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    OSBORNE, JOHN 2013. Connectedness properties of the set where the iterates of an entire function are bounded. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 155, Issue. 03, p. 391.


    Dijkstra, Jan J. and van Mill, Jan 2012. Negligible sets in Erdős spaces. Topology and its Applications, Vol. 159, Issue. 13, p. 2947.


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An explosion point for the set of endpoints of the Julia set of λ exp (z)

  • John C. Mayer (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700005460
  • Published online: 01 September 2008
Abstract
Abstract

The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[B1]P. Blanchard . Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85141.

[C]C. Caratheodory . Über die Begrenzung einfach zusammenhangender Gebiete. Math. Ann. 73 (1913), 323370.

[D2]R. L. Devaney . The structural instability of exp (z). Proc. Amer. Math. Soc. 94 (1985), 545548.

[DG]R. L. Devaney & L. R. Goldberg . Uniformization of attracting basins. Duke Math. J. 55 (1987) 253266.

[F]P. Fatou . Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337370.

[Mc]C. McMullen . Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300 (1987), 329342.

[P]G. Piranian . The boundary of a simply connected domain, Bull. Amer. Math. Soc. 64 (1958), 4555.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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