Some investigations on the dynamics of the family λz exp(z)

Rational families converging to a family of exponential maps

“…(5) For any a P C˚and for any ε ą 0, Jpf a q X B ε p0q ‰ H. Further, there exists a The next proposition gives some properties of the limiting parameter space. This result appears in [8]. Proposition 4.2.…”

“…In fact, the map f a pzq is conjugate to the more familiar map g λ pzq " λze z , via the involution on C 8 given by z Þ Ñ´1{z and with λ " 1 a . Therefore, since many of the properties of g λ are known [4,8], we show how they emerge as limiting properties of the family of maps f a,d .…”

“…The family g 1 a pzq " z a e z , which is conformally conjugate to the limiting map f a pzq " aze 1{z , has been well studied (see e.g., [3,4,8]). It is also useful to view each map f a as a map on the punctured plane, f a : C˚Ñ C˚with 0 and 8 as asymptotic values.…”