Real non-attractive fixed point conjecture for complex harmonic functions

TL;DR

This paper proves a conjecture about fixed points of complex harmonic functions, showing that certain fixed points have multipliers with real parts at least one, with implications for polynomial and rational harmonic functions.

Contribution

It establishes the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions, including cases without super-attracting conditions.

Findings

Every such harmonic function with a super-attracting fixed point has a fixed point with multipliers' real parts ≥ 1.

For polynomial harmonic functions, the result holds even without super-attracting assumptions.

Explicit examples and visualizations are provided, and the problem for transcendental harmonic functions is discussed.

Abstract

We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of degree at least 2. We show that every such function with a super-attracting fixed point has a $\mathfrak{h}$-fixed point $\zeta=\mu+\overline{\omega}$ such that the real parts of its multipliers satisfy $\text{Re}(\partial_z h(\mu)) \geq 1$ and $\text{Re}(\partial_z g(\omega)) \geq 1$. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions.