Iteration of a composition of exponential functions
Authors: Dong Xiaoying 
Published: June 2012

We show that for certain complex parameters

λ
1

,
…
,

λ

n
−
1

{\lambda _1},\ldots ,{\lambda _{n - 1}}

and

λ
n

{\lambda _n}

the Julia set of the function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript lamda 1 e Super Superscript Super Super Superscript period Super Super Super Superscript period Super Super Super Super Superscript period Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Super Subscript n minus 1 Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript e Super Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Superscript Super Super Super Super Super Super Super Super Subscript n Super Super Super Super Super Super Super Super Sub Superscript z Super Super Super Super Superscript Super Super Super Superscript Super Super Superscript">

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

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

\]
is the whole plane

C

\mathbb {C}

. We denote by

Λ
\Lambda

the set of

n
n

-tuples

(

λ
1

,
…
,

λ
n

)
,

λ
1

,
…
,

λ
n

∈

R

({\lambda _1},\ldots ,{\lambda _n}),{\lambda _1},\ldots ,{\lambda _n} \in \mathbb {R}

for which the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript lamda 1 e Super Superscript Super Super Superscript period Super Super Super Superscript period Super Super Super Super Superscript period Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Super Subscript n minus 1 Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript e Super Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Superscript Super Super Super Super Super Super Super Super Subscript n Super Super Super Super Super Super Super Super Sub Superscript z Super Super Super Super Superscript Super Super Super Superscript Super Super Superscript Baseline minus z equals 0">

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

−
z
=
0

{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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript lamda 1 e Super Superscript Super Super Superscript period Super Super Super Superscript period Super Super Super Super Superscript period Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Super Subscript n minus 1 Super Super Super Super Super Super Superscript Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript e Super Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Superscript Super Super Super Super Super Super Super Superscript Super Super Super Super Super Super Super Super Subscript n Super Super Super Super Super Super Super Super Sub Superscript z Super Super Super Super Super Super Super Super Subscript Super Super Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript Super Super Super Super Super Superscript Super Super Super Super Superscript Super Super Super Superscript Super Super Superscript Super Superscript Superscript Baseline comma">

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

,

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

\]
which is denoted by

q
q

. We also show that when

(

λ
1

,
…
,

λ
n

)

∈
Λ

({\lambda _1},\ldots ,{\lambda _n})\, \in \Lambda

, the Julia set of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript lamda 1 e Super Superscript Super Super Superscript period Super Super Super Superscript period Super Super Super Super Superscript period Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Super Subscript n minus 1 Super Super Super Super Super Superscript Super Super Super Super Super Super Superscript e Super Super Super Super Super Super Super Superscript lamda Super Super Super Super Super Super Superscript Super Super Super Super Super Super Super Super Subscript n Super Super Super Super Super Super Super Super Sub Superscript z Super Super Super Super Superscript Super Super Super Superscript Super Super Superscript">

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

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

\]
is the complement of the basin of attraction of

q
q

. The ideas used in this note may also be applicable to more general functions.