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On the arithmetic nature of various constants at the accumulation points of unimodal maps and in particular of the logistic map

Karamanos, Kostas and Kotsireas, Ilias (2003) On the arithmetic nature of various constants at the accumulation points of unimodal maps and in particular of the logistic map. [Preprint]

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      Abstract

      We have recently proposed a construction based on the symbolic dynamics of the class of unimodal maps, where the phenomenon of (weak) chaos can be mapped in the non-normal (and possibly transcendental) character of some relevant constants. In some cases one can show explicitly that the corresponding constants are transcendental. We analyze these cases. We also explain a more recent result based on the statistical analysis of their binary expansions, that it could be seriously envisaged that the Feigenbaum constants \alpha and \delta for the logistic map are also transcendental. We also show a way to attack the same problem for the control parameter value at the first accumulation point. Finally, we analyze the B4 point of the logistic map thus confirming two relevant conjectures by Bailey and Broadhurst.

      Item Type: Preprint
      Additional Information: pubdom TRUE
      Uncontrolled Keywords: Logistic map, bifurcation, Experimental mathematics, Feigenbaum constants
      Subjects: 82-xx Statistical mechanics, structure of matter
      Faculty: UNSPECIFIED
      Depositing User: Russell Luke
      Date Deposited: 27 Oct 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/16

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