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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
Posts: 684
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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 pari-gp or are interested in downloading pari-gp, I can post the pari-gp 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 pari-gp or are interested in downloading pari-gp, I can post the pari-gp 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
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