[MSE] Shape of orbit of iterations with base b on Shell-Thron-region

[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

[sheldonison]

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

[Gottfried]

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

Hi 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

[sheldonison]

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

Hi 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.

[Ember Edison]

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.  

Hi
I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples? 

And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)? 
I see the ihex_deriv() you use are not simple.

[sheldonison]

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.  

Hi
I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples? 

And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)? 
I see the ihex_deriv() you use are not simple.

l=1/exp(1); /* the fixed point */
b=exp(-exp(1)); /* the base */
f=b^b^(l+x)-l; /* this is the f(f(x); a function with a derivative=1 and a fixed point of ~zero */
f=strip0fx(f); /* strip off the x^0 term which was only approximately zero; it is required to be exactly zero */

ecalle(f,16); /* initialize formal series for ecalle assymptotic; 16 positive terms; two neg terms; one log term */
z1=0.1; z2=b^b^(z1+l)-l; /* test example; z1=0.1; z2=f(f(0.1)); */
ecalleu(z1) /* [abel_function,abel_function_derivative]; [44.892671576806680805639501168174, */
ecalleu(z2) /* [abel_function,abel_function_derivative]; [45.892671576806680693723673529703, */

the closer z1 and z2 are to zero, the more accurate the asymptotic series; in this case the abel function is accurate to 16 decimal digits.  Notice that f has fixed point of zero, derivative of 1, and no x^2 term

f = x*  1.0000000000000000000000000000000
+x^ 2* -4.324402048806904993 E-39   /* ecalle smart enough to ignore x^2 */
+x^ 3* -1.2315093498217750378717379100958
+x^ 4*  0.83689737179948615587202206894090
+x^ 5*  0.90996916721907065130183768671435

This leads to an ecalle assymptotic form for f with two neg terms; a_2/x^2 + a_1/x; a log term, and we calculated 16 terms of the assymptotic above.
- Sheldon

[Daniel]

Hello,
I was interested in Siegal disks in the exponential map. Dr. Robert Devaney told me in no uncertain terms that the exponential map produced no Siegal disks because the exponential function is an entire function and entire functions can't produce Siegal disks. I'm sorry that I have no personal insight into the matter.
Daniel

[Ember Edison]

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.  

Hi
I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples? 

And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)? 
I see the ihex_deriv() you use are not simple.

ecalleu(z1) /* [abel_function,abel_function_derivative]; [44.892671576806680805639501168174, */
ecalleu(z2) /* [abel_function,abel_function_derivative]; [45.892671576806680693723673529703, */


- Sheldon

1.Is the sexpeta() merged upper & low superfunction?
2.Can the cheta/incheta/sexpeta/slogeta normal work when base isn't eta?

[sheldonison]

1.Is the sexpeta() merged upper & low superfunction?
2.Can the cheta/incheta/sexpeta/slogeta normal work when base isn't eta?

1. no sexpeta; and cheta are both generated from the Ecalle assymptotic series solution for \( f(z)=\exp(z)-1 \), which has two sectors.  The Ecalle assymptotic generates two different solutions, depending on whether you approach the fixed point of zero from z>0 by iterating \( f^{-1}(z)=\log(z+1) \) before evaluating the series, which generates the cheta upper superfunction, or whether you approach the fixed point from z<0 by iterating f(z) before evaluating the series which generates the sexpeta lower superfunction.  That two different analytic functions can be generated from the same assymptotic series makes sense when you realize Ecalle's solution is a divergent assymptotic series, and you need to iterate f(z) or f^-1(z) enough times so that |z| of z is small enough to generate an accurate result.

The conjecture is that the limit of Kneser, as the base approaches eta from above would be sexpeta; I don't know if the conjecture has been proven.

2. no, the code for cheta/eta are only for exp(z)-1.   For any given assymptotic Ecalle series, you pretty much need to handcode evaluating that series, taking into account the required error terms and how much precision you want, and how many terms of the series to generate, and which of the 2n sectors of the f^n function you are interested in.  For the superfunction, you need approximations of the inverse of ecalle; the approximations I use are not generic. Also, it might get pretty tricky as the denominator of the derivative at the fixed point grows beyond small single digits ...

[Ember Edison]

1. no sexpeta; and cheta are both generated from the Ecalle assymptotic series solution for \( f(z)=\exp(z)-1 \), which has two sectors.  The Ecalle assymptotic generates two different solutions, depending on whether you approach the fixed point of zero from z>0 by iterating \( f^{-1}(z)=\log(z+1) \) before evaluating the series, which generates the cheta upper superfunction, or whether you approach the fixed point from z<0 by iterating f(z) before evaluating the series which generates the sexpeta lower superfunction.  That two different analytic functions can be generated from the same assymptotic series makes sense when you realize Ecalle's solution is a divergent assymptotic series, and you need to iterate f(z) or f^-1(z) enough times so that |z| of z is small enough to generate an accurate result.

The conjecture is that the limit of Kneser, as the base approaches eta from above would be sexpeta; I don't know if the conjecture has been proven.

1.What base can't merged? Is the all Shell-Thron-region rational base and Singularity base can't merged, or just Singularity base, or just eta?
2.What does "limit" mean? Is the \( {\lim_{\delta \to 0^+}}fatou.gp.sexp_{\eta+I*\delta}(z) \) is merge superfunction? upper? lower?