Abstract

Fatou, Julia, and escaping sets in holomorphic (sub)semigroup dynamics

Highlights

  • We confine our study to the Fatou, Julia, and escaping sets of a holomorphic semigroup and its subsemigroup

  • We take a set A of holomorphic functions and construct a semigroup S that consists of all elements that can be expressed as a finite composition of elements in A

  • In [13, Theorems 3.3], we proved that the escaping set of a transcendental semigroup S is the same as the escaping set of each of its particular functions if the semigroup S is generated by finite type transcendental entire functions

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Summary

Introduction

We confine our study to the Fatou, Julia, and escaping sets of a holomorphic semigroup and its subsemigroup. We prove the following assertion: Theorem 1.1 If a subsemigroup T has finite index or cofinite index in an abelian transcendental semigroup S , I(S) = I(T ), J(S) = J(T ) and F (S) = F (T ) . From [11, Theorem 3.1 (1) and (3)], we can say that Fatou and escaping sets of holomorphic semigroup may be empty.

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