, Volume 161, Issue 1, pp 347-371

Instability of exponential Collet-Eckmann maps

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Abstract

Given λ ∈ ℂ \ {0} let the entire function f λ: ℂ → ℂ be defined by the formula $$f_\lambda (z) = \lambda e^z $$ . The question of structural stability within this family is one of the most important problems in the theory of iterates of entire functions. The natural conjecture is that f λ is stable iff f λ is hyperbolic, i.e., if the only singular value 0 is attracted by a an attracting periodic orbit. We present some results positively contributing towards this conjecture. More precisely, we give some sufficient conditions of summability type which guarantee that the map f λ is unstable.

The research of both authors was supported in part by the NSF/PAN grant INT-0306004.
The research of the first author was supported in part by the NSF Grant DMS 0400481.
The research of the second author was supported in part by the Polish KBN Grant 2 PO3A 034 25.