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This is an expository plus research paper which mainly exposes preliminaryconnection and contrast between classical complex dynamics and semigroupdynamics of holomorphic functions. Classically, we expose some existing resultsof rational and transcendental dynamics and we see how far these resultsgeneralized to holomorphic semigroup dynamics as well as we also see what newphenomena occur.

A Short Comparison of Classical Complex Dynamics and Holomorphic Semigroup Dynamics