The Mandelbrot Set as a Modular Form

An exploration of the interior or the Mandelbrot Set and the appearance of functions appearing to be similar to modular forms. This provides yet another example of the close interconnection between the structure of the Modular Group SL(2,Z) and fractals. The relationship is demonstrated computationally and visually, and not from first principles; visually, the interior resembles the Weierstrass elliptic invariant g2. However, it is a resemblance only; the various explicit expressions that can be found are shown to not actually be modular forms. It is hypothesized that some simple but currently unknown transformation will convert them into modular forms. The construction of the interior is based on averaging together iterated values with a spectral-type summation, and then analyzing the asymptotic behavior of the sum. Leading divergence are easy to explain and remove; the remaining finite parts hint at modular symmetry. This is a work in progress. A final conclusion and analysis has not been reached. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. Updated and revised versions of this monograph can be found at http://www.linas.org/math/sl2z.html 1 Dedekind Eta and the Mandelbrot Set XXX This paper may be subject to occasional revision. XXX Modular forms are a particular kind of function on the complex upper half-plane studied in analytic number theory and the theory of elliptic curves. A precise definition of a modular form[?] will be given later in this paper. As a simple example, consider the Euler function φ(q) = ∞ ∏ k=1 (1−q) (1) on the complex plane. It is closely related to the Dedekind eta function, which is a modular form. Figure 1 shows φ(q) inside the unit disk |q| < 1. Graphs of most modular forms visually resemble this picture in one way or another. The exploration presented in this monograph will be mostly visual, not algebraic; none-the-less, various basic expressions will be developed to make the hypothesis as explicit as possible.