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Julia Sets of Certain Exponential Functions

✍ Scribed by Piyapong Niamsup

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
114 KB
Volume
250
Category
Article
ISSN
0022-247X

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✦ Synopsis

We characterize the Julia sets of certain exponential functions. We show that the Julia sets J F Ξ» n of F Ξ» n z = Ξ» n e z n where Ξ» n > 0 is the whole plane , provided that lim kβ†’βˆž F k Ξ» n 0 = ∞. In particular, this is true when Ξ» n are real numbers such that Ξ» n > 1 ne 1/n . On the other hand, if 0 < Ξ» n < 1 ne 1/n , then J F Ξ» n is nowhere dense in and is the complement of the basin of attraction of the unique real attractive fixed point of F Ξ» n . We then prove similar results for the functions

where Ξ» i ∈ -0 1 ≀ i ≀ n + 1 a j > 1, 1 ≀ j ≀ n, and m n β‰₯ 1.

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