Complex Tetration, to base exp(1/e)

[Ember Edison]

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

[sheldonison]

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. 

[Ember Edison]

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.

[sheldonison]

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)

[Ember Edison]

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.

[Ember Edison]

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).

[bo198214]

Sheldon, I am glad you helped out on this question, I am - like always - in limited time mode.

[sheldonison]

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."