On Almost Abelian Finite Type Transcendental Semigroups
Author(s):
Dal Bahadur Saud
Dal Bahadur Saud
Department of Mathematics,
Durgalaxmi Multiple Campus, Far West University,
Attariya, Godawari- 02, Kailali,
Nepal.
saudd5531@gmail.com
,
Bishnu Hari Subedi
Bishnu Hari Subedi
Central Department of Mathematics,
Institute of Science and Technology,
Tribhuvan University,
Kirtipur, Kathmandu,
Nepal.
subedi.abs@gmail.com
,
Ajaya Singh
Ajaya Singh
Central Department of Mathematics,
Institute of Science and Technology,
Tribhuvan University,
Kirtipur, Kathmandu,
Nepal.
singh.ajaya1@gmail.com
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.
2020 Mathematics Subject Classification
37F10, 37F20, 30D45, 30D05.