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Hi,
I was reading the article[1] and i can't reproduce it in mathematica.
I need some help, and very much need some code.
Edison
[1] https://arxiv.org/abs/1105.4735
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05/07/2019, 04:17 PM
(This post was last modified: 05/07/2019, 04:46 PM by sheldonison.)
(05/05/2019, 11:38 PM)Ember Edison Wrote: Hi,
I was reading the article[1] and i can't reproduce it in mathematica.
I need some help, and very much need some code.
Edison
[1]https://arxiv.org/abs/1105.4735
Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating \( z\mapsto\exp(z)1 \) which is congruent to iterating \( \eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}1; \)
The asymptotic series for the Abel equation for iterating z is given by equation 18. I have used this equation to also get the value of Tetration or superfunction for base \( \eta=\exp(1/e) \), by using a good initial estimate, and then Newton's method. If you use parigp or are interested in downloading parigp, I can post the parigp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.
\( \alpha(z)=\frac{2}{z}+\frac{1}{3}\log(\pm z)\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3\frac{71}{435456}z^4+... \)
If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.
\( \alpha(z)\approx\alpha(\exp(z)1)+1 \)
To get arbitrarily accurate results, we iterate \( z\mapsto\exp(z)1 \) enough times or for the repellilng flower, we can iterate \( z\mapsto\log(z+1) \) enough times so that z is small and the asymptotic series works well.
 Sheldon
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(05/07/2019, 04:17 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi,
I was reading the article[1] and i can't reproduce it in mathematica.
I need some help, and very much need some code.
Edison
[1]https://arxiv.org/abs/1105.4735
Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating \( z\mapsto\exp(z)1 \) which is congruent to iterating \( \eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}1; \)
The asymptotic series for the Abel equation for iterating z is given by equation 18. I have used this equation to also get the value of Tetration or superfunction for base \( \eta=\exp(1/e) \), by using a good initial estimate, and then Newton's method. If you use parigp or are interested in downloading parigp, I can post the parigp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.
\( \alpha(z)=\frac{2}{z}+\frac{1}{3}\log(\pm z)\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3\frac{71}{435456}z^4+... \)
If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.
\( \alpha(z)\approx\alpha(\exp(z)1)+1 \)
To get arbitrarily accurate results, we iterate \( z\mapsto\exp(z)1 \) enough times or for the repellilng flower, we can iterate \( z\mapsto\log(z+1) \) enough times so that z is small and the asymptotic series works well. Yes, I need it!
I think just has something wrong when i am definiting function. Source code will be helpful.
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05/08/2019, 04:50 PM
(This post was last modified: 05/08/2019, 05:38 PM by sheldonison.)
(05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it!
I think just has something wrong when i am definiting function. Source code will be helpful. [attachment=1343]
Code: \r baseeta.gp
initeta(); /* initeta initializes kecalle series; 25terms */
slog1=slogeta(1); /* renormalize so slog(1)=0; slog1=3.029297214418036; */
z=slogeta(2.5) /* 21.038456088895745460253062718325504556; */
ploth(t=1.5,25,sexpeta(t)); /* plot of sexpeta; sexpeta(0)=1 */
z2=invcheta(100) /* 0.79336896191958487417879655443666434028 */
z1=invcheta(4); /* 4.5049005907984782975089673142337641018 */
ploth(t=z1,z2,cheta(t)); /* plot of upper superfucntion of eta */
z=slogeta(I) /* 1.217279555798763 + 0.5193692007946583*I */
z=invcheta(I) /* 1.808671078843811 + 1.565868985090261*I */
z=cheta(1+I) /* 6.501975132474055 + 4.920389603877520*I */
baseeta.gp (Size: 6.4 KB / Downloads: 543)
 Sheldon
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(05/08/2019, 04:50 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it!
I think just has something wrong when i am definiting function. Source code will be helpful.
Code: \r baseeta.gp
initeta(); /* initeta initializes kecalle series; 25terms */
slog1=slogeta(1); /* renormalize so slog(1)=0; slog1=3.029297214418036; */
z=slogeta(2.5) /* 21.038456088895745460253062718325504556; */
ploth(t=1.5,25,sexpeta(t)); /* plot of sexpeta; sexpeta(0)=1 */
z2=invcheta(100) /* 0.79336896191958487417879655443666434028 */
z1=invcheta(4); /* 4.5049005907984782975089673142337641018 */
ploth(t=z1,z2,cheta(t)); /* plot of upper superfucntion of eta */
z=slogeta(I) /* 1.217279555798763 + 0.5193692007946583*I */
z=invcheta(I) /* 1.808671078843811 + 1.565868985090261*I */
z=cheta(1+I) /* 6.501975132474055 + 4.920389603877520*I */
Thank you! I am reading.
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(05/08/2019, 04:50 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it!
I think just has something wrong when i am definiting function. Source code will be helpful.
Code: \r baseeta.gp
initeta(); /* initeta initializes kecalle series; 25terms */
slog1=slogeta(1); /* renormalize so slog(1)=0; slog1=3.029297214418036; */
z=slogeta(2.5) /* 21.038456088895745460253062718325504556; */
ploth(t=1.5,25,sexpeta(t)); /* plot of sexpeta; sexpeta(0)=1 */
z2=invcheta(100) /* 0.79336896191958487417879655443666434028 */
z1=invcheta(4); /* 4.5049005907984782975089673142337641018 */
ploth(t=z1,z2,cheta(t)); /* plot of upper superfucntion of eta */
z=slogeta(I) /* 1.217279555798763 + 0.5193692007946583*I */
z=invcheta(I) /* 1.808671078843811 + 1.565868985090261*I */
z=cheta(1+I) /* 6.501975132474055 + 4.920389603877520*I */
Sorry, I think we need penteta, ipenteta, hexeta, ihexeta in fatou.gp because pentinit(etaB) is use sexpinit(etaB).
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Sheldon, I am glad you helped out on this question, I am  like always  in limited time mode.
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(08/13/2019, 08:27 PM)bo198214 Wrote: Sheldon, I am glad you helped out on this question, I am  like always  in limited time mode.
Thanks you for your kind comments Henryk. It has been a pleasure to learn more and more about the start of the art of complex dynamics. I still don't quite understand all of Shishikura's papers, "Bifurcation of parabolic fixed points", an in particular, how Shishikura used perturbed fatou coordinates in his other proofs. "In fact, in [Sh1], such a notion was already introduced and its second iterate played a crucial role in the proof of the fact that a parabolic point can be perturbed so that the Hausdorff dimension of the Julia set is arbitrarily close to 2."
 Sheldon
