Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F08, 30D05

Retrieve articles in all journals with MSC: 58F08, 30D05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1013329-2
PII: S 0002-9947(1991)1013329-2
Article copyright: © Copyright 1991 American Mathematical Society