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 Pentation fractal Daniel Fellow Posts: 206 Threads: 70 Joined: Aug 2007 07/22/2022, 04:14 AM This following fractals appear to be the most popular piece of mathematics I've done. Nomenclature is a bit of a problem as there is no standard way to name such fractals that are Mandelbrot sets where the iterated quadradic equation is replaced by power towers. So I think of them as tetration Mandelbrot fractals. Now I want to show what the pentation Mandelbrot set would look like, the pentation version of the above fractals. Big question is what tetration extensions should be used. I believe the method I developed for extending tetration to complex numbers would be a good candidate. My professional background is as a full stack developer, not a mathematician. So I'm in a good position to write code using my Taylor's series method. I expect I will try to write code for both Julia and Pari. My method has been seen as being based on Schroeder's function and for almost all complex values my method should produce the same results. Well, wouldn't that be appropriate to create pentation fractals? Of course what would be really cool would be software implementations of all the major methods of extending tetration. It might be a quick way of identifying which methods produce the same results. What do folks think, is complex based tetration appropriate for creating pentation fractals in the complex plane? JmsNxn was correct when I asked how to directly computer pentation fixed points and he pointed out what I meant to ask for was computing the tetration fixed points. What I currently need is a software approach to computing tetration fixed points because my Taylor's series method is based on fixed points. I figure I can either calculate fixed points by brute force if needed or use the fixed point associated with a pixel to begin a close approximation of the fixed point of the adjacent pixels. Thoughts? Daniel JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 07/22/2022, 05:23 AM First off, beautiful fractals as always, Daniel. I wish I understood in greater detail how you constructed them, but I'm a naive programmer. I'm better at moving hex code and numbers around; I suck at describing detailed graphing protocols/systems of construction. I'd also like to add that this is a very very deep problem. Even if we just think about Schroder, we have to choose a fixed point, create an iteration, then choose a fixed point of that iteration. Your method isn't exactly Schroder's method, it's closer to the regular iteration method; I have said though, that it's expressible through Schroder. So, the bell polynomial method (your method as I understand it) is similar to Gottfried's; and this generally creates Schroder's method about a fixed point; but a tad different--especially because it handles neutral fixed points. So, as I see it, you have two paths ahead of you. Take the regular iteration route, where Kouznetsov develops how to create pentation for pretty much every base. This is very similar to what Bo talked about with his Taylor expansions; but a full on book describing the regular iteration. Which incidentally, is just an alternative language for the bell-polynomial language (for the most part). Or, if you want to make graphs with $$b \in (1,\eta)$$, and all the hyper operators, and for real values, and learn how these iterations would work--keep talking to me. I assume you chose the first route. I highly suggest Kouznetsov's text book. If you Pm me I'll send you the link to the down load.  He writes about all sorts of iteration theory, and it's done through regular iteration. He has a good long section on pentation which I'm sure you'd love Daniel Fellow Posts: 206 Threads: 70 Joined: Aug 2007 07/22/2022, 09:11 AM (This post was last modified: 07/22/2022, 01:02 PM by Daniel.) (07/22/2022, 05:23 AM)JmsNxn Wrote: First off, beautiful fractals as always, Daniel. I wish I understood in greater detail how you constructed them, but I'm a naive programmer. I'm better at moving hex code and numbers around; I suck at describing detailed graphing protocols/systems of construction. I'd also like to add that this is a very very deep problem. Even if we just think about Schroder, we have to choose a fixed point, create an iteration, then choose a fixed point of that iteration. Your method isn't exactly Schroder's method, it's closer to the regular iteration method; I have said though, that it's expressible through Schroder. So, the bell polynomial method (your method as I understand it) is similar to Gottfried's; and this generally creates Schroder's method about a fixed point; but a tad different--especially because it handles neutral fixed points. So, as I see it, you have two paths ahead of you. Take the regular iteration route, where Kouznetsov develops how to create pentation for pretty much every base. This is very similar to what Bo talked about with his Taylor expansions; but a full on book describing the regular iteration. Which incidentally, is just an alternative language for the bell-polynomial language (for the most part). Or, if you want to make graphs with $$b \in (1,\eta)$$, and all the hyper operators, and for real values, and learn how these iterations would work--keep talking to me. I assume you chose the first route. I highly suggest Kouznetsov's text book. If you Pm me I'll send you the link to the down load.  