Dynamics of complex harmonic mappings
Gopal Datt, Ramanpreet Kaur
TL;DR
This paper explores the dynamics of complex harmonic mappings, focusing on their Fatou and Julia sets, revealing contrasting behaviors with holomorphic functions and demonstrating the existence of harmonic mappings with wandering domains.
Contribution
It introduces the study of Fatou and Julia sets for harmonic mappings, highlighting fundamental properties and novel behaviors not seen in holomorphic functions.
Findings
Contrasting behaviors of Fatou and Julia sets compared to holomorphic functions
Existence of harmonic mappings with wandering domains
Fundamental properties of harmonic Fatou and Julia sets
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.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Mathematical Modeling in Engineering