Talk:Böttcher's equation

Power or iterate?

The article has been edited to make it explicit that the RHS of the equation is the n-th iterate. I found this article: Carl C. Cowen (1982). "Analytic solutions of Böttcher's functional equation in the unit disk" (PDF). Aeq. Math. 24: 187–194. Zbl 0526.30033. This reference makes it clear that you start with an analytic function h with an n-fold zero at zero [the wiki article considers the trivial variation h(z-a)+a] and look for a function F such that F(h(z)) = F(z)ⁿ where the RHS is indeed the n-th power of F(z). In fact the reference observes that if n is fixed by the condition on h having an n-fold zero at zero, then the only value of k in an equation F(h(z)) = F(z)^k that gives an interesting result is k=n. So, power or iterate? Spectral sequence (talk) 01:54, 20 July 2012 (UTC)

Of course it is the power: cf Schröder's equation and the transformation that produced it. Cuzkatzimhut (talk) 10:19, 20 July 2012 (UTC)

I thought so — I have added a link to Schröder's equation, thanks for that. Spectral sequence (talk) 15:07, 20 July 2012 (UTC)

new info

Hi, Here are some interesting links:

• http://math-functions-1.watson.jp/sub4_math_020.html (It is in Japan but you can translate it. It works fine )
• http://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html
• http://www.mndynamics.com/indexp.html#XR

HTH --Adam majewski (talk) 14:54, 10 October 2017 (UTC)