Dal Bahadur Saud, Bishnu Hari Subedi, and Ajaya Singh,
On Almost Abelian Finite Type Transcendental Semigroups,
Indian Journal of Mathematics,
Volume 67,
Issue 02,
2025,
Pages 281--305,
ISSN 0019-5324,
Abstract:Let $ S $ be a finite type semigroup generated by transcendental entire functions. Define escaping set of semigroup $ S $ by $ I(S)= \{ z \in \mathbb{C}:S \text{ is iteratively divergent at }  z \} $. In this paper, we investigate the following results: Suppose $  I(S) = I(h)  $ for generator $ h \in S $ such that there exists a conformal map $  \phi(z) = -z  $ satisfying $  h = h \circ \phi  $ and $  \phi \circ h = g  $ with $  g \in S  $ or there exists a conformal map $ \phi (z)$= $az+b $ for $a,b\in \mathbb{C}$ with $ a\neq 0 $ satisfying $ h \circ g = \phi \circ g \circ h $ for all generators $ h, g \in S $. Then the semigroup $ S=\langle h,g\rangle $ is almost abelian. Also under the similar condition involving finite generators of $ S $ and a conformal map, the subsemigroup generated by some generators among the finite generators is almost abelian. Moreover for every arbitrarily finitely generated semigroup $  S = \langle h_1, h_2, \ldots, h_n \rangle  $ of finite type, neither the semigroup $S$ itself nor any of its subsemigroup(s) is necessarily an almost abelian. However, for every arbitrarily infinitely generated semigroup $  S = \langle h_1, h_2, h_3, \ldots \rangle  $ or finitely generated semigroup $  S = \langle h_1, h_2, \ldots, h_n \rangle  $ of finite type, if $  I(S) = I(h_i)  $ for $  h_i \in S  $ satisfying $  h_i = h_i \circ \phi  $ and $  \phi \circ h_i = h_j \in S  $ under the conformal map $  \phi(z) = -z  $  or satisfying $ h_i \circ h_j = \phi \circ h_j \circ h_i $ for $ h_i, h_j \in S $, where $ h_i \neq h_j $ (for $ i, j\in\mathbb{N} : $ $ i \neq j $) under conformal map $ \phi (z)$= $az+b$ for $a,b\in \mathbb{C}$ with $a\neq 0 $, the full semigroup or its subsemigroup(s) or both become necessarily almost abelian.
Keywords: Conformal map, conjugate map, commutator, almost abelian semigroup, Fatou set, escaping set.