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 )
- Schroder's equation - wikipedia
- Koening coordinates by Inigo Quilez
- "Help on Koenigs coordinates II " discussion on sci.fractals
- Critical points and Fatou theorem by Evgeny Demidov
- Gabriel Koenigs - Biographie at The Mathematics Genealogy Project
- Five Proofs of the Theorem of Koenigs by Lennart Carleson and Theodore W. Gamelin
- complex plot
- P.S. Bourdon, Convergence of the Koenigs sequence, Contemporary Math. 213 (1997), 1-10.
- P.S. Bourdon and J.H. Shapiro, Mean growth of Koenigs eigenfunctions, Journal Amer. Math. Soc., 10 (1997), 299-325.
- P.S. Bourdon, Essential angular derivatives and maximum growth of Koenigs eigenfunctions, Journal of Functional Analysis, to appear.
- E. Clarkson, Complex Plot: A visual aid in examining analytic functions, Washington and Lee Journal of Science, to appear.
- D. S. Bennett and J. M. Carr, Researching Linear Fractional Models, W&L J. of Science 2, (1992), 20-22.
- Poggi-Corradini, Norm convergence of normalized iterates and the growth of Koenigs maps, Arkivor Matematik, to appear.
- Poggi-Corradini, The Hardy class of geometric models and the essential spectral radius of composition operators, J. Funct. Anal., 141 (1997), 129-156.
- J. H. Shapiro, "Composition Operators and Classical Function Theory," Springer-Verlag, New York, 1993.
- G. Koenigs, Recherches sur les integrales de certaines equationes functionelles, Annales Ecole Normale Superior (3) 1 (1884), Supplement, 3-41.
- C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions," CRC Press, Boca Raton, 1995.
- I. N. Baker and Ch. Pommerenke, On the iteration of analytic functions in a half-plane, J.London Math. Soc. (2) 20 (1979), 255-258.
- C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), 69-95.
- Ch. Pommerenke, On the iteration of analytic functions in a half-plane, J. London Math Soc. (2) 19 (1979), 439-447.
- G. Valiron, Sur l'iteration des fonctions holomorphes dans un demi-plan, Bull des Sci. Math. (2) 55 (1931), 105-128.
- P. Bourdon and J. Shapiro, Cyclic phenomena for composition operators, Memoirs of the Amer. Math. Soc., Number 596, 1997.
- Mark Elin, Victor Goryainov, Simeon Reich, and David Shoikhet : Fractional Iteration and Functional Equations for Functions Analytic in the Unit Disk. Computational Methods and Function Theory Volume 2 (2002), No. 2, 353–366
- Studies on composition operators: proceedings of the Rocky Mountain Mathematics Consortium, July 8-19, 1996, University of Wyoming. Contemporary mathematics (American Mathematical Society) ; v. 213
July 8-19, 1996, University of Wyoming, Farhad Jafari
Jafari, Rocky Mountain Mathematics Consortium
Ed.: Farhad Jafari, Rocky Mountain Mathematics Consortium
Publisher : AMS Bookstore, 1998
ISBN 0821807684, 9780821807682
Autor: Adam Majewski adammaj1-at-o2-dot-pl
Feel free to e-mail me. (:-))