Denjoy-Wolff like set for rational semigroups

TL;DR

This paper introduces the Denjoy-Wolff like set for rational semigroups, explores its properties, and uses it to classify and partition specific classes of these semigroups, extending understanding of their dynamic behavior.

Contribution

It defines the Denjoy-Wolff like set for rational semigroups, analyzes its properties, and applies it to classify and partition certain classes of semigroups.

Findings

Denjoy-Wolff like set is countable for finitely generated Abelian rational semigroups

The set helps classify a special class of rational semigroups into three sub-classes

Semigroups in this class can be partitioned into k parts, where k equals the cardinality of the Denjoy-Wolff like set

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.