Modular forms: iteration fractal of the Dedekind eta function
The image below is my first effort at generating an iteration fractal from a modular form, in this case the Dedekind eta function:
The image shows points, s, in the section of the complex plane between -1 < Re(s) < 25 and 0 < Im(s) < 25. The zeros are located along the real axis, which forms the bottom edge of the image.
The main block is 24 units wide, and is divided into two rather different columns, each 12 units wide. The main block repeats as you move along the real axis.
The blue / cyan coloured regions are those points, s, where iteration of η(s) leads to undefined values (i.e. values with a negative imaginary part). The green / yellow / red regions are those points, s, where iteration of η(s) leads to the attractive fixed point at approximately 0.8252 + 0.3133i.
Of all the functions I have looked at to date, only the Dedekind eta function has a complex fixed point.