He writes about all sorts of iteration theory, and it's done through regular iteration. He has a good long section on pentation which I'm sure you'd love How about I take both routes. As far as graphing the higher hyper operators go, I'm quite interested. Define the hyper etas $$\eta_k$$ with $$\eta=\eta_2$$ as the parabolic case of $$\eta_k \uparrow^k x$$ where the multiplier is 1 and described by the Abel functional equation. I have the following conjectures. $$\lim\limits_{k \to \infty}\eta_k=2$$ also that for $$b \in (1,\eta]$$ $$j b \uparrow^k x$$ I call these "slack" functions because they grown slower without bound. $$\lim\limits_{k \to \infty}b\uparrow^k x = b$$ Daniel JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 07/24/2022, 01:27 AM (This post was last modified: 07/24/2022, 01:28 AM by JmsNxn.) (07/22/2022, 09:11 AM)Daniel Wrote: (07/22/2022, 05:23 AM)JmsNxn Wrote: First off, beautiful fractals as always, Daniel. I wish I understood in greater detail how you constructed them, but I'm a naive programmer. I'm better at moving hex code and numbers around; I suck at describing detailed graphing protocols/systems of construction. I'd also like to add that this is a very very deep problem. Even if we just think about Schroder, we have to choose a fixed point, create an iteration, then choose a fixed point of that iteration. Your method isn't exactly Schroder's method, it's closer to the regular iteration method; I have said though, that it's expressible through Schroder. So, the bell polynomial method (your method as I understand it) is similar to Gottfried's; and this generally creates Schroder's method about a fixed point; but a tad different--especially because it handles neutral fixed points. So, as I see it, you have two paths ahead of you. Take the regular iteration route, where Kouznetsov develops how to create pentation for pretty much every base. This is very similar to what Bo talked about with his Taylor expansions; but a full on book describing the regular iteration. Which incidentally, is just an alternative language for the bell-polynomial language (for the most part). Or, if you want to make graphs with $$b \in (1,\eta)$$, and all the hyper operators, and for real values, and learn how these iterations would work--keep talking to me. I assume you chose the first route. I highly suggest Kouznetsov's text book. If you Pm me I'll send you the link to the down load.  He writes about all sorts of iteration theory, and it's done through regular iteration. He has a good long section on pentation which I'm sure you'd love How about I take both routes. As far as graphing the higher hyper operators go, I'm quite interested. Define the hyper etas $$\eta_k$$ with $$\eta=\eta_2$$ as the parabolic case of $$\eta_k \uparrow^k x$$ where the multiplier is 1 and described by the Abel functional equation. I have the following conjectures. $$\lim\limits_{k \to \infty}\eta_k=2$$ also that for $$b \in (1,\eta]$$ $$j b \uparrow^k x$$ I call these "slack" functions because they grown slower without bound. $$\lim\limits_{k \to \infty}b\uparrow^k x = b$$ So, I cannot help you with the indifferent case, as that went beyond my purview. But: $$\lim_{k\to\infty} b \uparrow^k x = b\,\,\text{for}\,\, \Re(x) > 0\\$$ And your identity $$b \uparrow^{k-1} x > b \uparrow^{k} x$$ is absolutely true. Though you should start your index at $$k=2$$, because multiplication kinda throws a wrench in the gear and violates this rule (k=0). To be fair, I have never graphed these functions, because the recursion required becomes absolutely bonkers. I had originally found the solution using integral transforms and fractional calculus, so if you are efficient at calculating integrals, be my guest to evaluate those formulas. The much more salient way is to just calculate Schroder functions, which is unfortunately a very very time consuming process. But, everything works out very well with the bounded analytic hyper operators--"slack" functions. There exists a decreasing sequence of functions $$1 < \omega_k(\alpha) < e$$ for $$k \ge 1$$ and a decreasing sequence of functions $$0 < \lambda_k(\alpha) < 1$$ and a decreasing bilateral sequence of functions $$a^k_n(\alpha) > 0$$. Such that: $$\alpha \uparrow^k z = \omega_{k-1}(\alpha) - \sum_{n=1}^\infty a_n^k(\alpha) \lambda_k(\alpha)^z\\$$ This can define a recursive process where if you have: $$\alpha \uparrow^{k-1} z\\$$ Then: \begin{align} \omega_{k-1}(\alpha) & =\alpha \uparrow^{k-1}\omega_{k-1}(\alpha)\\ \lambda_k(\alpha) &= \frac{d}{dz}\Big{|}_{z=\omega_{k-1}}\alpha \uparrow^{k-1} z\\ \end{align} Where upon, the $$a_n$$ can be found using a Taylor expansion of the functional