Koenig coordinates




Text from Complex Plot www pages :

" ... one of these intertwining maps, called the Koenigs eigenfunction after the mathematician Gabriel Koenigs. Koenigs proved that for an analytic(complex-differentiable) function f mapping the unit disc into itself, having 0 as an attractive fixed point, and having f'(0) not equal 0, that there exists a function g that satisfies the equation g°f = cf, where c is a complex scalar. In addition, if we define f[n] to be the n-th iterate of f (e.g., f ° f ° f would be the 3rd iterate of f), Koenigs proved that sequence f [n](z) / f '(0)n converges uniformly to g. We term g the Koenigs eigenfunction of f. Thus Koenigs has given us a numerical method of plotting the output points under these kinds of eigenfunctions, and that is exactly what Cplot does. Similar iterative processes exist for the three other types of intertwining maps that Cplot handles, and Cplot uses those iteration formulas in a similar manner. For more complete information, see the Complex Plot manual or several of the papers cited under Related Stuff. "


Using program mandel : "The modulus of the Koenigs coordinate is already shown in mode 0, when the period is at most 4." ( Wolf Jung )


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Autor: Adam Majewski adammaj1-at-o2-dot-pl
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