Israel Journal of Mathematics

, Volume 161, Issue 1, pp 347–371

Instability of exponential Collet-Eckmann maps


DOI: 10.1007/s11856-007-0082-y

Cite this article as:
Urbański, M. & Zdunik, A. Isr. J. Math. (2007) 161: 347. doi:10.1007/s11856-007-0082-y


Given λ ∈ ℂ \ {0} let the entire function fλ: 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 fλ is stable iff fλ 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 fλ is unstable.

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North TexasDentonUSA
  2. 2.Institute of MathematicsWarsaw UniversityWarszawaPoland