A Short Comparison of Classical Complex Dynamics and Holomorphic Semigroup Dynamics

TL;DR

This paper compares classical complex dynamics with holomorphic semigroup dynamics, highlighting similarities, differences, and new phenomena in the generalization from classical to semigroup settings.

Contribution

It provides an exposition and initial research insights into how classical results extend and differ in the context of holomorphic semigroup dynamics.

Findings

Classical results are partially generalized to semigroup dynamics.

New phenomena emerge in holomorphic semigroup dynamics.

The paper identifies key contrasts and connections between the two areas.

Abstract

This is an expository plus research paper which mainly exposes preliminary connection and contrast between classical complex dynamics and semigroup dynamics of holomorphic functions. Classically, we expose some existing results of rational and transcendental dynamics and we see how far these results generalized to holomorphic semigroup dynamics as well as we also see what new phenomena occur.