Home > Mathematica, Mathematics > Climbing the exponential ladder

This post is motivated/inspired by Professor Bender’s book and lectures.  The following graphics relate to the continued exponential. The color coding is a first attempt at color coding the orbits of the iterated exponential:

$a_0=1$

$a_n=e^{z a_{n-1}}$

For example, $a_3=e^{ze^{ze^z}}$

The cardioid delimits the zone of convergence (to unique limit). It’s mottled appearance  is due to areas of convergence slower than my arbitrarily chosen iteration number (and they way I scripted partitioning the  orbits). The other colors are orbits of varying oscillatory/cyclic behaviour (varying points). The color schemes are the Gradient schemes (51) from Mathematica.  The distribution of limit cycle size was skewed so the cycle  size was logarithmically transformed to display the  range.

The animated  gif is lower resolution grid of complex plane [-2,2] x [-2,2]. The higher resolution images take longer to generate and with time I hope to write a better algorithm, chose a most discriminating color scheme. I am ‘happy’ that it resembles the images in the video lectures…they are beautiful to my eyes aesthetically as well as fascinating with with respect to iterated functions.