equation (which can be a bit tedious). Note, this isn't the way I constructed them, but it is equivalent. I'd suggest taking a look at this paper (it's pretty short, but that's because I wanted it to be efficient, it's a little rough around the edges). The trouble is that it's not conducive to computation because it's a whooooollllleeeee mess of mellin transforms, lol. https://arxiv.org/pdf/2106.03935.pdf This paper is based on an undergraduate thesis I did at U of T about 7 years ago, I just cleaned it up and shortened it a good amount. bo198214 Administrator Posts: 1,595 Threads: 101 Joined: Aug 2007 07/24/2022, 02:37 PM (This post was last modified: 07/24/2022, 02:47 PM by bo198214.) Just as a background, let me give a short overview about the mathematically founded (as opposed to numerically good looking) approaches: There is basically two constructions: at one fixed point, and at two fixed points. Iteration at one fixed point - regular iteration:   The idea is here to make the iterated functions analytic/holomorphic at that fixed point (that is where the word "regular" comes from - in those times one used "regular" as a synonym for what we today call analytic/holomorphic) or at least asymptotically analytic (in the parabolic case). And one can show for the typical cases (hyperbolic/parabolic) that an (asymptotically) analytic solution exists as well as that it is unique by this demand - the method for hyperbolic fixed points is typically called Schröder iteration. This can be approached from the side of (formal) power series, but also there are (proven) limit formulas. The downside is that if the t-iterate is analytic at one fixed point it is not at any other for most t. Iteration at two fixed points - crescent iteration (this term is my invention, do you have better ones, or will we make it our standard term?):   Kneser's approach is a particular construction of the more general Perturbed Fatou Coordinates. For the latter under certain conditions one can show that, there exists a holomorphic Abel function on a fundamental region/crescent  (in the case of Kneser it is constructed via the Riemann mapping theorem, in the case of Perturbed Fatou-Coordinates the measurable Riemann mapping theorem is used.) and it is unique by demanding that it is injective on the fundamental region. Another uniqueness theorem comes from Paulsen, which states that is unique if sequences tending to the first fixed point are mapped by the Abel function to sequences with imaginary part going to infinity, and for the second fixed point to negative infinity (though it is only proven in the context of exponentials, I think it could be a general criterion). The corresponding iteration $$f^t(z) = \alpha^{-1}(t+\alpha(z))$$ is not analytic at both fixed points. While the regular iteration is quite well studied and there are practical formulas for iteration, for the crescent iteration it is a bit more vague (due to the difficulties numerically handling (measurable) Riemann mappings). But typically people identify the Levenstein (Sheldon), Kouznetsov and the Paulsen numerical methods with the crescent iteration. Both methods are not identical, we know that for $$b>e^{1/e}$$ regular iteration at one of the primary fixed points is not real valued on the real axis, and vice versa we know (by computation of Paulsen) that the (b-continued) crescent iteration for bases $$b0} / B \times \mathcal{A}_0/C \to \mathcal{A}_0\\  Where here \(\mathcal{A}_0$$ is the immediate basin about the fixed point $$2$$ of $$f(z) = \sqrt{2}^z$$--where $$B$$ is a branch cut, and $$C$$ is another branch cut spawning from the fixed point $$2$$. So this wouldn't be holomorphic at $$z= 2$$. This looks a lot like the regular iteration, but it perturbs the regular iteration so that instead of period $$-2\pi i/\log\log(2)$$ it now has period $$2 \pi i$$. And by consequence adding a lot fractal branching and singularities. You can construct this with a theta mapping on the original regular iteration. This is sort of just an off shoot of the regular iteration though, and isn't related to the theta mappings of crescent iteration. It is also holomorphic on a smaller domain than the regular iteration or the crescent iteration. Just adding that theta mappings can make uncountably many iterations--not just the crescent iteration. Though it is by far the most important one. Daniel Fellow Posts: 206 Threads: 70 Joined: Aug 2007 07/26/2022, 10:50 AM Ask and you shall receive. Thanks Bo, this is very helpful! I'm impressed with what can be done with two fixed points. In trying to make a contribution to real tetration I considered what could be done with the entire set of fixed points. Couldn't it be possible to construct real tetration using a countable infinity of fixed points to fix the interval between $$^nb \textrm{ and } ^{n+1}b$$ at a countable infinity of points on the real line and thus fixing all points on the real line. As far as regular iteration from a single single fixed point not being holomorphic, I did a test which indicated that the