Fractalforums

Hi,

There is intersting pdf file about complex dynamics:
http://www.math.titech.ac.jp/~kawahira/courses/mandel.pdf
but in japan

Is it possible to translate it to english?\

I have tried google translate. It works but the result is html file without images

TIA



Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/pdf-file-in-japan/562/

Thanks for the link, some interesting English papers on M-set on his website. I asked him if he has an English version, will let you know if he does.

This Boettcher function is interesting but most math papers I've read are satisfied that they can prove it exists and leave it at that, rather than figuring out how to compute it.

Maybe he can join us. We need more mathematicians. 

Thanks for the link, some interesting English papers on M-set on his website. I asked him if he has an English version, will let you know if he does.

This Boettcher function is interesting but most math papers I've read are satisfied that they can prove it exists and leave it at that, rather than figuring out how to compute it.

Look at :
https://gitlab.com/adammajewski/parameter_external_angle

Comments and upgrades are welcome (:-)

Look at :
https://gitlab.com/adammajewski/parameter_external_angle

Comments and upgrades are welcome (:-)

Thanks for that. In the series expansion for the external angle, what is \( f^n_c(c) \)?

\[ f^n_c(c) \] is n-th iteration of z=c under f function. Is it an error ?

\[ f^n_c(c) \] is n-th iteration of z=c under f function. Is it an error ?

I don't know if it's an error, never seen that formula. How is it derived?
Puzzled that any Misiurewicz point with preperiod 1 (I think those are all terminal points?) will encounter a division by zero when \( f^n_c(c)=c \).

I don't know if it's an error, never seen that formula. How is it derived?

\( \Phi_M(c) = \Phi_c(c). \)
so put c instead of z in the equation  \( \Phi_c(z) =  \) which can be found  here by  Wolf Jung http://www.mndynamics.com/indexp.html#XR 
 

Puzzled that any Misiurewicz point with preperiod 1 (I think those are all terminal points?) will encounter a division by zero when \( f^n_c(c)=c \).

Right. But this equation is for the external angle. It mens for the points from the exterior of Mandelbrot set only!

 Misiurewicz points belong to Mandelbrot set.

I'm not telling that I know much about it. I made this repo to collect informations about it and learn it. Help is welcome. 

\( \Phi_M(c) = \Phi_c(c). \)
so put c instead of z in the equation  \( \Phi_c(z) =  \) which can be found  here by  Wolf Jung http://www.mndynamics.com/indexp.html#XR 

He states his equation (3) is valid for large z, so I don't think you can set z=c (assuming c is not large).
 

He states his equation (3) is valid for large z, so I don't think you can set z=c (assuming c is not large).

Right, but

When the modified function arg(z/(z-c)) is used instead of the principal value for computing argc(z) according to (3), this formula will be valid not only for large |z| but everywhere outside the Julia set.

and Wolf uses the same function for both Julia and Mandelbrot set
https://en.wikibooks.org/wiki/Fractals/mandel#External_angle

Right, but 
"When the modified function arg(z/(z-c)) is used instead of the principal value for computing argc(z) according to (3), this formula will be valid not only for large |z| but everywhere outside the Julia set."

Can you explain why? (3) derives from (2) which is valid for large z, so how can redefining the range of arg(z) make it valid for all z? I don't doubt he's right, but find it hard to follow.

But my original question is answered, \( f^n_c(c) \) is the usual M-set iteration but starting from c instead of 0, thanks.