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Extending external rays throughout the Julia sets of rational maps

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Abstract

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z n + λ/z n where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Boston University, 111 Cummington St., Boston, MA, 02215, USA

    Figen Çilingir & Robert L. Devaney

  2. Department of Mathematical Sciences, United States Military Academy, West Point, NY, 10996, USA

    Elizabeth D. Russell

Authors
  1. Figen Çilingir
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  2. Robert L. Devaney
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  3. Elizabeth D. Russell
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Correspondence to Robert L. Devaney.

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Çilingir, F., Devaney, R.L. & Russell, E.D. Extending external rays throughout the Julia sets of rational maps. J. Fixed Point Theory Appl. 7, 223–240 (2010). https://doi.org/10.1007/s11784-010-0003-2

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  • DOI: https://doi.org/10.1007/s11784-010-0003-2

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