Iteration of a composition of exponential functions

Dong, Xiaoying

doi:10.1090/S0002-9947-1991-1013329-2

Iteration of a composition of exponential functions
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by Xiaoying Dong

Trans. Amer. Math. Soc. 328 (1991), 517-526

DOI: https://doi.org/10.1090/S0002-9947-1991-1013329-2

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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 \[ {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}\] 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 \[ {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}} - z= 0\] has exact two real solutions. In fact, one of them is an attracting fixed point of \[ {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}},\] which is denoted by $q$. We also show that when $({\lambda _1},\ldots ,{\lambda _n}) \in \Lambda$, the Julia set of \[ {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}\] 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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