7

In the Mandelbrot Set, c=0 has a periodicity of 1 and is surrounded by a cartoid of non-periodic points that asymptotically approach periodicty 1. Extending left along the real axis are connected lobes of respective periodicity $2^1,2^2,2^3,2^4,2^5...$ Where exactly does that series of lobes end?

MandelLobes

The center of the $2^8$-periodic lobe is -1.401153290849923, and there is a Misiurewicz Point (2-periodic after iteration 4) at -1.43036, so the end of the series of connected lobes must be somewhere between those two points.

The wiki article on Bifurcation Constants gives some detail on this $2^n$-periodic portion of the Mandelbrot but doesn't discuss the limit.

I am interested in the properties of this point more than a numerical estimate of its value. Is it transcendental, rational, a polynomial root, non-periodic, periodic, pre-periodic (Misiurewicz), a limiting case of pre-periodic?

Jerry Guern
  • 2,764
  • Are you asking about the end of the period doubling cascade? If so, I think the period 16 lobe is centered approximately at -1.3969453597045611. Once you know a few of them, you can use Feigenbaum's first constant to get a decent approximation to the next which can be used as a seed to get a higher precision approximation with Newton's method. This allows you to get a pretty good estimate of the limit. Also, why do you think there's a Misiurewicz point point at the end of all that? – Mark McClure Feb 04 '16 at 02:36
  • @MarkMcClure I did something like that and got an okay estimate, but it's always limited by machine precision. I'm more interested in the properties of that point than a more precise numerical estimate. I don't think it will be periodic or Misiurewicz but a limiting case of both. I listed a Misiurewicz point just because that was the rightmost real point I could find that I KNEW was outside the "period doubling cascade". – Jerry Guern Feb 04 '16 at 02:46
  • 1
    The estimate you have, -1.42625, does in fact yield a nearly super-attractive orbit of period 16, but it's not part of the period doubling cascade. Rather, it's the center of a baby-brot well to the left of the period doubling cascade. I agree that Misiurewicz points are certainly outside the period doubling cascade but you should be able to find them arbitrarily close by. You might find information on the $c$-value you seek under the name Feigenbaum attractor. – Mark McClure Feb 04 '16 at 02:52
  • @MarkMcClure Oops, forgot to say i fixed that and thank you. – Jerry Guern Feb 04 '16 at 02:54
  • @MarkMcClure BTW, on you comment about finding Misiurewicz Points arbitrarily close. I posted a question here about finding Misiurewicz Points (before I knew they were called that) along with the code I used. In practice, I could only find a certain number of Misi pts before solving (2^16)-order polynomials became impractical. The one I listed above was the right-most one I found, but I checked higher orders. Any thoughts on that other Question? – Jerry Guern Feb 04 '16 at 07:50
  • You might find this paper interesting: http://digital.csic.es/bitstream/10261/8916/3/Pastor02_pre.pdf "Operating with external arguments in the Mandelbrot set antenna" which uses properties of external ray tuning to mention in passing that the end of the period doubling cascade has irrational external angle, and is thus not a Misiurewicz point. So the "rightmost Misiurewicz point" is like the "smallest positive number", a kind of oxymoron. – Claude Feb 05 '16 at 10:01
  • Correct me if I'm wrong, but considering that the Mandelbrot set can be zoomed into infinitely, if you travel along the real axis, there will always be a point further along the real axis with more lobes. It may come really close to some number, but I don't think you can specify an exact number. – Byte11 Nov 27 '17 at 23:41
  • @Claude What I mentioned the right-most real-axis Misi point thatIcouldfind* above only because it serves as a left bound for the value I was looking for. I wasn't claiming it's the actual right-most Misi point, nor claiming that one exists. – Jerry Guern Dec 04 '17 at 18:59
  • @Byte11 Your reasoning about fractals is not correct here. Imagine building a snowman but setting down a sphere of diameter 1, putting a diameter 1/2 on top of it, a diameter 1/4 on top of that, and so forth to infinity. That's clearly a fractal you could zoom in on forever. But it's height assymptotically approaches 2. – Jerry Guern Dec 04 '17 at 19:02
  • Allow me to clarify, imagine you are at a point inside inside the mandelbrot set, say 1.5+0i. If you step to the left by one, you will be outside of the mandelbrot set, but if you step to the left by 0.1, you will be in it. Now pick a point close to the edge of the mandelbrot set (but still in it). Now step 0.1 over, that may carry you out of the set, but stepping by 0.01 will keep you in it. What I'm saying is that If you are in a point inside the mandelbrot set, you can always step some infinitesimally small amount and still be in the set. – Byte11 Dec 04 '17 at 22:16
  • Considering the Mandelbrot set is still complex at small scales and has miniature versions of the structure you're talking about, I don't think there is a finite number of lobes. To me, this question is like asking what's at the end of infinity. – Byte11 Dec 04 '17 at 22:18
  • @Byte11 First, I'm guessing you meant -1.5 in your example. And of course there are infinite lobes in the series I'm talking about, and no one ever disputed that. Finally, my question is simply what value an infinite series converges to, not equivalent to 'what's at the end of infinity'. I think you're not understanding some basic concepts here. – Jerry Guern Dec 04 '17 at 22:26
  • @JerryGuern Ah, okay. I didn't know you meant where it converges at. – Byte11 Dec 04 '17 at 22:31
  • You can transform the logistic map $x_{n+1} = rx_n(1-x_n)$ into the quadratic map $z_{n+1} = z_n^2+c$ by a linear change of variables. Complete the square, $x_{n+1} = r(1/4-(x_n-1/2)^2)$ and let $z_n = r(1/2-x_n)$. Then $c$ and $r$ are related by the equation $c = r(1/2-r/4)$. This means that the behavior of the Mandelbrot set along the real axis lines up with the logistic map (there is a good figure demonstrating this on Wikipedia). The 'accumulation point' of the logistic map is well known, you can find its decimal expansion at https://oeis.org/A098587, and the corresponding value of $c$ is – Daniel Widdowson Apr 21 '18 at 03:45

0 Answers0