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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Iteration of a composition of exponential functions

Author: Xiaoying Dong
Journal: Trans. Amer. Math. Soc. 328 (1991), 517-526
MSC: Primary 58F08; Secondary 30D05
MathSciNet review: 1013329
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Abstract: We show that for certain complex parameters $ {\lambda _1},\ldots,{\lambda _{n - 1}}$ and $ {\lambda _n}$ the Julia set of the function

$\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$

is the whole plane $ \mathbb{C}$. We denote by $ \Lambda $ the set of $ n$-tuples $ ({\lambda _1},\ldots,{\lambda _n}),{\lambda _1},\ldots,{\lambda _n} \in \mathbb{R}$ for which the equation

$\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}} - z= 0$

has exact two real solutions. In fact, one of them is an attracting fixed point of

$\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}},$

which is denoted by $ q$. We also show that when $ ({\lambda _1},\ldots,{\lambda _n})\, \in \Lambda $, the Julia set of

$\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$

is the complement of the basin of attraction of $ q$. The ideas used in this note may also be applicable to more general functions.

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Article copyright: © Copyright 1991 American Mathematical Society