Denjoy-Wolff like set for rational semigroups
Subham Chatterjee, Gorachand Chakraborty, Tarun Kumar Chakra
TL;DR
This work generalizes the Denjoy-Wolff paradigm to rational semigroups by introducing the Denjoy-Wolff like set $DW(G)$ and studying its dynamics on the unit disk. It proves that for finitely generated Abelian rational semigroups, $DW(G)$ is at most countable, and it explores a conjecture that $DW(G)$ is either empty or a singleton. Through concrete examples, the paper shows DW behavior can be empty, finite, or infinite, and it introduces a special class $\Psi$ of semigroups where every map contracts the disk to a Denjoy-Wolff point, enabling a partition of the semigroup into $|DW(G)|$ blocks via an equivalence relation on limit points. The results provide a framework for classifying rational semigroups by their Denjoy-Wolff like dynamics and suggest a structured approach to understanding Julia/Fatou dynamics in semigroup settings, including invariance under conjugation and the dichotomy of absorbing, dispersing, and hybrid behaviors within class $\Psi$.
Abstract
In this paper, we introduce the concept of Denjoy-Wolff set in rational semigroups. We show that for finitely generated Abelian rational semigroups, the Denjoy-Wolff like set is countable. Some results concerning the Denjoy-Wolff like set and the Julia set are also discussed. Then we consider a special class of rational semigroups and discuss various properties of Denjoy-Wolff like set for this class. We use the concept of Denjoy-Wolff like set to classify the class into 3 sub-classes. We also show that for any semigroup in this class, the semigroup can be partitioned into k partitions where k is the cardinality of the Denjoy-Wolff like set.