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We investigate to what extent Fatou set, Julia set and escaping set oftranscendental semigroup is respectively equal to the Fatou set, Julia set andescaping set of its subsemigroup. We define partial fundamental set andfundamental set of transcendental semigroup and on the basis of this set, weprove that Fatou set and escaping set of transcendental semigroup $ S $ arenon-empty.

Fatou Set, Julia Set and Escaping Set in Holomorphic Subsemigroup Dynamics