Animations in Complex Dynamics
A transcendental family with Baker domains
The function f(z) = z - 1 + (1 - 2z)ez has 0 as superattracting fixed point, and also has a Baker domain. The latter is seen as a large region of the left half plane stretching out to infinity along the negative real axis. The Julia set divides the Riemann sphere into the basin of attraction of 0, the Baker domain, and all preimages of the basin and of the Baker domain.
In this animation, the Julia set is shown in white on the Riemann sphere.
The function f(z) = z - 1 + (1 - 2z)ez can be approximated by the polynomials pd(z) = z - 1 + (1 - 2z)*(1+z/d)d. In fact, it makes perfect sense to say that it is a dynamically nice approximation. The following two animations show sequences of Julia sets of the polynomials for increasing degrees d before reaching a picture of the limit Julia set, that is to say, the Julia set of the limit function f. All pictures are made on the Riemann sphere in a chart near infinity. Note that all the functions pd as well as f have the origin as an attracting fixed point. Its basin of attraction is colored red. For the polynomials, infinity is an attracting fixed point and its basin of attraction is marked blue. Note that in the limit these basins will eventually vanish. The approximating polynomials have further attracting fixed points. The green area marks the union of their attracting basins. Note that the union of these attracting basins converges to the Baker domain of f which is also colored green. Loosely speaking, the Baker domain of the limit function is grown by certain basins of attracting fixed points of the approximating polynomials.
Click HERE for an animation showing first the images of the Julia sets of the polynomials pd where d=22, . . . , 210 and finally the image of the Fatou set of the limit function.
Click HERE
for an animation showing first the images of the Julia sets of the polynomials
pd where
d=3*22, . . . , 3*210
and finally the image of the Fatou set of the limit function.
Acknowledgement
These animations and the mathematical research behind them are joint work of Aimo Hinkkanen (UIUC), Hartje Kriete (Göttingen, Germany), and Bernd Krauskopf (Bristol, U.K.). Special thanks to Stuart Levy (NCSA) and George Francis (UIUC). This work was partially supported by the National Center for Supercomputing Applications and the National Science Foundation under grant DMS990001N and utilized the SGI CRAY Origin2000 at the National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign.