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 [MSE] Shape of orbit of iterations with base b on Shell-Thron-region Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 08/17/2019, 02:08 PM (This post was last modified: 08/17/2019, 02:11 PM by Gottfried.) In MSE I've started a discussion providing also some pictures on the properties of shapes of the orbits 0 \to 1 \to b \to b^b \to ... and in which way there occurs "divergence". Such "divergence" has been proven for classes of bases b on boundary of the Shell-Thron-region by I.N.Baker & Rippon in 1983 and according to Sheldonison has been thoroughly        investigated by J.C.Yoccoz .                       See here: https://math.stackexchange.com/q/3323851                 Possibly I can take some pictures here, but the transfer of the mathjax-formulae is a visual horror for me... Conclusions I might transfer to this place if the discussion has some sufficient finishing.                      Gottfried Additional material:                  http://go.helms-net.de/math/tetdocs/_equ...quator.pdf       an earlier article of mine as first approximation to the problem                       https://math.stackexchange.com/q/1820410/1714                                         https://math.stackexchange.com/questions...it-chaotic Gottfried Helms, Kassel sheldonison Long Time Fellow Posts: 626 Threads: 22 Joined: Oct 2008 08/17/2019, 02:28 PM (This post was last modified: 08/17/2019, 02:38 PM by sheldonison.) (08/17/2019, 02:08 PM)Gottfried Wrote: In MSE I've started a discussion providing also some pictures on the properties of shapes of the orbits 0 \to 1 \to b \to b^b \to ... and in which way there occurs "divergence". Such "divergence" has been proven for classes of bases b on boundary of the Shell-Thron-region by I.N.Baker & Rippon in 1983 and according to Sheldonison has been thoroughly        investigated by J.C.Yoccoz .                      Hey Gottfried, The unbounded values in the orbits of $\exp_b^{[\circ n]}(0)$ for these values of b in your MSE post is really cool. $\phi=\frac{\sqrt{5}+1}{2};\;\;\;\;\lambda=\exp\left(\frac{2\pi i}{\phi}\right);\;\;\;b=\exp\left(\lambda\exp(-\lambda)\right);\;\;\;l=\exp(\lambda);\;\;\;l=b^L$ Just a clarification on $\lambda=\frac{2\pi i}{c}$ multiplies with c real, and/or rational.  I think Yoccoz's work primarily involved iterating $z \mapsto \lambda z + z^2$ and his proof of the sharp convergence; non-convergence of the Schroeder $\Psi(f(z))=\lambda\Psi(z)$ when c is an irrational number.  Yoccoz proved that if c has a continued fraction that doesn't misbehave super badly, the the $\Psi(z)$ series converges, and that $\Psi(z)$ doesn't converge if the continued fraction misbehaves, see https://en.wikipedia.org/wiki/Brjuno_number  I don't know if Yoccoz's proof has been extended to proof it also applies to iterating exponentials, but the conjecture would be that it applies.  So when you find Siegel disc's pictures, they typically like to use a value of $c=\phi=\frac{\sqrt{5}+1}{2}$ since the golden ratio has the most ideally behaved continued fraction so it tends to show nice easily computable fractal behavior. I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_disc - Sheldon Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 08/18/2019, 08:17 AM (08/17/2019, 02:28 PM)sheldonison Wrote: I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_discHi Sheldon - in the meantime I've aggregated more data for plots based on c=silver constant, and I've to recompute accordingly data for the other types of c - just to apply the new insights to the small paper that I'd linked to, and hopefully extend that small paper towards becoming a better "catalogue".             However, I need a break for a couple of days and surely cannot be much productive in this matter. A question which is coming up at the moment is how to characterize the interior of the $z_0=1$ fractal shape, say the set of curves produced by $0.5 < z_0 < 0.99$, somehow like a gradient-field, displaying little arrows instead of dots, perhaps including directions of fractional iteration-height (as far as this might be meaningfully applicable). That's just the desire to embed the observation into known phenomena in other areas, hopefully of the physical world.            Moreover, I think I've to meditate first a bit about that Yoccoz-work and Siegel-discs which you have directed me to and what this shall give me for the understanding of the whole phenomen. Cordially -   Gottfried Gottfried Helms, Kassel sheldonison Long Time Fellow Posts: 626 Threads: 22 Joined: Oct 2008 08/18/2019, 02:22 PM (08/18/2019, 08:17 AM)Gottfried Wrote: (08/17/2019, 02:28 PM)sheldonison Wrote: I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_discHi Sheldon - in the meantime I've aggregated more data for plots based on c=silver constant, and I've to recompute accordingly data for the other types of c - just to apply the new insights to the small paper that I'd linked to, and hopefully extend that small paper towards becoming a better "catalogue".             However, I need a break for a couple of days and surely cannot be much productive in this matter. A question which is coming up at the moment is how to characterize the interior of the $z_0=1$ fractal shape, say the set of curves produced by  $0.5 < z_0 < 0.99$, somehow like a gradient-field, displaying little arrows instead of dots, perhaps including directions of fractional iteration-height (as far as this might be meaningfully applicable). That's just the desire to embed the observation into known phenomena in other areas, hopefully of the physical world.            Moreover, I think I've to meditate first a bit about that Yoccoz-work and Siegel-discs which you have directed me to and what this shall give me for the understanding of the whole phenomen. Cordially -   Gottfried Gottfried, I just read your equater.pdf; very nice.  There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.  Considering what happens as the continued fraction becomes less well behaved is more difficult, as the irrational number gets closer and closer to behaving like a rational number.  Most cases with an irrational multiplier still have a Schroeder function so there is still an infinite number of copies where $z_n$ gets arbitrarily close to zero and there is a logarithmic singularity at zero so $z_{n-1}$ gets arbitrarily large so the fractal should still be unbounded, but it might take an uncountable number of iterations to show that behavior...  The nice thing about using a multiplier of the golden ratio is that one can actually compute the Schroeder function and get good numerical results for the orbits for  $0.5 < z_0 < 0.99$, which gives the same gradient curves as iterating $z\mapsto b^z$.   The Schroder function has a 1-1 mapping from the unbounded fractal to a circle.  One can also study the Julia set for these iteration mappings, but I haven't done it.  There are an infinite number of other pre-images of the fractal since the $b^z$ exponential function has a period of $\frac{\2\pi i}{\ln b}$.   Thanks for posting a delightful topic, both on MSE, and here. - Sheldon « Next Oldest | Next Newest »

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