Convergence properties of the Gronwall area formula for quadratic Julia sets

TL;DR

This paper investigates the limitations of the Gronwall area formula for quadratic Julia sets, demonstrating that finite sum approximations along the Mandelbrot set's boundary are inadequate, with implications for complex dynamics analysis.

Contribution

It reveals the convergence failure of the Gronwall area formula when approximated by finite sums for quadratic Julia sets, using parabolic enrichment techniques.

Findings

Finite sum approximations are insufficient for the Gronwall area formula.

The boundary of the Mandelbrot set affects the approximation accuracy.

Parabolic enrichment provides insights into convergence issues.

Abstract

Using parabolic enrichment, it is shown that Gronwall area formula for the filled Julia set along the boundary of the main cardioid of the Mandelbrot set cannot be well approximated by replacing it by a finite sum. A revised version of this article will appear in the Journal of the London Mathematical Society.