Fractals/Iterations in the complex plane/Fatou coordinate

Parabolic iteration Tetration Forum : Parabolic Iteration Tetration Forum : Parabolic Iteration, again Fatou function Fatou function :stackexchange : half-iterate-of-x^2 + c is defined only inside petal ( attracting petal or repelling ), not on the whole neighbourhood of the fixed point is a conformal function which satifies Abel's equationS. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda : Holomorphic Dynamic. January 13, 2000 | 0521662583 | 978-0521662581wiki : Abel%27s_equation transforms f(z) to unit translation : "These are coordinates in which f looks like a translation." Małgorzata StawiskaLiouville theorem with parameters: asymptotics of certain rational integrals in differential fields by Małgorzata Stawiska maps petal to right half of plane in u coordinate. unrolls invariant curvs ( orbits ) : maps "circles" to straight lines Normalization Fatou coordinate can be normalized = it maps critical point to zero :Dynamics in one complex variable: introductory lectures by John W. Milnor, page 7-6 Parabolic fixed point is mapped to point at infinity on Riemann sphere Fatou coordinate Fatou coordinate u : where: is the Fatou function Description at Hyperoperations Wiki what we call "Abel function",wikipedia : Abel function they call it "Fatou coordinates".new results from complex dynamics at Tetration Forum Fatou coordinates Minicourse "Analytic classification of germs of generic families unfolding a parabolic pointFatou coordinate at Hyperoperations Wiki Shishikura perturbed Fatou coordinates Shishikura perturbed Fatou coordinates Computing Fatou_coordinate for f(z)=z/(1+z) Fatou_coordinate for f(z)=z+z^2 Fatou_coordinate for f(z)=z^2 + c Programs QFract QFract by INOU Hiroyuki and pictures To build from the source code, you need : Qt 4.5 (or later?) from http://qt.nokia.com/ and install. Boost C++ Libraries (http://www.boost.org/). Download source files from this page : First unpack the archive as follows tar zcvf qfract-110725_2-src.tar.gz Go to the program directory : cd qfract-110725_2 and edit files : Makefile, config.h, plugins/Makefile to adjust your environment. For example in config.h change : #define PLUGIN_PATH "/Users/inou/prog/qfract4/plugins" #define COLORMAP_PATH "/Users/inou/prog/qfract4/colormaps" for your own settings. Then to compile everything run from console : make To run the program from console : ./qfract Search arxiv search: fatou+coordinate math.stackexchange search: fatou+coordinate mathoverflow search: fatou+coordinate References