regular iteration of the exponential map provides not only the correct multiplier at the fixed point, it predicted the location of the next fixed point and it's multiplier. Of course I would need to recreate the experiment if folks were to believe me. My goal in generating fractals is to make the underlying mathematics transparent. My professional background is as a programmer, so I expect I can write terse clean code that is easy to read and understand. I trying doing some stuff with Julia but most of the packages were broken. On the other hand with Pari-GP I found the code for computing Bell polynomials which I can make good use of. Daniel JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 07/28/2022, 12:39 AM (07/26/2022, 10:50 AM)Daniel Wrote: Ask and you shall receive. Thanks Bo, this is very helpful! I'm impressed with what can be done with two fixed points. In trying to make a contribution to real tetration I considered what could be done with the entire set of fixed points. Couldn't it be possible to construct real tetration using a countable infinity of fixed points to fix the interval between $$^nb \textrm{ and } ^{n+1}b$$ at a countable infinity of points on the real line and thus fixing all points on the real line. As far as regular iteration from a single single fixed point not being holomorphic, I did a test which indicated that the regular iteration of the exponential map provides not only the correct multiplier at the fixed point, it predicted the location of the next fixed point and it's multiplier. Of course I would need to recreate the experiment if folks were to believe me. My goal in generating fractals is to make the underlying mathematics transparent. My professional background is as a programmer, so I expect I can write terse clean code that is easy to read and understand. I trying doing some stuff with Julia but most of the packages were broken. On the other hand with Pari-GP I found the code for computing Bell polynomials which I can make good use of. Regular iteration at a single fixed point is absolutely holomorphic..? I'm confused. What was the problem, is that if we perform regular iteration about $$z \approx L$$ the fixed point, then it won't be real valued on the real line, but it's still holomorphic. Also, if you perform the regular iteration on the real line, as Kouznetsov refers to it, you get the crescent iteration, which is again holomorphic. The absolute power of regular iteration is that it's always holomorphic. Perhaps you got confused by what I wrote, using $$\theta$$ mappings you can perturb the regular iteration, but then it is no longer holomorphic in a neighborhood hood of the fixed point, is that what you meant to say? bo198214 Administrator Posts: 1,595 Threads: 101 Joined: Aug 2007 07/31/2022, 08:31 PM (07/28/2022, 12:39 AM)JmsNxn Wrote: Regular iteration at a single fixed point is absolutely holomorphic..? I'm confused. What was the problem, is that if we perform regular iteration about $$z \approx L$$ the fixed point, then it won't be real valued on the real line, but it's still holomorphic. Also, if you perform the regular iteration on the real line, as Kouznetsov refers to it, you get the crescent iteration, which is again holomorphic. The absolute power of regular iteration is that it's always holomorphic. Perhaps you got confused by what I wrote, using $$\theta$$ mappings you can perturb the regular iteration, but then it is no longer holomorphic in a neighborhood hood of the fixed point, is that what you meant to say? Now I am confused JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 08/01/2022, 10:56 PM Lmao! Sorry, Bo! It appears you and Kouznetsov refer to regular iteration differently! That's what's screwing up. Correct me if I'm wrong, but when you refer to regular iteration about $$L$$ for base $$b= e$$, let's say, you are referring to the Taylor solution about $$z\approx L$$; I.e: the iteration that looks like: $$f^{\circ t}(z) = L + \lambda^t(z-L) + O(z^2)\\$$ This function is entire in $$t$$, and there exists some $$t_0$$ such that $$f^{t_0}(z_0) = 1$$, thereby, we can make a tetration $$f^{t+t_0}(z_0)$$ which is a superfunction to $$f(z) = \exp(z)$$, but it is not real valued. $$f^{1/2 + t_0} \not \in \mathbb{R}$$. I.e. it's a tetration function, but it's not real valued Kouznetsov refers to the regular iteration about $$b = e$$, as the Kneser iteration. This is based on his manner of constructing the function. Which I'm sure you're well aware of. His manner on the real line is real valued, and is what he always calls regular iteration. I think I've been conflating regular iteration and Kouznetsov regular iteration. I apologize again, English is my first language, but I'm terrible with vernacular. So in that sense, your english is better than mine, lol. Woe is the trouble of standardizing terminology, lol. I apologize, and if you're still confused, please please please explain what I've said to make you confused. Regards. « Next Oldest | Next Newest »

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