     0.0000417496147766831044556
z18    -1.54311265996486419402     0.0000154599913144917732600

window -1.54389150716018304821 to -1.54354172740913823104
z17    -1.54388090052770975288     0.000142892198283435926925
z18    -1.54357443364491605528     0.0000510913999908299379366
z19    -1.54375717344627227934     0.0000180484964150937955160
z20    -1.54364836925566699769     0.00000642435636305903336541
z21    -1.54381025766971924338     0.00000537595932368788837315
z22    -1.54361666220344091840     0.00000192022411050607955763

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).

  • $\begingroup$ tic.itefi.csic.es/gerardo/publica/Pastor96.pdf [ Pastor96 ] = G. Pastor, M. Romera and F. Montoya, "An approach to the ordering of one-dimensional quadratic maps", Chaos, Solitons and Fractals, 7/4 (1996), 565-584. Preprint $\endgroup$
    – Adam
    Commented Mar 18, 2023 at 10:51

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