A Study of Fatou Set, Julia set and Escaping Set in Conjugate Transcendental Semigroup

arXiv1808.03275MaRDI QIDQ6305251

Ajaya Singh, Bishnu Hari Subedi

Publication date: 9 August 2018

Abstract: We define commutator of a transcendental semigroup, and on the basis of this concept, we define conjugate semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given semigroup is nearly abelian. We also prove that image of each of escaping set, Julia set and Fatou set under commutator (affine complex conjugating maps) is equal to respectively escaping set, Julia set and Fatou set of conjugate semigroup. Finally, we prove that every element of the nearly abelian semigroup $S$ can be written as the composition of an element from the set generated by the set of commutators $Phi(S)$ and the composition of the certain powers of its generators.

Mathematics Subject Classification ID

Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets (37F10) Holomorphic families of dynamical systems; holomorphic motions; semigroups of holomorphic maps (37F44)

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