04/10/2012, 10:37 PM
(This post was last modified: 04/10/2012, 11:01 PM by sheldonison.)
I found a good paper online by Mitsushiro Shishikura and Hiroyuki Inou titled, The renormalization for parabolic fixed points and their perturbation. I'm reading both this paper, as well as Shishikura's chapter "Bifurcation of Parabolic Fixed Points", in the book "The Mandelbrot Set, Theme and Variations". The online paper is more recent, and probably improved. I haven't gotten very far yet, but I see some concepts that seem similar to the 1-cyclic theta transformations I have used in my posts on this forum, \( z+\theta(z) \) which seems to correspond to Shishikura's horn map. Then the Tetration programs I have posted here which are iteratively calculating \( \theta(z) \) and sexp(z), are hopefully calculating what is mathematically equivalent to Shishikura's horn map, for \( f(z)=\exp_b(z) \). The notation used by Shishikura for the horn map is:
\( E_f(z)=E_f(z+1)+1 \), where \( E_f(z)-z \) is periodic with period 1, and decays to a constant as \( \Im z \to \infty \).
- Sheldon
\( E_f(z)=E_f(z+1)+1 \), where \( E_f(z)-z \) is periodic with period 1, and decays to a constant as \( \Im z \to \infty \).
- Sheldon
