Dynamics of complex harmonic mappings
Gopal Datt, Ramanpreet Kaur
TL;DR
This work extends Fatou--Julia theory to complex harmonic mappings under the direct-composition operation by defining $f=h+\overline{g}$ and analyzing the resulting normality, Fatou set $F(f)$, and Julia set $J(f)$. It develops two main threads: fundamental properties linking $F(f)$ and $J(f)$ to the holomorphic parts $h$ and $g$, and existence results showing that harmonic dynamics can exhibit wandering domains, including escaping and oscillating types, with constructions and growth conditions that bound Fatou components. Key contributions include inclusions such as $F(h)\cap F(g)\subseteq F(f)$ and the invariance $F(f^{p,\circleddash})=F(f)$, demonstrations of wandering domains for transcendental harmonic mappings (including an oscillating example), and, for polynomial-harmonic maps, results on compact Julia sets and their uniform perfectness, plus criteria ensuring all Fatou components are bounded in certain cases. Overall, the paper highlights distinctive dynamical behavior for harmonic mappings compared to holomorphic ones and lays groundwork for further study of harmonic dynamics with potential applications in fluid mechanics and related fields.
Abstract
This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a holomorphic function. The existence of harmonic mapping with a wandering domain is also shown.