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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.

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, Denton, TX, 76203-1430, USA

    Mariusz Urbański

  2. Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland

    Anna Zdunik

Authors
  1. Mariusz Urbański
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  2. Anna Zdunik
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Corresponding author

Correspondence to Mariusz Urbański.

Additional information

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.

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Urbański, M., Zdunik, A. Instability of exponential Collet-Eckmann maps. Isr. J. Math. 161, 347–371 (2007). https://doi.org/10.1007/s11856-007-0082-y

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  • DOI: https://doi.org/10.1007/s11856-007-0082-y

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