Israel Journal of Mathematics

, Volume 161, Issue 1, pp 347–371

Instability of exponential Collet-Eckmann maps



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.


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  1. [DMS]
    P. Dominguez, Makienko and G. Sienra, Ruelle operator and transcendental entire maps, Discrete and Continuous Dynamical Systems 12 (2005), 773–789.MATHMathSciNetCrossRefGoogle Scholar
  2. [Kra]
    I. Kra, Automorphic forms and Kleinian groups, W.A. Benjamin, Inc., Reading, Mass., 1972.MATHGoogle Scholar
  3. [Le]
    G. Levin, On an analytic approach to the Fatou conjecture, Fundamenta Mathematicae 171 (2002), 177–196.MATHMathSciNetGoogle Scholar
  4. [M1]
    P. Makienko, Remarks on Ruelle Operator and Invariant Line Field Problem, Preprint 2001.Google Scholar
  5. [M2]
    P. Makienko, Remarks on Ruelle Operator and Invariant Line Field Problem II, Ergodic Theory and Dynamical Systems 25 (2005), 1561–1581.MATHCrossRefMathSciNetGoogle Scholar
  6. [PU]
    F. Przytycki and M. Urbański, Fractals in the Plane — the Ergodic Theory Methods, available at, to appear.

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

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