Julia Sets of Certain Exponential Functions

AbstractWe characterize the Julia sets of certain exponential functions. We show that the Julia sets J(Fλn) of Fλn(z)=λnezn where λn>0 is the whole plane C, provided that limk→∞Fkλn(0)=∞. In particular, this is true when λn are real numbers such that λn>1ne1/n. On the other hand, if 0<λn<1ne1/n, then J(Fλn) is nowhere dense in C and is the complement of the basin of attraction of the unique real attractive fixed point of Fλn. We then prove similar results for the functions[formula] where λi∈C−{0}, 1≤i≤n+1, aj>1, 1≤j≤n, and m, n≥1