#dedekind eta function

[primepatterns]

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.

[primepatterns]

Modular forms: iteration fractal of the Dedekind eta function, update

My previous post featured a first image of the iteration fractal of the Dedekind eta function. Subsequent analysis of the data has led me to revise the colour scheme. The Dedekind eta function has two attractive fixed points: a complex fixed point at approximately 0.8252 + 0.3133i and a real fixed point at 0.

As with images of iteration fractals of Hurwitz zeta functions that have two attractive fixed points, I have revised the colour scheme for the iteration fractal of the Dedekind eta function to render pixels that converge on the second fixed point in shades of grey.

This new colour scheme is illustrated in the following images:

-1 ≤ Re(s) ≤  25, 0 ≤ Im(s) ≤ 25, 100 pixels per unit

-0.5 ≤ Re(s) ≤  0.5, 0 ≤ Im(s) ≤ 0.6, 1000 pixels per unit

-0.1 ≤ Re(s) ≤  0.1, 0 ≤ Im(s) ≤ 0.2, 20000 pixels per unit

Given the links between modular forms and L functions, I am hopeful of finding further interesting fractals.